text:
\section{Introduction:  formulation of the problem \label{Intro}}

The mechanism for the emergence of the classical behavior of a macroscopic body  has been largely clarified as due to the
entanglement  of the body studied with the environment (any kind of fluxes, light, air molecules, cosmic ray background, etc.)  and the consequent decoherence.
Quantitative analysis of these effects and characteristic features of the decoherence  have been  discussed in  many papers  \cite{JoosBis,Joos,Zurek,Arndt1}.

It appears reasonable to assume that such a mechanism, applied to the experimental devices,  entangled in general with the environment (the rest of the world), does explain the classical state of the measurement apparatus after each measurement  -  the uniqueness of each measurement outcome -, providing us with  one of the indispensable ingredients for    properly understanding the measurement processes.

However,  this is but one of the many distinct  issues about the interpretation and the fundamental aspects of quantum mechanics,  which go generally under the name of  the ``quantum measurement problems". The latter include different interpretations of quantum mechanical predictions,  the mechanisms or models of the so-called wave-function collapse,   the relation between Born's rule and  the Schr\"odinger equation, as well as various
 apparent puzzles, paradoxes and  conceptual difficulties.

   Earlier references and discussions on those issues can be found  in  \cite{WheelerZ,Bell,Peres}.

 The solution to most of these problems  has been discussed recently   \cite{KK} from a new perspective. It is proposed that the fundamental law of quantum mechanics  is various expectation values (the fluctuation averages) of the dynamical variables, which describe the relative frequencies of various outcomes in case experiment is done, and which replace Born's rule in the conventional approach. One of the key observations in \cite{KK}   is that the fundamental building blocks of our world are various (elementary or composite) particles, and hence  that all elementary interactions and accordingly all measurement processes at their heart, are spacetime  pointlike events.  This leads to an effective spacetime localization of each measurement event,  a fact perceived by us as a wave-function collapse. Another key observation is the interplay and crucial roles  {\it factorization} and {\it  entanglement} play  in
the conceptual basis of quantum mechanics in general,   and in particular, in the actual time evolution of  the measurement processes.  The new perspective proposed in \cite{KK} that the quantum fluctuations described by the wave function are real, and that they are there, independently of any experiments or  of human presence, seems to dispel the final doubt about the fundamental theory of physics being unable to make a unique prediction of each single experiment  necessarily.  Our point of view seems to be consistent with what we know from cosmology today, that  all the structures of the universe, the galaxies,   stars and planets, and {\it  ourselves},  have grown out of an uncontrollable quantum density fluctuations at some stage of the inflationary universe.

 The central issue of the present work,   to understand  how the classical mechanics, and more generally classical physics,  emerges  in quantum mechanics,  for which systems, for which variables,  and under which conditions or  limits,  is  actually  a question largely unrelated, logically,
to the  solution of these  quantum measurement problems and various subtle issues accompanying  them.  Thus  it is important  to disentangle the two classes of the problems as clearly as possible
and try to understand the way Newton's mechanics and  classical physics  emerge
   for macroscopic bodies in quantum mechanics,  without necessarily bringing into the discussion many  unrelated (especially, philosophical) aspects  of  the quantum measurement problems.

Still,   there is one  key issue  linking  the two problems.   It is the very definition of the position and momentum of the centre-of-mass of a macroscopic body,
which  necessarily involves a
 measurement, both in the quantum-mechanical and classical contexts.
  There,  the well known difference between quantum mechanics and classical mechanics
makes appearance in a significant way.

   It is the purpose of this work to improve our understanding of the mechanism for the emergence of classical mechanics in quantum mechanics.
    In particular,  we ask a simple question:  does a macroscopic body  in the vacuum, i.e., without a pre-existent environment,   behave classically or quantum mechanically?
However,  we must first better define the  various terms and conditions, and formulate the problem itself more precisely.

What is a macroscopic body?  Is a large molecule  such as a DNA macroscopic?
Does it need to contain  a large macroscopic number of atoms or molecules?   Is $N=1000$ sufficiently macroscopic, or should it be of the order of
the Avogadro number $N_A \sim 6 \cdot  10^{23}$?
Also, what does a ``body" mean?  Should it be a solid, such as a piece of metal, having a crystaline structure inside?
Or could it be a drop of  liquid,  or even a volume of gas?  What about a dust particle, or any  piece of a biological body?
  Also, should the body have some particular  mass,  energy, density or temperature $T$,   to behave classically?

This  last question brings us actually to   the very starting point of our analysis.  As we shall be  studying the behavior of a ``body in the vacuum",  from here on  we mean by $T$  the temperature of the body, and not of the environment (which is absent).
 Because  the temperature of the surrounding space  (the vacuum)  is zero,
clearly   the problem is not a static one,  unless $T=0$.   It will be seen below that the (large) mass and body temperatures will play essential roles for a macroscopic body to behave classically.

Now at $T=0$,  whatever body,  microscopic or macroscopic,  and independently of the state of matter, gas, liquid, solid  or Bose-Einstein condensed (BEC)  gas or liquid,  is in a quantum mechanical state. This is because, by definition,  at $T=0$  all  the atoms and molecules  composing the body, as well as their collective motions,  are in their quantum-mechanical ground state  \footnote{This includes the cases the body consists of many identical fermions, where the many-body ground state must obey the Fermi-Dirac statistics, a condition which leads to various kinds of so-called degenerate Fermi gas.
 Examples in Nature abound,  they go from ordinary metal conductors, to the neutron stars, and the white dwarfs.
 }.   The whole system is in a unique ground state, which is  the content of  the third law of thermodynamics: the entropy is zero  at $T=0$   (which is known as  Nernst-Planck's law)
 \footnote{Actually, small quantum mechanical systems may have degenerate ground states; however it is well known that due to tunnelling effects, the true vacuum (the lowest-energy eigenstate) is unique.  In systems with an infinite number of degrees of freedom,  however,   this argument does not work.  The system may have  more than one degenerate  ground states, without  tunnelling among them.
In these systems
 Nernst-Planck's law  still applies, because  the system necessarily chooses, at $T=0$,  one of the lowest-energy states as its physical, unique  ground state. Spontaneous symmetry breaking is one of the manifestations of  this phenomenon. }.    The specific heat of any matter approaches  to $0$ as $T\to 0$.

We must also make a qualification on the concept of {\it  the vacuum}, before we proceed.        In quantum field  theory  the ``vacuum"  is not  an empty, sterile space.   It represents the ground state with respect to all possible field excitations in the theory.    Under normal conditions  such a zero-point energy  uniformly distributed  all over the space is  not observable \footnote{A good illustration of this point is  Planck's formula for the black-body radiations.  It expresses the energy distributions  among various electromagnetic excitations  above the vacuum, at temperature $T$.
 It is described by the numbers of photons each carrying the excitation energy quantum  $h\nu=\omega  \hbar $. The zero-point energy  $\tfrac{1}{2} \omega  \hbar $  of each oscillator
filling the whole space,  is not observable under normal conditions. }.
  However,  under certain local disturbance  such as a strong gravity and spacetime curvature such as those  near a blackhole,  or extremely strong electromagnetic fields applied to a restricted region, particle pairs are produced from the vacuum, leading to observable effects  (the vacuum polarization,  the Unruh effects,  Hawking radiation, etc.).  Another well-known  manifestation of  the quantum fluctuations of the vacuum is the so-called Casimir effect,  which exerts attractive forces between two macroscopic parallel conducting plates.  We assume that it is a good approximation to disregard, {\it   for the purpose of the following discussion, }   these nontrivial quantum properties of the  vacuum
  (see for instance  \cite{Birrell})   surrounding our macroscopic body.

 This said,  our starting point is  this.   Any matter, at $T=0$, or at temperatures {\it   sufficiently close to it},  is quantum mechanical.    The problem is to
understand   what  ``sufficiently close to  $T=0$"  means.
In general,  the matter will maintain quantum-mechanical behavior   at $T >  0$,  until $T$  reaches some critical value. How high can  $T$ be above the absolute zero temperature, before the system loses  the characteristic
quantum mechanical behaviors,  depends on the  type of atoms and molecules, the kind and state of matter,  and their  density.   The details depend also on the type of  interactions among the
constituent atoms and molecules and  on the nature of  their collective excitation modes.  See Sec.~\ref{QCL}  for a discussion  on these questions.

In the case of a gas composed of  identical bosonic atoms,  the manifestation of their  (collective) quantum-mechanical properties has been
shown  through the so-called Bose-Einstein condensation (BEC),  a phenomenon in which a macroscopic number of atoms occupy the same lowest-energy quantum states, and as a result   is  described by a macroscopic wave function, $\Psi ({\bf r})$.   BEC  was first realized for the gas of $Rb^{87}$  atoms  at  $T \le 1.7 \cdot  10^{-7} $  K \cite{Cornell}, and for  $N\!a^{23}$  at   $T \le 2 \cdot 10^{-6} $ K    \cite{Ketterle}.
The dilute  gas of $H\!e^4$  was  laser-cooled down to $4.7 \cdot 10^{-6}$ K,  to form a BEC  involving  $ \sim  5 \cdot 10^5$  $H\!e$ atoms \cite{Cohen}.

 In high-energy physics context  (i.e.,  in the physics of the fundamental interactions), the same phenomenon  (BEC)
has been well understood, since the sixties.  The condensation of identical (elementary or composite) bosons   at $T=0$   and  the consequent spontaneous symmetry breaking
of global \cite{NambuJona,Goldstone}  or local symmetries   \cite{Higgs,Englert},  have been some of the main theoretical ingredients which made possible  the construction of   the standard   $SU(3)_{QCD} \times ((SU(2)\times U_Y(1)))_{GWS} $ theory \footnote{QCD and GWS  stand respectively for Quantum Chromodynamics and Glashow-Weinberg-Salam.} of the strong and electroweak interactions,  around 1970-1974 \cite{Weinberg,Salam,Glashow}.
This is one of the most precisely tested  theories in physics known  today.

Detailed studies by using lattice simulations of  QCD  show  that the quark-antiquark chiral-symmetry breaking condensates, $\langle  {\bar q}  q \rangle$,   drops  to $0$ at a critical temperatures
$T_c \simeq  190$ MeV  ($\simeq 10^{12}$ K), signalling a chiral-symmetry-restoration phase transition.

Coming back to atomic physics,
 it is an active research area to study the properties of  Fermi gas  (atomic)
of various types, e.g.,    $He^3$,     at  low temperatures.

Superconductivity occurring  in some metal   at temperature  below a critical temperature,   e.g., below $T_c=9.7^o K$ for $Nb$,  is another, extremely interesting  phenomenon \cite{LL9}.

On the other hand,  the electrons inside  the normal metals behave collectively  approximately as a degenerate quantum Fermi gas even at room temperatures.  For instance,  electric conductivity  is  a typical quantum-mechanical phenomenon involving the tunnel effects.

Another, special  example is light. Light is a  gas of noninteracting, free bosons (the photons).  It is  quantum mechanical at any temperature.  Even though Maxwell's theory is able to describe successfully myriad of macroscopic electromagnetic phenomena,  it met with unsurmountable difficulties in explaining the behavior of the  blackbody radiation spectra and its temperature dependence.
Alongside the related problems of explaining the behaviour  of the specific heat of various matter,    which turned out to depend on the detailed nature of the microscopic quantum-mechanical excitation modes  inside  materials,
clarifying the nature and solution of  the blackbody problem has led Planck to the breakthrough discovery of
quantum physics in 1900.

Various kinds of liquid also  show  extraordinary quantum  behaviors  at very low temperatures.  Liquid  $H\!e^4$  exhibits superfulidity  at temperatures  below  $\sim 1 \, $ K, a phenomenon  already known around 1930 \cite{LL9}.     Superfluidity phase  in  liquid  $H\!e^3$  occurring around  $\sim 2 \cdot 10^{-3} $ K   was discovered only in 1972 \cite{He3}.

Even the solids exhibit  quantum-mechanical behavior at sufficiently low temperatures.    A recent fabrication of a  mechanical resonator
\cite{Aaron}  kept at $T\sim 2\cdot  10^{-2} $ K    was  regarded by some as  a promising start for realizing in the laboratory an analogue of a  linear superposition of the
dead and alive cat states.

{\noindent  \bf Remark:  } Actually, this might have been an excess of enthusiasm, not to be taken literally.   On the one hand, a quantum superposition of macroscopic bodies require  their  body temperatures  close to the absolute zero.  On the other,
a living cat needs nonzero  body temperatures for  its biological functions, thus  it is necessarily a mixed state  (see below).    The  conclusion is  that  a ``quantum superposition of dead and alive cat"  is a self-contradictory notion,  neither realizable in practice nor  in principle.

~~~

This brief survey should be sufficient to illustrate  that   ``how does classical mechanics  emerge from quantum mechanics?"  is, generally speaking, not  a very  well-formulated question.   Also, a fact that clearly came into focus in the overview is the prominent role the temperature plays in the whole discussion.

Keeping  all these subtleties in mind,
we set ourselves to study in this work
   the time-evolution of the  center-of-mass (CM)  position and momentum
of    {\it    a generic  macroscopic body in the vacuum   and  at nonzero body  temperatures},  by using the Schr\"odinger equation (the Ehrenfest theorem).

After identifying the elements which ensure the center of mass of  a macroscopic body to have a classical trajectory (Sec.~\ref{basic}),  we proceed to  find  that the CM motion of such a body is indeed described by
 Newton's equations  (Sec.~\ref{Ehrenfest}).   We will show this for the body in  weak  external gravitational  forces,  under a harmonic potential,   as well as under weak, slowly varying external electromagnetic fields.

  Effects due to the finite size of the body, such as the tidal effect, well known in classical physics, also appear correctly from the Ehrenfest theorem, as will be verified by applying perturbation theory.

Sec.~\ref{QCL} discusses briefly the temperatures characterizing  the boundaries between the quantum and classical descriptions, drawing lessons from some experimental work in the literature.

Summary and conclusion are in
Sec.~\ref{Summary}.

\section{Center of mass  of a macroscopic body  \label{basic}}

There are four elements which enter  the derivation of Newton's equations for  a macroscopic body in quantum mechanics.  Three of them  are
\begin{description}
  \item[(i)]   Heisenberg's uncertainty relations for the CM of the macroscopic body;
  \item[(ii)]   Absence of the diffusion of the wave packet; and
  \item[(iii)]   The mixed-state nature of a macroscopic body at nonzero temperatures, and decoherence.
\end{description}
These three elements,  taken together, imply that
the center-of-mass position and momentum of  (the narrow wave packet representing) a macroscopic body  is  well defined  at each instant of time.

Note that this is so even if, in general, the microscopic degrees of freedom inside a  macroscopic body under consideration  (such as the electrons, photons, atoms and  molecules inside a metal ball, a rock,  planets and stars) may continue to maintain characteristic  quantum-mechanical behaviors.

Once the classical, unique position and momentum of a macroscopic body  is identified
with the vacuum expectation values of the center-of-mass position and  momentum operators  in a narrow, diffusionless  wave packet,
 it is in principle a straightforward exercise to check  that
 \begin{description}
  \item[(iv)]    Ehrenfest's  theorem for such a wave packet
 coincides with Newton's force law,  in appropriate limits the macroscopic body can be regarded as pointlike.
 \end{description}
 Due to the composite, extended  nature of a macroscopic body,  however,  classically known  effects such  as  polarization, the tidal effects, etc.,  also appear naturally  from the multipole  expansion of the external potential.
 See  Sec. \ref{Ehrenfest}.

\subsection{Heisenberg's uncertainties    \label{Heisenberg} }

For a microscopic particle, an atom or molecule,   the derivation of Heisenberg's uncertainty relation is well known.
The canonical position and momentum operators of a particle,   $x, p$, satisfy
\begin{equation}   [x,p] =i\hbar\;.
\label{commut}
\end{equation}
For compactness of expressions we write the formula for one-dimensional particles, but  the generealization is  understood.
The uncertainty of $x$ or of  $p$  is defined as their dispersions
\begin{equation} \Delta x \equiv  \sqrt{\langle (x-x_0)^2 \rangle}, \qquad   \Delta p \equiv  \sqrt{\langle (p-p_0)^2 \rangle}\;,
\end{equation}
where
\begin{equation}   x_0 =\langle x \rangle =\langle \psi|x|\psi \rangle  \;,
\quad  p_0 =\langle p \rangle =\langle \psi|p|\psi \rangle  \end{equation}
are  their mean values  in the state $\psi$.
  One has
\begin{equation}  \langle \psi|A^{\dagger}A | \psi\rangle    \ge 0  \end{equation}
 for any operator of the form,
\begin{equation}   A= x-x_0  + i \alpha (p-p_0)\;, \end{equation}
where $\alpha$ is an arbitrary real number,
and for any state  $\psi$. That is
\begin{equation}  \langle (x-x_0)^2 \rangle - \alpha \hbar + \alpha^2 \langle (p-p_0)^2 \rangle \ge
0\;.  \end{equation}
It means that  the discriminant  of the quadratic form above  is semi-negative definite,
that is
\begin{equation} \Delta x \cdot  \Delta p  \ge  \frac{\hbar }{ 2}\;. \label{indeterm}\end{equation}
The case of a particle in three-dimensional space is  a straightforward extension of these discussions.

Let us now consider the case of a macroscopic body, made of  a large number $N$ of atoms and molecules, e.g.,  $N\simeq 10^{23}$,   $N\simeq 10^{57}$, etc.
The center-of-mass position, the total mass and the total momentum  are  defined as
\begin{equation}     {\bf R} =  \frac{1}{M}    \sum_{i=1}^N   m_i   {\bf r}_i\;;  \qquad  M=  \sum_{i=1}^N  m_i\;,\qquad    {\bf P}=  \sum_i {\bf p}_i\;. \label{simple1}
\end{equation}
It follows that   (${\mathbf R}= (X_1, X_2, X_3)$)
\begin{equation}  [{X}_k, {P}_{\ell} ] =      \frac{1}{M}    \sum_{i=1}^N   m_i  \,   ( i\hbar   \,  \delta_{k \ell})   =   i\hbar    \,  \delta_{k \ell} \;, \qquad    k, \ell =1,2,3\;. \label{simple2}
\end{equation}

For simplicity let us consider the total Hamiltonian of the form,
\begin{equation}  H= H_{CM} +   H_{int}\;,
\label{TotHam}  \end{equation}
\begin{equation}       H_{CM}=  \frac{{\bf P}^2}{2M} +  V({\mathbf R})\;;
  \qquad     H_{int}= \sum_{i=1}^{N-1}    \frac{{\tilde p}_i^2}{2\mu_i} +   V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;,
\label{Hamiltonians}  \end{equation}
and the
 interaction potential  depending  only on the  internal, relative positions
\begin{equation}     V ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) =   V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})  \;\label{form}
\end{equation}
\begin{equation}    {\bf r}_i = {\bf R} +  {\tilde {\bf r}}_i\;, \quad  i=1,2,\ldots, N\;; \qquad
    \sum_{i=1}^N    m_i   {\tilde {\bf r}}_i = 0\;,  \label{relative}
\end{equation}
and $\mu_i $ are the reduced masses \footnote{In two-body  ($N=2$) systems, it is customary to take as the internal position operator as   ${\mathbf r}\equiv    {\mathbf r}_1 - {\mathbf r}_2$,
rather than in (\ref{relative})  here, i.e.,    $ {\tilde {\bf r}}_1 =   {\bf r}_1 -  {\mathbf R} =  (m_2/M)  ( {\mathbf r}_1 - {\mathbf r}_2) $.   This explains the apparently different expression for  the reduced mass  from that found usually  in the textbooks.
}
\begin{equation}    \frac{1}{\mu_i}   =  \frac{1}{m_i} - \frac{1}{M}\;,\qquad      i=1,2,\ldots, N-1\;.
\end{equation}
Such a  Hamiltonian allows us to write the wave function in a factorized form,
\begin{equation}   \Psi({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) =   \Psi_{CM}({\bf R}) \psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;.   \label{factorization}
\end{equation}

Following the same  procedure as for  a particle,  (\ref{commut}) - (\ref{indeterm}),       by using this wave function   $\Psi$   instead of  $\psi$,
 one finds  straightforwardly
\begin{equation} \Delta X_k  \cdot  \Delta P_{\ell}   \ge  \frac{\hbar }{ 2}   \, \delta_{k \ell}     \; \label{indetMacro}   \end{equation}
for a macroscopic body.

Needless to say, we are using the nonrelativistic approximation here, for simplicity.
Take for instance the hydrogen atom.  The mass of a hydrogen atom in the ground state,  is not equal to  $m_e + m_P$, but
$m_H= m_e + m_P -  14 \,{\rm e\!V}/c^2 $,  by taking into account the binding energy.  However the error we make by   approximating the total mass by the sum of the electron and proton mass  is of the order of
\begin{equation}             \frac  {14 \, {\rm eV}/c^2 }{m_H} =    \frac  {14 \, {\rm eV}  }{0.94  \, {\rm GeV}}      \sim   O(10^{-8})\;.
\end{equation}
The same kind of errors will be even  smaller  in the case of the molecules, as their binding energies are in general smaller than the case of atoms.

Clearly this kind of argument cannot be applied to the level of the atomic nuclei,  where the binding energy is  relatively much larger.
The problem might look even more serious if  a similar consideration is  extended to a more fundamental level.  It is known today that  the protons and neutrons which  make up the atomic nuclei are actually bound states made of the quarks and gluons,  and described by Quantum Chromodynamics  (QCD).  The mass of the nucleons, $\sim  940  \, {\rm MeV}/c^2\sim   1.67 \cdot 10^{-24}$  g,
is mostly the effect of the  strong gauge interaction dynamics, with the contribution of the  up and down quark masses    ($m_u\sim  5 \, {\rm MeV}/c^2$, $m_d\sim  10 \, {\rm MeV}/c^2$)  being only a small part of it.

Clearly, the problem is that the protons and neutrons are  highly relativistic, quantum mechanical bound states,  for which the simple  classical  discussion such as (\ref{simple1}), (\ref{simple2}) cannot be  applied.

Nevertheless,  nothing prevents  us from considering a macroscopic body as made up of  $N\simeq 10^{23}$ or    $N\simeq 10^{51}$    {\it   atoms or molecules},
which are in turn bound states of the electrons and the atomic nuclei (protons and neutrons),     and for the purpose of the discussion  here,  from disregarding  how the protons and neutrons
are themselves made of  the quarks and gluons \footnote{Today  we know that  quantum chromodynamics  (QCD)  describes correctly the properties of the proton, neutron, and other strongly-interacting particles such as pions and $\rho$  mesons. }.
 The understanding of how a macroscopic body is made out of atoms and molecules does not require - say, most of time -   the knowledge of how  the physics at smaller scales explain the properties of nuclei, atoms and molecules: it is  a general character of physics laws.

With this understanding,  (\ref{indetMacro}) is the correct Heisenberg's uncertainty relation for the center-of-mass position and momentum of a macroscopic body  \footnote{This fact is  often simply assumed.  It leads to the limits  in the resolution  in various precision measurements involving macroscopic bodies and
optical instruments,   known as  the standard quantum limit  (SQL)   \cite{SQL}. }.

What is most remarkable is the fact that a macroscopic body, e.g., made of $N= 10^{23}$ or  $N= 10^{51}$     atoms and molecules, satisfies the same Heisenberg's uncertainty relations
for its CM position and momentum,    as that for each of the constituent atoms. {\it  Heisenberg's uncertainties - the effects of quantum fluctuations -  do not pile up. }

But this means that  for an experimental  determination of the position and momentum of the CM of a macroscopic body, Heisenberg's uncertainty  constraints  may be regarded
unimportant.
To have some concrete ideas,  one may take for instance, somewhat arbitrarily,    the  experimental uncertainties   $   \Delta X,  \Delta P$    of the order of
\begin{equation}     \Delta P \sim    10^{-2}  ({\rm g}) \times   \frac{10^{-4}   {\rm cm}}{\rm sec}  \sim   10^{-6}  \,  \frac  {{\rm g} \, {\rm cm} }{\rm sec}\;;   \qquad     \Delta X \sim  10^{-4} \, {\rm cm}\;.
\label{RuleofT}
\end{equation}
  Their product is
\begin{equation}   \Delta P \cdot \Delta X \sim  10^{-10}   {\rm erg} \,  {\rm sec}    \;,
\end{equation}
many orders of magnitude   larger than the  quantum mechanical lower bound,    $ \hbar  \sim 1.05  \times 10^{-27}   \,   {\rm erg} \, {\rm sec}.$
Thus even allowing for  much  better precision by several  orders of magnitudes than the rule-of-thumb (\ref{RuleofT}),   both for the position and for the momentum,  one can still regard
the Heisenberg uncertainty relations insignificant.  In other words one can effectively take
\begin{equation}  \Delta P\approx 0\;, \quad  \Delta X \approx 0\;  \label{Definite}
\end{equation}
for macroscopic bodies.

In other words,    the position and momentum  of a macroscopic body can be measured and determined  simultaneously
 with  an ``arbitrary" precision,  at   macroscopic scales.

\subsubsection{Observation of the gravitational waves}

An alert reader  might wonder how such an (obviously  very simple-minded)  consideration can resist a scrutiny in view of the extraordinarily    precise observation of the gravitational waves  at LIGO and VIRGO
laboratories  made recently  \cite{GW150914}.
In the first observation of GW150914  performed at the two LIGO detectors,   to fix the idea,   $\sim  40$ Kg  mirrors used in the Michelson-Morley interferometer   oscillate  about  $\pm 10^{-16}$ cm,  with frequency around $100$ Hz.
Thus  the momentum and position uncertainties of the mirrors can be taken to be  of the order of
\begin{equation}  \Delta P \sim  4\cdot 10  \cdot 10^3 \cdot  10^{-16} \cdot 100 \sim   4 \cdot 10^{-10} {{\rm g\, cm}}/{{\rm sec}}  \;, \qquad   \Delta X \sim 10^{-16} \, {\rm cm}\;,
\end{equation}
such that their product is
\begin{equation}      \Delta X \cdot  \Delta P \sim   4   \cdot   10^{-26}  \, {\rm erg}\,\, {\rm sec}  \;,
\end{equation}
a factor $\sim 10^2 $ above the quantum-mechanical  lower bound   (\ref{indetMacro}),   i.e.,     $\hbar /2  \sim 5 \cdot 10 ^{-28}$   erg sec.
 See Ref. \cite{Braginsky} for a more careful analysis, where  it is shown that  the main  source of the  sensitivity limit for the gravitational wave detection  comes from the effects related to  the quantum fluctuations of the photons in the Fabry-Perot cavity, related to the radiation pressure and  laser shot noise, rather than the Heisenberg uncertainty of the mirror CM position and momentum.      In other words,
  the observed gravitational-wave-induced oscillation of the mirrors  is a classical motion.

\subsection{Diffusion of a wave packet   \label{Diffusion}  }

Another textbook result in quantum mechanics  is  that  the wave function of a free particle
 (e.g., an electron, an atom)  diffuses in time.  For instance,
a  Gaussian wave packet of size $\sim a$   at time $t=0$,
\begin{equation}    \psi(x, 0) = {\rm const}  \, e^{- x^2 /  a^2}
\end{equation}
evolves,  according to the Schr\"odinger equation,
\begin{equation}    i \hbar   \frac{d}{dt} \psi(x, t) =   H\,   \psi(x, t) \;,   \qquad  H =  \frac{ p^2}{2 m} \;
, \qquad  p= - i \hbar \frac{\partial}{\partial x}
\end{equation}
to a wider distribution,
\begin{equation}   |\psi(x, t)|  =   {\rm const}  \, e^{- x^2 /  a(t)^2} \;, \qquad     a(t) =  \sqrt{   a^2 +   \frac{4 \hbar^2 t^2}{m^2  a^2}  } \sim    \frac{ 2\hbar t }{m a}
\end{equation}
after time $t   (\gg  \tfrac{m a^2}{2 \hbar})$.   Note that the initial  momentum uncertainty
\begin{equation}      \Delta p \sim \frac{\hbar }{\Delta x} \sim \frac{\hbar }{a}
\end{equation}
is responsible for the spreading of the wave packet.

{\it   The mass plays an essential  role in the  diffusion of a wave packet. }  To have a concrete idea,   a free electron with mass $m \sim 10^{-27}$ g
described  by a wave packet of size $\sim 1$ $\mu$m,  will double its wave packet size   in   $t=10^{-8}$ sec.   It will spread to the size of $\sim100$ m  in one second.
For a hydrogen atom, of mass $m \sim 10^{-24}$ g,  described by the wave packet of  size $\sim 1$ $\mu$m,  the time needed to double its wave function spread is about
 $t=10^{-5}$ sec.

In comparison, the center of mass of a  macroscopic  body  of  $1$ g,  described by the free Schr\"odinger equation
\begin{equation}      i \hbar  \frac{d}{dt} \Psi =  - \frac{  \hbar^2  \nabla^2}{2M}   \Psi\;;
\end{equation}
and satisfying the same Heisenberg's relations,  (\ref{indetMacro}),  as for the electron,   initially described by a wave packet of size    $\sim 1$ $\mu$m   for its center-of-mass wave function,
will require
\begin{equation}       T \sim 10^{19}  {\rm sec}   \sim 10^{11}  {\rm yrs}  \;,
\end{equation}
to double its size, which is longer than the age of our universe.

The conclusion is that  the diffusion of a  free wave packet - quantum fluctuations - of the center-of-mass position is negligible   for  a macroscopic object, simply due to its large mass.

{\it   This difference explains an apparent (and  loosely defined) puzzle often discussed  in the literature \cite{Joos},  why  microscopic particles  (e.g., electrons) are ``usually"  in momentum eigenstates,  whereas a macroscopic body is
``always"  in a position eigenstate.   }

Note that  the  (negligible)  spread of its CM wave packet, has absolutely nothing to do with the size  (i.e. a diameter) of a macroscopic body,   $L_0$.  Let us recall  also that the diffusion of the wave function recalled above refer to a free particle.
Thus  even though the atoms and   the  electrons inside the atoms  composing a macroscopic body,  remain  quantum mechanical systems,  their wave functions do not diffuse:    they are
described by stationary bound-state wave functions. Their size  $L_0$   (of a macro molecule, a crystal ball,    a
piece of a stone, the earth, the sun,  etc.)  is determined by the dynamics which bind the constituent electrons, atoms  and molecules  in
a bound state, the lattice structure,  etc.,   in other words  by the extension (i.e., spatial support)   of the wave functions   describing the bound state.

In the discussions below, it is indeed fundamental not to confuse the size of a macroscopic body $L_0$    as defined above,  with  the extension of the wave packet of its center of mass,    $\Delta_{CM}$.   Clearly, any experimentalist  capable of  determining  the size of the body
 $\sim L_0$  should be able to measure  its CM position  with precision,
\begin{equation}     \Delta_{CM} \ll     L_0\;.   \label{always}
\end{equation}
This inequality will be understood  throughout.

To conclude,  once the position and momentum of the CM of a macroscopic body are given by a measurement  done with arbitrary precision  (see Sec.~\ref{Heisenberg}),   the diffusion of its wave packet  is negligible.
Its position and momentum will be always well defined.  A macroscopic body  (its CM, not its internal degrees of freedom)  traces a classical trajectory.

 \subsubsection{An extended wave packet is a quantum concept   \label{impossible}}

 The fact that a macroscopic  body has a negligible  quantum-mechanical diffusion, means that  an extended wave packet  (e.g., with  $\Delta_{CM} \ge  L_0$)   for the center of mass of a macroscopic body,  is a meaningless concept.

Both in quantum mechanics and in classical mechanics, the concept of the position and/or momentum is  defined as the result of  a measurement.
In the quantum case,  that is, for a microscopic system,    Heisenberg's uncertainty relations  do not admit  the simultaneous   determination of the
measured  values of  the canonically conjugate  variables, $x, p$.   If one desires to prepare the particle with  a reasonably well-defined position,
for instance in the direction perpendicular to its motion,  it is sufficient to let the beam impinge  upon  a film with a hole  of diameter $d$   in it.  If it passes
its position has been determined with the precision (uncertainty), accompanied by the uncertainty in its momentum  \cite{Heisenberg},
\begin{equation}  \Delta x \sim  d\;; \qquad .^.. \quad    \Delta p \ge   \frac{\hbar}{ d} \;.
\end{equation}
The particle will then diffuse   (Fig.~\ref{Diffuse}), and if there are many holes in the sheet,  the particle waves diffusing  from different holes interfere.

\begin{figure}
\begin{center}
\includegraphics[width=5in]{Diffuse}
\caption{\small  A microscopic particle going through a slit. The passage, which amounts to a position measurement, induces momentum uncertainty, hence diffusion.   }
\label{Diffuse}
\end{center}
\end{figure}

 The  image  of the electron wave ripples,  shown in Fig. ~\ref{Tonomurawaves},
 which look identical to  water ripples  on a puddle surface  produced  by rain drops,  but are smaller by a factor
$\sim 10^{-6}$ in scale, have been obtained by A. Tonomura, by letting $50$ kV electron beams go through a thin collodion film with tiny holes.

  As the  law of  diffusion of a free particle wave is known,  an initial wave packet of an arbitrary size can be  experimentally prepared, and used as the initial condition.

\begin{figure}
\begin{center}
\includegraphics[width=4in]{TonomuraRipples}
\caption{\small   Diffusion and interferences of the electron waves.   They have been produced by letting a $50$ kV  electron beam
fall on a thin collodion (celluloid) film.  Courtesy by A. Tonomura.   }
\label{Tonomurawaves}
\end{center}
\end{figure}

In sharp contrast,   a macroscopic particle  position  determination  requires a hole of diameter $D$   larger than its size, $ L_0$.  If it passes its transverse position will have been measured with the precision,   $\Delta X =  D$, but it does not coincide with the size of its center-of-mass wave packet, even though it will provide an upper limit for the latter,    see (\ref{always}).   Also,   as  the latter  does not diffuse,
there is no way  to prepare its wave function with   a desired,  extended wave packet.  Note that the  diameter  $D$  of the hole must be larger than   $L_0$ (otherwise
the body will not pass);   after the passage   its position is determined    with a (classical) uncertainty  \begin{equation}  \Delta X =  D   >   L_0, \end{equation}
 but it is simply  due to the lack of information which (classical)  path the body has taken inside the hole  (Fig.~\ref{Single}).
The momentum uncertainty caused by  such a position measurement would be  given by
\begin{equation}    \Delta P \sim  \frac{\hbar}{D}   <    \frac{\hbar}{L_0} \;.\label{wouldbe}
\end{equation}
But this is much less than the  original momentum uncertainty of the  CM   wave packet,
\begin{equation}      \Delta P_{CM}  \sim     \frac{\hbar}{\Delta_{CM}}   ( \gg      \frac{\hbar}{L_0} ) \;.\label{momunc}
\end{equation}
In other words, the position measurement does not affect the original momentum uncertainty, in contrast to the case of a microscopic particle. The passage through a hole
does not induce a diffusion of the wave packet for a macroscopic particle.

\begin{figure}[h]
\begin{center}
\includegraphics[width=4.5in]{Singleslit}
\caption{\small  The passage of a macroscopic particle of size $L_0 \gg  \Delta_{CM}$,   through a hole of size  $D> L_0$.  The passage does not increase momentum uncertainty:
there is no diffusion.   }
\label{Single}
\end{center}
\end{figure}

\subsection{Decoherence:  mixed-state nature of a macroscopic body at finite temperatures  \label{Decoherence}}

The third  condition for a macroscopic body to behave classically,  is decoherence. It is here that the finite body-temperature  plays the essential role,  which makes the body
 in the vacuum under considertion a metastable, radiating,  mixed state.

The absence of the diffusion of the CM of a macroscopic body  (Sec.~\ref{Diffusion})  was  regarded  by Born  as a sufficient reason for such an object to behave classically.
  Einstein disputed it   by observing  \footnote{This debate is reported, e.g., in  \cite{Joos}.}  that a  superposition of the state
\begin{equation}   |\Psi \rangle   = c_1  |  \Psi_1 \rangle  +   c_2   | \Psi_2 \rangle\;, \label{nopure}
\end{equation}
where $\Psi_1$ and $\Psi_2$ are two narrow wave packets  centered at  ${\bf R}={\bf R}_1$ and at ${\bf R}={\bf R}_2$, separated by a macroscopic distance,  {\it is}  allowed by quantum mechanics, i.e., as a solution of  the Schr\"odinger equation.  A state of this kind  certainly contradicts the notion of a classical particle, with its unique trajectory,  at a definite position at each  instant of time.

One way to explain away  this apparent paradox is to note that a macroscopic body, {\it   at nonzero temperature}, is   a mixed state, as it radiates.
The nature, and the solution,  of the problem  would then be  the same as the one for the so-called Schr\"odinger cat paradox \cite{KK}.
 A macroscopic body  at finite temperatures  is  a mixed state:  it is, at best, a metastable state,
 \begin{equation}  |  \Psi_1 \rangle  =    |  \Psi_1^{(0)}  \rangle   +    |  \Psi_1^{(1)}, \gamma  \rangle   +   |  \Psi_1^{(2)}, \gamma  \gamma  \rangle + \ldots   \label{macrobody}
 \end{equation}
 where  $ |\Psi_1^{(0)}  \rangle $   is the effective  "wave function" of the metastable state describing  systems with atoms and molecules in excited states;
 $  |  \Psi_1^{(1)}, \gamma  \rangle  $ with one emitted photon flying away, with the rest of the system still containing atoms and molecules
 in excited states,    $  |  \Psi_1^{(2)}, \gamma  \gamma  \rangle  $ with two emitted photons,  etc.   Similarly for  $ |  \Psi_2 \rangle$,
  \begin{equation}  |  \Psi_2 \rangle  =    |  \Psi_2^{(0)}  \rangle   +    |  \Psi_2^{(1)}, \gamma  \rangle   +   |  \Psi_2^{(2)}, \gamma  \gamma  \rangle + \ldots   \label{macrobody2}
 \end{equation}
  The coefficients
  of different terms  in (\ref{macrobody}) and in  (\ref{macrobody2})   are omitted, as the norm of each term  depends on time anyway.    In the case of a single atom in a metastable, excited state, it is still possible to write a reasonably precise expression for    its (effective) wave function (see for instance \cite{KKGP});     in the case of a macroscopic body at finite temperature, with  many atoms and molecules in the metastable, excited states, such a description is neither available, nor really needed. The only thing needed is the fact that each of  the ``states",
  $ |  \Psi_1 \rangle  $ and  $ |  \Psi_2 \rangle  $, is actually a mixed state,  so that their  quantum interference  is  negligible.      The density matrix  corresponding to  $\Psi$ of  (\ref{nopure}) in the coordinate representation is diagonal, in good approximation.

It is interesting to  compare the situation with the wave function of an $Ag$ atom, in the Stern-Gerlach experiment,
\begin{equation}     |\psi \rangle   = c_1  |  \psi_{1}  \rangle  +   c_2   | \psi_{2}  \rangle\;, \qquad    \label{SGatom}
\end{equation}
where  $\psi_1$ and   $\psi_2$  represent the spin up and down states, respectively.  In an inhomogeneous magnetic field  in the ${\hat z}$ direction,    the two components
$\psi_{1}$ and $\psi_{2}$  get separated in the vertical direction,  as the atom proceeds towards (say) the ${\hat x}$ direction.
The two wave packets might well be separated  by a macroscopic distance, before  the atom impinges upon the photographic plate.
The wave function  (\ref{SGatom})  still represents a pure quantum state, and  the phase coherence  between $\psi_{1}$ and $\psi_{2}$  can be
verified and measured, if    the two beams are reconverged  by another magnetic field, before they hit the photographic plate, in

 an experimental setup known as the quantum eraser.

The contrast  between the classical ``wave packets"  (\ref{nopure}) and the quantum-mechanical pure state  (\ref{SGatom})   (with a macroscopic separation between the two peaks of the wave packets),  may be made manifest,    by assuming that  such a superposition  has been somehow prepared  at time $t=0$, and by  studying the time evolution  of the density matrix in the position representation.    The ideas of  decoherence  is that because of the entanglement of the particle  with the environment, nondiagonal elements of the density matrix  in the case of a macroscopic body,  get rapidly damped in time, leaving the diagonal components, showing that the
system  is indeed  a mixed state   \cite{JoosBis,Joos,Zurek}.

    In the case of  large molecules,  such as $C_{70}$,   as a model for   (\ref{nopure}),
a related study has been performed by  Arndt et. al. \cite{C70}, where   the environment is replaced  \footnote{This expression is not quite exact.   Here the laser beam is used to excite the $C_{70}$  molecules which then radiate, a process distinct  from the scattering by the environment particles, as considered in   \cite{JoosBis,Joos,Zurek}.} by a laser beam    bombarding  the
$C_{70}$  molecules,  before the latter enter a double slit region  (for detecting their quantum-mechanical feature, such as the diffraction and interference).
This experiment verifies clearly, quantitatively,  the decoherence due to the finite-temperature mixed state nature of the body.

The main observation in the present work is that, actually,  it is  sufficient that the body radiates, i.e., the body temperature is nonzero,  for it to be a mixture, and  to decohere.
There is no need for a pre-existent environment.   Concrete calculations  of the radiation spectrum  from highly  excited, isolated small particles   have been
performed by Hansen and Campbell  \cite{Hansen}.
The result of this analysis was then used by Arndt et.al. \cite{C70} to derive  the
decoherence \`a la   \cite{JoosBis,Joos,Zurek}   for  $C_{70}$   beams,  i.e., calculating the effects of the entanglement with the environment (in this case, the photons it emitted)  on the density matrix in the position representation, and showing that  the two  (or more) nondiagonal peaks quickly die out  in time.  See Sec.~\ref{QCL}  for more comments on this $C_{70}$  experiment.

To do a quantitative analysis of decoherence of this sort   for a macroscopic body made of $N \sim 10^{23}$ atoms, in the vacuum and at  the temperature $T\ne 0$,  might look  quite a nontrivial  challenge.

\subsubsection{Split wave packets for a   (finite $T$) macroscopic body:  an impossible notion}

Actually,   there is a simpler answer,  logically sharper than such an argument,  to the question why a pure state as (\ref{nopure}) does not occur in nature, for a macroscopic body at nonzero temperatures.   The answer is,  as it follows from the  discussions of Sec.~\ref{Heisenberg} $\sim$  Sec.~\ref{Decoherence},   that  {\it    it is not possible to prepare  a doubly (or multiply)  split wave packet  such as  (\ref{nopure}),  experimentally.  Such a ``state"  represents  an impossible, thus meaningless,  initial condition for a macroscopic body with  finite body-temperatures.}

Let us note that both in quantum and classical physics,  the initial condition to be used as the input of the dynamical evolution  of the system, is the result of a precedent measurement.  And it is here that   quantum and classical mechanics differ significantly.

We can now paraphrase the discussion made already, for the  impossibility of preparing a macroscopic wave function with  an arbitrary extension,    in Sec.~\ref{impossible},     to see  that it is not possible to  prepare  a split wave packet of the form, (\ref{nopure}).

Let us recall, for comparison,  that  a  beam  of {\it microscopic} particles such as the electron or an atom,  can be  sent to a  double slit, to yield a wave function  of the form, (\ref{SGatom}), after the passage.  Famous experiments
such as  \cite{Tonomura}  showing beautiful interference fringes \`a la Young,  demonstrate clearly  the pure-state nature of  $|\psi\rangle$.
 Or if the particle carries a spin, a wave packet can be  split in  two or more  (two if  $s=\tfrac{1}{2}$,   three if  $s=1$, etc.)  components  in  an inhomogeneous magnetic field,  as in the familiar Stern-Gerlach experiment.
There is no problem in preparing experimentally  an initial pure  state having the form (\ref{SGatom}); it is a perfectly
good  initial condition.

Vice versa,  for a classical particle (a macroscopic body at finite temperatures) there is no way to prepare experimentally  a coherent superposition of the form,
(\ref{nopure}). The fact that both $|\Psi_1\rangle$ and   $|\Psi_2\rangle$ represent  mixed states,  see (\ref{macrobody}), (\ref{macrobody2}),  means that
 one is here dealing with a  particle {\it   either}  near  ${\bf R}={\bf R}_1$  {\it  or } near  ${\bf R}={\bf R}_2$.   Such a state is represented as a
 diagonal density matrix, and not by  a wave function of the form, (\ref{nopure}).

 A double slit  wide enough  for a macroscopic body to pass, will simply leave the particle either behind the one  slit or  the other.
See Fig.~\ref{Double}.  It may pass through one or the other if the experiment is repeated \footnote{Obviously, here the original transverse  momentum  uncertainty is assumed to be such that one cannot predict
which path the particle takes.
}.  It would be a statistical mixture, not a pure state.

The Stern-Gerlach  setting would  not help  either, to ``prepare"  a  doubly (or multiply) split wave packet for a macroscopic body.   As such an object has a well-defined  position and momentum for its center
of mass  (see Sec.~\ref{Heisenberg} and Sec.~\ref{Diffusion})  always,    it would simply follow a classical  trajectory  in a magnetic field,  in a definite, generic direction (and not  up {\it  or}  down),
 depending on the orientation of its magnetic moment and on the gradient of the  magnetic field. This is what we know from classical physics, what is known as an empirical fact, and what indeed follows from quantum mechanics: see   (\ref{NoSG})
below.

 \begin{figure}
\begin{center}
\includegraphics[width=5in]{Doubleslit}
\caption{\small  The passage of a macroscopic particle of size $L_0$,   through a  double slit, each hole with width  $D> L_0$.  The passage
will leave the particle behind one of the slits, each time.}
\label{Double}
\end{center}
\end{figure}

 \subsubsection{Settling the Born - Einstein dispute}

This discussion on the impossibility of a split wave packet for a macroscopic body,    concerns  radiating, mixed-state, thus classical,  body, at finite temperatures.    As recalled  in the Introduction,  however,
at temperatures sufficiently close to the absolute zero, any matter,  microscopic or macroscopic,  is quantum mechanical.  A macroscopic
 coherent superposition (\ref{nopure})   (a pure state of this form)  is certainly possible, for such systems.
  The pair of the SQUID states with macroscopic fluxes of opposite signs, with possible tunnelling between them \cite{Leggett},  are  a perfect analogue  of the split wave packet
   (\ref{nopure}).    Apart from technical issues there should not be any problem in constructing explicitly a doubly split wave packet by using e.g.,   the macroscopic wave functions
   of the Bose-Einstein condensed gas of some atoms.

 We  conclude that Born and Einstein were both partly right: at finite temperatures   a macroscopic body is a mixed state and the absence of the diffusion
 (due to their large mass)  is indeed the basic reason for it to possess a classical trajectory;   at the same time,  coherent superposition of macroscopically distinct states is simply  not possible.   On the other hand,  at temperatures sufficiently close to absolute zero,  any system is quantum mechanical, and a macroscopic,  a split-wave-packet state (\ref{nopure})  is  certainly possible.  Such a state might look counter-intuitive from our daily experiences, nonetheless is nothing special in quantum mechanics.   The only problem is that, {\it   for a macroscopic body},  it requires extraordinarily  low temperatures, and  a highly sophisticated experimental work to realize one  (such as those  in \cite{Leggett} -\cite{Brand};    Sec.~\ref{QCL} for a further discussion).

\section{The Ehrenfest theorem and Newton's equation  \label{Ehrenfest}}

Once the concept of a dispersionless center-of-mass wave packet of a macroscopic body is  established, with the initial condition $({\mathbf R}_{CM},
{\mathbf P}_{CM})$  as a well-defined and experimentally realizable input   (Sec.~\ref{Heisenberg}  - Sec.~\ref{Decoherence}),
 the derivation of Newton's equation  itself is  a straightforward (in principle)  application of Ehrenfest's theorem.

The Ehrenfest theorem for a  microscopic particle  (electron, atom, molecule)  reads
\begin{equation}   \frac{d}{dt}  \langle \psi   |  {\bf r}  |\psi \rangle  =   \frac{1} {i\hbar}   \langle \psi   |  [{\bf r}, H]  |\psi \rangle =   \langle \psi  | \frac{\bf p}{m}  |\psi \rangle  \;; \label{Ehr1}
\end{equation}
\begin{equation}   \frac{d}{dt}  \langle \psi   |  {\bf p}  |\psi \rangle  =   \frac{1} {i\hbar}   \langle \psi   |  [{\bf p}, H]  |\psi \rangle =   - \langle \psi  |   \nabla V |\psi \rangle  \;.  \label{Ehr2}
\end{equation}
If   $ |\psi \rangle $ represents a wave packet of more or less well-defined position and momentum,
this looks almost like  Newton's equation.  As is well known, a straightforward interpretation of  the wave packet as something representing a  sort of matter distribution inside a classical particle,  and the conclusion that  (\ref{Ehr1}) and (\ref{Ehr2})
constitute the derivation of Newton's equation in quantum mechanics,
 however,
are not tenable.     For instance, the wave packet of an electron hitting a  potential barrier easily breaks into two, one transmitted
and the other reflected;  on the other hand, the electron itself never breaks up.   A classical particle does not
 break either:   it gets either transmitted or reflected, depending on its energy and the barrier height.

This  kind of consideration forces  upon us  the
standard interpretation of the wave function as an amplitude for relative frequencies  \cite{KK}   (probability amplitude),   not
as something representing a real matter density.

\subsection{Ehrenfest's  theorem for a macroscopic body   \label{EhrMac}}

Let us now generalize (\ref{Ehr1}) and  (\ref{Ehr2})
to  a    multiparticle bound state described by the wave function,
\begin{equation}  \psi({\bf r})   \to    \Psi({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N)\;,
\end{equation}
where $N$ is some large, macroscopic number, typically  $N\sim 10^{23}$ for a  piece of material of  $\sim1$g,    $N \sim 10^{51}$ for the earth.
The CM position, momentum  and the masses are given by
\begin{equation}     {\mathbf R} =  \frac{1}{M}    \sum_{i=1}^N   m_i   {\mathbf r}_i\;;\qquad    {\mathbf P}= \sum_{i=1}^N  {\bf p}_i\;, \qquad   M=  \sum_{i=1}^N  m_i\;,\label{CMposition}
\end{equation}
The Hamiltonian is
\begin{equation}    H \to  H=      \sum_i    \frac{{\bf p}_i^2}{2m_i}  +  V ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N)\;.  \label{macro}
\end{equation}

All nuclei, atoms, molecules, crystals and condensed matter  states   are described by the Hamiltonian of this form.    We assume that also a general macroscopic body under consideration (a metal piece, a dust particle, a stone,  a rock, planets and stars, galaxies)   is also  described by
the Hamiltonian,   (\ref{macro}).
    For a neutral atom  (Comlombic bound states, with a definite size of the order of Bohr's radius,    $r_B \sim 10^{-8} cm$),  all interactions are internal interactions, the Coulomb interactions between the electrons and the nucleus, and pairwise  interactions among the  electrons.

      For simplicity,   we neglect writing explicitly  relativistic corrections and   spin-dependent terms in   $H$.

In the study of the atomic spectra  (the internal excitation levels),   one usually neglects the  dependence on the center of mass ${\bf R}$ in the potential, and the atom as a whole is assumed to move as a free particle.    A similar comment can be made about the nuclear and molecular spectra: the main interest is in the internal excitation  modes and energy spectra.

Here,    we are  interested in the motion of the center of mass,  and  the forces the composite body  feels as a whole.

The first  Ehrenfest's theorem  (\ref{Ehr1}) easily  generalizes to the multiparticle composite body.
As
\begin{equation}     \frac{d}{dt}  \langle \Psi   |  {\bf R}  |\Psi \rangle  =  \sum_i  \frac{d}{dt}  \langle \Psi   |  \frac{ m_i {\bf r}_i}{M}    |\Psi \rangle  =      \frac{1} {i\hbar}  \sum_i     \langle \Psi   |    \frac{ m_i}{M}      [{\bf r}_i, H]  |\Psi \rangle =      \langle \Psi   |    \frac{1}{M}  \sum_i   {\bf p}_i |\Psi \rangle \;,
\end{equation}
one finds
\begin{equation}     \frac{d}{dt}  \langle \Psi   |  {\bf R}  |\Psi \rangle  =  \langle \Psi  | \frac{\bf P}{M}  |\Psi \rangle  \;; \qquad  {\bf P}= \sum_i  {\bf p}_i \;.\label{New1}
\end{equation}
If we define  the classical CM  position and momentum of the body as
\begin{equation}      {\bf R}_{CM}(t)  \equiv     \langle \Psi   |  {\bf R}  |\Psi \rangle \;, \qquad   {\bf P}_{CM}(t)   \equiv     \langle \Psi   |  {\bf P}  |\Psi \rangle \;.      \label{PosMom}
\end{equation}
then (\ref{New1})  reads
\begin{equation}      \frac{d}{dt}   {\bf R}_{CM}(t)   =    \frac {{\bf P}_{CM}(t)}{M} \;,       \label{New11}
\end{equation}
which  is the familiar relation between the position and momentum for a classical point particle.

To consider  the  second Ehrenfest equation  (\ref{Ehr2}), let us write the potential as
\begin{equation}    V (  {\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) =     V (  {\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1},   {\bf R})\;\end{equation}
by introducing the relative coordinates ${\tilde  {\bf r}}_i$,
\begin{equation}    {\bf r}_1 = {\bf R} +  {\tilde {\bf r}}_1\;; \quad  {\bf r}_2 = {\bf R} +  {\tilde {\bf r}}_2\;; \quad \ldots,   \quad    {\bf r}_N = {\bf R} +  {\tilde {\bf r}}_N\;;    \label{invert}
\end{equation}
\begin{equation}     \sum_i  m_i   {\tilde {\bf r}}_i = 0\;. \label{relative2}
\end{equation}
Any $N-1$ of   $ {\tilde {\bf r}}_i $'s can be taken to be linearly  independent, e.g.,    $ {\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}. $

We need to compute
\begin{equation}     [ {\bf P}, H]  =\sum_i  [ {\bf p}_i, H]  =   -i \hbar \sum_i    \nabla_i
\end{equation}
Inverting  (\ref{invert}),  one has
\begin{equation}         {\bf R} =  \frac{1}{M}    \sum_i    m_i   {\bf r}_i\;;\quad    {\tilde {\bf r}}_1 = {\bf r}_1   - {\bf R}   \;; \quad \ldots   \quad   {\tilde {\bf r}}_{N-1} = {\bf r}_{N-1}  - {\bf R} \;.
\end{equation}
and a simple calculation shows that

\begin{equation}       \sum_{i=1}^N    \nabla_i=    \nabla_{\bf R}\;.
\end{equation}
as is well known from classical mechanics.      (\ref{Ehr2})  then  also simplifies:
\begin{equation}     \frac{d}{dt}  \langle \Psi   |  {\bf P}  |\Psi \rangle  = \frac{1}{i \hbar}   \sum_i  \langle \Psi   |  [{\bf p_i}, H]  |\Psi \rangle =
 -  \langle \Psi   |  \sum_i  \nabla_i   V     |\Psi \rangle  =  -  \langle \Psi   |  \nabla_{\bf R}    V     |\Psi \rangle  \;. \label{New2}
\end{equation}
(\ref{New11}) and (\ref{New2})  taken together look almost like Newton's equation.
As the left hand side of  equation (\ref{New2})    is
\begin{equation}    \frac{d}{dt}   {\bf P}_{CM}(t) \;;
\end{equation}
one would have to  show
\begin{equation}    \langle \Psi   |  \nabla_{\bf R}    V     |\Psi \rangle   =     \nabla_{ {\bf R}_{CM}} V_{class}( {\bf R}_{CM})\;,     \label{RHS}
\end{equation}
with an appropriately defined classical potential   $V_{class}( {\bf R}_{CM})$,
 in order to derive Newton's equation for a  pointlike particle of mass $M$ at   ${ {\bf R}_{CM}}$.
But  such a relation cannot be taken for granted, actually, as will be seen below.

A natural definition for  $V_{class}( {\bf R}_{CM})$  is
\begin{equation}   V_{class}( {\bf R}_{CM})  =     \langle \Psi  |  V (  {\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1},   {\bf R})   |  \Psi \rangle\;.   \label{CMV}
\end{equation}
Now  the wave function for the center of mass of a macroscopic body can be taken as an arbitrarily  narrow wave packet (Sec.~\ref{basic}).   The width of the  CM wave function packet  indeed corresponds to an  experimental measurement (preparation) of the  CM position
made with an ``arbitrary" precision, see   (\ref{Definite}),   with  $\Delta X_{CM}$  certainly much smaller than  body's size, $L_0$:
\begin{equation}   \Delta  X_{CM}  \ll   L_0\;.    \label{evenif}
\end{equation}

Actually,   the finite size of the body  $L_0$  affects    significantly  the  calculation of the  right hand side of the second Ehrenfest theorem,  (\ref{New2}),
except in few special cases  which will be discussed    below,  in Sec.~\ref{special1}
and Sec.~\ref{special2}. In more  general cases, such as  those discussed in Sec.~\ref{general1},     the corrections  from the  Newton equation for a pointlike mass $M$,   (\ref{RHS}),
arise  from the study of Ehrenfest's equations.    These deviations  reproduce  the effects well known in classical physics for macroscopic bodies,  such as the gravitational tidal forces.
The related effects, for microscopic systems,   are well known and  discussed in many quantum mechanics textbooks  for microscopic systems (e.g., deriving the polarization effects for atoms and molecules).
Our discussion below  in  Sec.~\ref{general1} adds absolutely nothing new, in principle,    from this point of view
 \footnote{There is however some novelty from the formal point of view.  Usually,  perturbation theory  in quantum mechanics is used  to compute either the shift in the energy eigenvalues (time-independent perturbation theory), or  the transition amplitudes (time-dependent perturbation theory),  caused by a perturbative potential,  $V^{\prime}$. Here we apply perturbation theory to
the Ehrenfest theorem, to study  how the expression for the ``forces"  gets modified with respect to the classical expectation for a pointlike particle.
This is perhaps not very familiar:  we discuss it in Appendix  A.

  }.   It is  nonetheless satisfactory
  that our efforts to identify Newton's equations for a macroscopic body  in quantum mechanics    (Sec.~\ref{Heisenberg} - Sec.~\ref{Decoherence}  and Sec.~\ref{Ehrenfest})      naturally leads to appropriate corrections due to the finite-size of macroscopic bodies, which are also well known in   classical physics.

\subsubsection{Mixed-state nature of a macroscopic body and Ehrenfest's theorem}

Before embarking on the actual application of the Ehrenfest theorem for  macroscopic bodies in some concrete cases, let us make a brief pause, to discuss an apparently minor, but actually a fundamental,  question

One of the essential ingredients for the emergence of the classical physics is decoherence  \cite{JoosBis}-\cite{Arndt1},  discussed
in Sec.~\ref{Decoherence} in our context.    It means that the macroscopic body under consideration has $T>0$ and is  radiating: it is a mixed state, which looks like  (\ref{macrobody}), with many infrared photons emanating from it.   How can a  na\"ive treatment such as  in Sec.~\ref{EhrMac}
in which the state of the macroscopic body is represented by  a pure-state wave function of $N$  atoms and molecules,  justified?

There are at least two good  reasons why such an approach is indeed correct.
The first  is the fact that  the angular distribution of the emitted photons can be considered random statistically,   and therefore regarded as   spherically symmetric. The average recoil from the  different emitted photons on the   CM  motion of the macroscopic body will cancel out, to become  negligible.

     But perhaps more significantly,
the effects of recoil are  negligibly small,   for an emission from excited atoms and molecules {\it  bound  in  a solid  (crystal)},
due to the fact that   the recoil kinematics involve whole body's  large mass, $M$.  This phenomenon is well known: it  is the essence of the famous M\"ossbauer effect.

Because of  (at least) these two properties of the infrared emissions from a macroscopic body,  the actual motion of the CM of a macroscopic body is not significantly affected by them.
The CM motion of the body described by each term of (\ref{macrobody})  is identical.

Keeping this understanding in mind,   we proceed below  to derive Newton's equations  by using  the Ehrenfest theorem for a macroscopic body, treating it as if it were a pure state described by a wave function.

{\noindent \bf  Remark:}    A moment of reflection shows that this apparently minor observation is actually at the heart of the whole discussion of the present work,   on  how the classical physics emerges from quantum mechanics.  What is not identical and, actually, intractably different in various terms in  (\ref{macrobody})   is the {\it  phases}  and  the  internal wave functions describing {\it  the quantum states of the  microscopic particles composing the body and of  the infrared photons flying away}. This is what makes  a  macroscopic body under consideration a mixed state.  The interference terms among the  different terms in
 (\ref{macrobody}),  among those in  (\ref{macrobody2}),   and {\it a fortiori},    between   $|\Psi_1\rangle$ and $|\Psi_2\rangle$, are absent.   No coherent  quantum superposition of  macroscopically distinct states is possible at finite temperatures.

\subsection{Gravitational potential on the  earth  \label{special1} }

As the first concrete example  let us  consider the case of external gravitational force on a massive body on the earth. The total potential is
\begin{equation}     V ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) =   V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) + V_{grav}( {\bf R})\;\label{form}
\end{equation}
where the first term is independent of  the CM position ${\bf R}$.   The Hamiltonian is the sum
\begin{equation}  H= H_{CM} +   H_{int}\;,
\label{TotHam}  \end{equation}
\begin{equation}       H_{CM}=  \frac{{\bf P}^2}{2M} +    V_{grav}( {\bf R})\;;\qquad     H_{int}= \sum_i    \frac{{\tilde p}_i^2}{2\mu_i} +   V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;,
\label{Hamiltonians}  \end{equation}
where
\begin{equation}     V_{grav}( {\bf R})=g    \sum_i   m_i z_i  =      M  g   Z\;,  \label{gravPot}
\end{equation}
where $g\simeq 980 \, {\rm cm}/{\rm sec}^2$ is the acceleration  on the surface of the earth.

The  form of the Hamiltonian  (\ref{form})   allows us  to take the wave function in the factorized form,
\begin{equation}      \Psi =    \Psi_{CM}({\bf R}) \psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;.
\end{equation}
The time independent Schr\"odinger equation
\begin{equation}   H \Psi = (  H_{tot}  +   H_{int} ) \Psi   =  E_{tot} \,  \Psi   \;,
\end{equation}
gives
\begin{equation}     H_{int}\,  \psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})  =  E_{int}  \,\psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;;
\end{equation}
\begin{equation}  H_{CM}  \Psi_{CM}({\bf R})  =  E_{CM}  \Psi_{CM}({\bf R}) \;,
\end{equation}
and
\begin{equation}     E_{tot}=    E_{CM}   +   E_{int}\;,
\end{equation}
where   $H_{int}$,   $\psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})$ and  $ E_{int}$ refer to the internal excitation modes
of the body under consideration
\footnote{In the case of certain  rigid macroscopic bodies (e.g., crystals),     $H_{int}$ and $E_{int}$  contain the  total angular momentum contribution of the rotational modes of the massive body as a whole (and the related energy), which  should  not be considered as the internal excitation modes.    In  the present work, however,  we will not separate these three degrees of freedom  from the internal excitations.} .

 (\ref{New2})  can be replaced  by
\begin{equation}     \frac{d}{dt}  \langle \Psi_{CM}  |  {\bf P}  |\Psi_{CM} \rangle  =    -      \langle \Psi_{CM}  |  \nabla_{\bf R}   V_{grav}( {\bf R})     |\Psi_{CM} \rangle  \;. \label{New222}
\end{equation}
and (\ref{PosMom}) and (\ref{CMV})  by
\begin{equation}      {\bf P}_{CM}(t)   \equiv     \langle \Psi_{CM} |  {\bf P}  |\Psi_{CM}  \rangle\;;  \qquad    {\bf R}_{CM}(t)  \equiv     \langle \Psi_{CM}    |  {\bf R}  |\Psi_{CM}  \rangle \;,     \label{CMR2}
\end{equation}
and
\begin{equation}   V_{class}( {\bf R}_{CM})  =   \langle \Psi_{CM}  | V_{grav} ({\bf R})   |  \Psi_{CM} \rangle\;, \label{CMV2}
\end{equation}
where   $\Psi_{CM}$  is an arbitrarily narrow  (see  (\ref{Definite})) wave packet describing the CM wave function,  centered at
${\bf R}_{CM}(t)$ and  ${\bf P}_{CM}(t)$.

For such a macroscopic body to obey Newton's law,
\begin{equation}      \frac{d}{dt}   {\bf P}_{CM}(t)   =
 -        \nabla_{{\bf R}_{CM}}   V_{class}( {\bf R}_{CM})
 \;.  \label{New22222}
\end{equation}
we need an approximate equality,
\begin{equation}  \langle \Psi   |  \nabla_{\bf R}   V_{grav}({\bf R})     |\Psi \rangle \simeq         \nabla_{{\bf R}_{CM}}   V_{class}( {\bf R}_{CM})  \langle \Psi   |  \Psi \rangle     =     \nabla_{{\bf R}_{CM}}   V_{class}( {\bf R}_{CM})  \;.   \label{derivation}
\end{equation}
Actually, for the gravitational potential on the earth,   (\ref{gravPot}),  one has
\begin{equation}  -   \nabla_{\bf R}   V_{grav}({\bf R})    =     -   \nabla_{\bf R}    M  g  Z =    -   M   g \,   {\bf  \hat   k} \;,
\end{equation}
a constant    (where   $ {\bf  \hat   k} $ is the unit vector in the   positive $z$  direction).  Also,
\begin{equation}    V_{class}( {\bf R}_{CM}) =  V_{grav}( {\bf R}_{CM})= M g  Z_{CM}\;,
\end{equation}
where the dispersion of  $ {\bf R}_{CM}$  has been neglected.
The derivation of Newton's equation from quantum mechanics,   (\ref{New22222}),     is therefore  exact.
No details about the internal structures  of the body  ($H_{int} $  and  $\psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})$) are needed to get this result.

\subsection{Motion in a gravitational potential due to a distant mass \label{general1} }

Let us now consider the gravitational force  on a body with mass $M= \sum_i  m_i$, exerted by  a distant mass $M_0$ at ${\mathbf R}_0$.
The gravitation potential is   in this case
\begin{equation}   V_{grav} ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) =    \sum_i    \frac{ G_N    \, m_i   M_0}{|{\bf r}_i  -   {\bf R}_0|}\;,
\end{equation}
where $G_N$ is Newton's  gravitational constant,
\begin{equation}  G_N \simeq  6.674   \times  10^{-11}   \,  {\rm m}^3  {\rm Kg}^{-1}   {\rm s}^{-2}\;.
\end{equation}
We assume that the gravitational force is weak as compared to the  internal  (electromagnetic) forces which bind the system into the macroscopic body. In other words,   in the total Hamiltonian,
\begin{equation}    H =    H_{CM}   +   H_{int}    +    V_{grav} ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) \;,
\end{equation}
\begin{equation}   H_{CM} = \frac{{\bf P}^2}{2M}\;,\qquad    H_{int}= \sum_i    \frac{{\tilde p}_i^2}{2\mu_i} +   V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;,\end{equation}
the gravitational potential  $V_{grav}$  can be regarded as a perturbation.  That means that   the   macroscopic body under consideration is a bound state described by  the time-independent Schr\"odinger equation,
\begin{equation}     H_{int}\,  \psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})  =  E_{int}  \,\psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;.   \label{unpert1}
\end{equation}
{\it Before}   taking into account the gravitational potential,
 the wave function thus has a factorized form,
\begin{equation}      \Psi =    \Psi_{CM}({\bf R}) \psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;. \label{unpert2}
\end{equation}
In (\ref{unpert1}) and (\ref{unpert2}) a suffix $(0)$    (``unperturbed")   is used to indicate that the wave function   $ \psi^{(0)}$  stands for the macroscopic body before
the gravitational potential is taken into account.

We assume  also  that  the distance from the body to the mass $M_0$  is much larger than the  linear  size of the body (the spatial extension of  $ \psi^{(0)}$), $L_0$.   We may then assume that
\begin{equation}      \forall i    \;,        \quad        | {\tilde   {\mathbf r}}_i  |  =     |{\mathbf r}_i -  {\mathbf R}| \sim    L_0 \;,    \qquad    \frac{L_0 }{| {\mathbf R} -   {\mathbf R}_0 |}       \ll   1   \;.  \label{approx}
\end{equation}
Actually, some care is needed  in such an approximation.   The reason is that   ${\mathbf r}_i$'s and   $ {\mathbf R}$  in  (\ref{approx})  are position {\it  operators},  and as such, have no \`a priori definite values.  However,  we remember that  these relations   are to be   used between the bra and ket  of the  state  vector of the  macoscopic body, $\Psi$.  As  for  $ {\tilde {\bf r}}_i$'s,       their extensions are effectively limited by the support of  the
 internal wave  function $ \psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}),$   hence by the size $L_0$ of the body.   In the case of   the center of mass position  ${\mathbf R}$  it is a bit  subtler,  as $  \Psi_{CM}({\bf R}) $     satisfies a free Schr\"odinger equation.  The consideration  in Sec.~\ref{Heisenberg} and Sec.~\ref{Diffusion},  however allows us to use for the CM wave function $  \Psi_{CM}({\bf R}) $ a well-defined, narrow wave packet, with an extension much smaller than  $L_0$  (see (\ref{evenif})).
The  approximation (\ref{approx})  is therefore justified.

The gravitational potential can now be multipole expanded in powers of   $1/ {|{\mathbf R}  -   {\bf R}_0|}$:
\begin{eqnarray}    && V_{grav}  ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) =    \sum_i    \frac{ G    \, m_i   M_0}{| {\tilde  {\bf r}}_i  +   {\mathbf R}   -   {\bf R}_0|}
\nonumber  \\
&&   =       \frac{ G  M   M_0}{|{\mathbf R}  -   {\bf R}_0|}    -    G \sum_i  \frac{  m_i   {\tilde  {\bf r}}_i^2}{ |  {\mathbf R}   -   {\mathbf R}_0|^3}
+   3 G \sum_i   \frac{  m_i   \left(  \tilde {\mathbf r}_i  \cdot ({\mathbf R} -  {\mathbf R}_0 )  \right)^2  }   { | {\mathbf R}   -   {\mathbf R}_0 |^5 }
+ \ldots     \label{quadr}
\end{eqnarray}
The first term corresponds to the gravitational potential for the pointlike mass $M$ at ${\mathbf R}$.

Note that the first, order $O((\frac{1}{|{\mathbf R}  -   {\bf R}_0|})^2 )$   correction term,   the ``dipole term",
\begin{equation}  -   \sum_i   \frac{ G \, m_i  {\tilde {\mathbf r}_i}  \cdot ( {\mathbf R} -  {\mathbf R}_0)     } { |  {\mathbf R}   -   {\bf R}_0|^3}
\end{equation}
is actually  absent in (\ref{quadr}),   as is well known for the gravity,   as    $\sum_i   m_i  {\tilde {\mathbf r}_i} =0$.

 The second and third term of  (\ref{quadr})   represent   the quadrupole  corrections,  the magnitude of which depends on the  internal  wave function, $\psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})$, i.e., the non-spherically-symmetric form of the body.

An important point  to note however   is that  the quadrupole (and higher) corrections will appear, even if the unperturbed wave function $\psi^{(0)}$
(the body without the influence of gravity), is perfectly spherically symmetric.  In such a  case,  the contribution of the  second and the third term of
(\ref{quadr})   on the right hand side of the second  Ehrenfest's theorem,
\begin{equation}      \frac{d {\bf P}_{CM} }{dt}=
 -  \langle  \Psi   |  \sum_i  \nabla_i   V     |\Psi \rangle
\end{equation}
might  appear  to   vanish,    if the factorized wave function (\ref{unpert2}) is used, with a  spherically-symmetric  internal wave function  $ \psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})  $.

Actually,   a nonvanishing quardupole  appears as the result of the
perturbation  to the wave function: the deformation of the body due to the external gravitational force.   In order to see these effects,   it is necessary to study the consequence of the perturbation due to gravity systematically in the context of the Ehrenfest theorem.  In Appendix~\ref{PertEh}   a standard perturbation theory  is  applied to Ehrenfest's theorem,
to  derive  modifications to the force systematically, order by order  in perturbation.

In order to apply perturbation theory to Ehrenfest's theorem,  as sketched in   Appendix~\ref{PertEh},  it is  however
more convenient to split the Hamiltonian  differently.  Namely, we write
\begin{equation}       V_{grav}  ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N)=    \frac{ G  M   M_0}{|{\mathbf R}  -   {\bf R}_0|}   +   V_{mp}({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N)\;,  \label{multipole1}
\end{equation}
and consider the multipole expansion terms
\begin{equation}    V_{mp}({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N)  =    -    G \sum_i  \frac{  m_i   {\tilde  {\bf r}}_i^2}{ |  {\mathbf R}   -   {\mathbf R}_0|^3}
+   3 G \sum_i   \frac{  m_i   \left(  \tilde {\mathbf r}_i  \cdot ({\mathbf R} -  {\mathbf R}_0 )  \right)^2  }   { | {\mathbf R}   -   {\mathbf R}_0 |^5 }
+ \ldots     \label{multipole2}
\end{equation}
as perturbation.   The total Hamiltonian is written as
\begin{equation}    H =  H_0 +    V_{mp} ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) \;,\qquad H_0=  {\tilde H}_{CM}    +   H_{int} \;,
\end{equation}
\begin{equation}   {\tilde H}_{CM}  = \frac{{\bf P}^2}{2M}  +     \frac{ G  M   M_0}{|{\mathbf R}  -   {\bf R}_0|}     \;,\qquad    H_{int}= \sum_i    \frac{{\tilde p}_i^2}{2\mu_i} +   V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;,\end{equation}
and now  only the multipole expansion terms $ V_{mp} $  are treated as perturbation.

Unperturbed system  $H_0$    is still described by a factorized wave function,
\begin{equation}      \Psi =   {\tilde  \Psi}_{CM}({\bf R}) \psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;. \label{unpert2}
\end{equation}
where  the CM wave function satisfies now the Schr\"odinger equation for  a pointlike mass $M$
\begin{equation}       {\tilde H}_{CM}      {\tilde  \Psi}_{CM}({\bf R})    =     E_{CM}  \,  {\tilde  \Psi}_{CM}({\bf R})    \;   \label{NewtonCM}
\end{equation}
  (which is Newton's law,   see below, (\ref{zeroth})),
whereas  the internal wave function is given  by (see (\ref{unpert1})),
\begin{equation}     H_{int}\,  \psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})  =  E_{int}  \,\psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;,\label{unpert1111}
\end{equation}
 which describes  the bound state without the effects of gravity.

The first Ehrenfest theorem takes the classic form,
\begin{equation}     \frac{d}{dt}  \langle \Psi   |  {\bf R}  |\Psi \rangle  =  \langle \Psi  | \frac {\bf P}{M}  |\Psi \rangle  \;; \qquad  {\bf P}= \sum_i  {\bf p}_i \;,\label{New111}
\end{equation}
that is,
\begin{equation}      \frac{d}{dt}   {\bf R}_{CM} =      \frac { {\bf P}_{CM}}  {M} \;, \qquad     {\bf R}_{CM} \equiv   \langle \Psi   |  {\bf R}  |\Psi \rangle  \;; \quad   {\bf P}_{CM} \equiv   \langle \Psi   |  {\bf P}  |\Psi \rangle  \;;  \label{classic}
\end{equation}
to all orders of perturbation.

The left hand side of the second Ehrenfest's theorem also contains
$  \frac{d}{dt}   {\bf P}_{CM} $ to all orders of perturbation, see  Appendix~\ref{PertEh}.  The  right hand side, which can thus  be interpreted as the force,   has an expansion in powers of  perturbation.

Let us assume,  for definiteness,    that, before taking into account    $V_{mp}$,  our macroscopic body is described by a spherically symmetric internal wave function, $\psi^{(0)}$.

We transcribe the formulas  (\ref{Ehr222}) $\sim$ (\ref{force02})  in  Appendix~\ref{PertEh}  to our three-dimensional macroscopic body with $N$-atoms (or molecules),
\begin{equation}   F \to{\bf F}\;,  \qquad     \frac{d}{dx} \to    \nabla_{{\bf R}} \;, \qquad   V_0 \to    \frac{ G  M   M_0}{|{\mathbf R}  -   {\bf R}_0|}    +    V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;;      \label{context1}
\end{equation}
\begin{equation}       H^{(0)} \to     {\tilde H}_{CM}    +   H_{int}   \;,      \qquad   V^{\prime}  \to       V_{mp} ({\bf r}_1,  {\bf r}_2, \ldots  {\bf r}_N) \;,   \label{context2}
\end{equation}
and
\begin{equation}   \psi^{(0)} \to   \Psi =   {\tilde  \Psi}_{CM}({\bf R}) \psi^{(0)}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;.      \label{context3}
\end{equation}
To the zeroth order in  $V_{mp}$,   (\ref{force00}) says that
\begin{equation}   \frac{d}{dt}   {\bf P}_{CM}  =   {\bf F}^{(0)}   = -       \bra  {\tilde  \Psi}_{CM}| \nabla   \frac{ G  M   M_0}{|{\mathbf R}  -   {\bf R}_0|}  |  {\tilde  \Psi}_{CM}  \rangle
\simeq    -   \nabla_{{\bf R}_{CM}}     \frac { G  M   M_0}  {  |{\mathbf R}_{CM}  -   {\bf R}_0|}   \;, \label{zeroth}
\end{equation}
where we have taken into account the fact that for a macroscopic body the CM  wave function can be taken as a wave packet, as narrow as we wish.
The equation (\ref{zeroth}), together with (\ref{classic}),   is just the Newton equation for a pointlike mass of $M$ at  ${\mathbf R}_{CM}$.

A quick inspection of   (\ref{force01})  in our context   (\ref{context1})-(\ref{context3})  shows   that the  effect of    $V_{mp}$   on the motion of the macroscopic body vanishes
to first order,
$ {\bf F}^{(1)}   =0$,
because of the angular momentum conservation (or by use of the Wigner-Eckart theorem).

The first nonvanishing effects of $V_{mp}$  on the motion of the body
 arise at the second order,  $ {\bf F}^{(2)}$,       see  (\ref{force02}).    The operator  $  \frac{dV_0 }{dx} $  with the translation
  (\ref{context1})   is  a zero-rank spherical tensor with respect to the internal degrees of freedom    ${\tilde {\bf r}}_i$.  On the other hand,    the  first-order correction to the wave function
 $|\psi^{(1)} \rangle$,   (\ref{wfcorr1}),   contains states of angular momentum $2$ or higher   (see (\ref{multipole2})).     $|\psi^{(1)} \rangle$  would be exactly  $\ell=2$ state if only the quadrupole term (explicit in   (\ref{multipole2}))    were kept,  but in general   contains all  higher angular momenta  $\ell=2,4,6,\ldots$.
 The second-order wave function correction    $|\psi^{(2)} \rangle$   can be seen from  (\ref{wfcorr2}) to contain angular momentum states    $\ell=0, 2,4,6,\ldots$.

Physically, the nonvanishing  effect of quadrupole (and higher) moments  on the  motion of the CM can be  interpreted as due to the first order  deformation of the  body  (first order
correction to the internal wave function), and the resulting nonvanishing elements  in the second-order correction to the force,   (\ref{force02}),   due to the now non-spherically-symmetric  distribution of the atoms and molecules inside the body.

 \subsubsection{Moon and the ocean tides on the earth}

 A familiar phenomenon related to these considerations  is the ocean tides  on  the earth, due to the gravitational force of the moon.
 A caricature situation of  the earth and moon, to the zeroth-order  in  $V_{mp}$,  is shown  in Fig.~\ref{EarthMoon1}.  The earth, including the ocean, is assumed to have  a spherically symmetric form, before  $V_{mp}$ is taken into account.
 The earth and moon  move according to Newton's equations between the pointlike masses  $M_{earth}$ and $M_{moon}$,  see (\ref{zeroth}), (\ref{classic}).
 When the multipole perturbation $V_{mp}$ is taken into account,  the  earth gets deformed.   Note that the deformation of the  earth's  wave function  is mainly  due to
 the shifts of the water molecules  in the ocean:    their bindings are  much weaker than the binding of the molecules and atoms in the solid earth crust.  The earth
 and moon now  look like  Fig.~\ref{EarthMoon2}.  This gives rise to the daily high and low ocean tides  (roughly twice each)  on the earth \footnote{According to Feynman \cite{Feynman} it was Newton who has first correctly identified  the cause of the daily ocean tides. }.

 Though done with a crude,  simplifying assumptions,   this discussion may be regarded   as the first  derivation of a familiar daily experience (ocean tides),
 by a direct use of  the Schr\"odinger equation  for a  macroscopic body (the earth)   described by  a wave function with  $N\sim 10^{51}$  atoms !

\begin{figure}[h]
\begin{center}
\includegraphics[width=5in]{EarthMoon1}
\caption{\small  The earth and moon, in the approximation in which  the multipole expansion terms  $V_{mp}$ of  (\ref{multipole1}),  (\ref{multipole2})  are neglected.   The earth is assumed to be
perfectly spherically symmetric, including its ocean.  }
\label{EarthMoon1}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\includegraphics[width=5in]{EarthMoon2}
\caption{\small  The earth's wave function is now deformed by the perturbation, $V_{mp}$.  The main effect is in the change of distribution of the water molecules which are more loosely bound than the atoms and molecules in the earth's crust.  }
\label{EarthMoon2}
\end{center}
\end{figure}

\subsection{Harmonic oscillator potential     \label{special2} }

Next consider the case of a harmonic potential acting  uniformly on all the components of the body. The potential is given by
\begin{equation}     V = \sum_{i=1}^N       \frac{   m_i \omega^2     {\mathbf r}_i^2 }{2}   \;.
\end{equation}
In this case the Hamiltonian does not simply factorize as a sum of the CM  part and the internal Hamiltonian,  as  in   (\ref{Hamiltonians}). Nevertheless, Ehrenfest's theorem simplifies again.

The first Ehrenfest theorem reads,
\begin{equation}     \frac{d}{dt}  \langle \Psi   |  {\bf R}  |\Psi \rangle  =  \langle \Psi  | \frac {\bf P}{M}  |\Psi \rangle  \;; \qquad  {\bf P}= \sum_i  {\bf p}_i \;,\label{New111}
\end{equation}
that is,
\begin{equation}      \frac{d}{dt}   {\bf R}_{CM} =      \frac { {\bf P}_{CM}}  {M} \;.
\end{equation}
The second simplifies as well:
\begin{eqnarray}   &&        \frac{d {\bf P}_{CM} }{dt}=        \frac{d}{dt}  \langle \Psi   |  {\bf P}  |\Psi \rangle  =   \frac{1}{i \hbar}   \sum_i  \langle \Psi   |  [{\bf p_i}, H]  |\Psi \rangle =
 -  \langle  \Psi   |  \sum_i  \nabla_i   V     |\Psi \rangle   = \nonumber \\
 &&    -   \langle \Psi   |   \sum_{i=1}^N     m_i \omega^2     {\mathbf r}_i    |\Psi \rangle   =   -  M  \omega^2    \langle \Psi | {\mathbf R}      |\Psi \rangle =     -  M  \omega^2   {\mathbf R}_{CM}    \;. \label{New222}
\end{eqnarray}
The center of mass of the whole body  obeys Newton's equation with the harmonic potential.

Again,  the knowledge about the internal interactions, which bind and determine the wave functions of the body
is not required.

\subsection{External electromagnetic  fields    \label{EM}   }

It is in principle straightforward to generalize these discussions to the case of a macroscopic body  in an external electromagnetic fields,  especially
in the limit the body can be considered pointlike,  and under the uniform and constant and  slowly varying  electromagnetic fields.

For  a macroscopic particle of total charge $Q$
 the Hamiltonian is given by
\begin{equation}    H \to  H=      \sum_i   \left[  \frac{({\bf p}_i - \frac{e_i}{c} {\bf A}({\bf r}_i) )^2}{2m_i}  +  e_i \Phi({\bf r}_i) - {\bf \mu}_i \cdot {\bf B}({\bf r}_i)    \right]   +    V_{int}({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1}) \;, \label{macroEM}
\end{equation}
\begin{equation}  \sum_i e_i = Q\;; \qquad     {\bf B} =   \nabla \times   {\bf A};\qquad      {\bf E}   =   \nabla   \Phi({\bf r})\;,
\end{equation}
and $\mu_i$ is the intrinsic magnetic moment of the $i$-th constituent.
The interactions with the external electromagnetic fields are  included in the expression inside the square brackets;    the last term of (\ref{macroEM})
represents all other interactions among the atoms and molecules describing the bound state.
 $V_{int}$ is assumed to have no dependence on the CM coordinate,
 \begin{equation} \nabla_{\bf R} V_{int} =0\;: \label{noR}
 \end{equation}
  the deviation  from the free motion for the  CM  of the body under study, is
due only to the external electromagnetic fields.
 For simplicity, we have left  the dependence of   $V_{int}$ on  the spin variables and on the momenta  ${\tilde {\bf p}_i}$   (such as in the relativistic corrections and spin-orbit interactions)   implicit.

\subsubsection{Motion in  a constant electric field \label{constE}}

Let us start with the simplest case of a constant electric field,  ${\bf E}$:  we take
\begin{equation}      \Phi({\bf r}) =   {\bf E} \cdot {\bf r}\;, \qquad   {\bf A}=  {\bf B}=0\;.
\end{equation}
The first Ehrenfest equation for the CM position reads simply
\begin{eqnarray}     \frac{d}{dt}  \langle \Psi   |  {\bf R}  |\Psi \rangle  &=&  \sum_i  \frac{d}{dt}  \langle \Psi   |  \frac{ m_i {\bf r}_i}{M}    |\Psi \rangle  =      \frac{1} {i\hbar}  \sum_i     \langle \Psi   |    \frac{ m_i}{M}      [{\bf r}_i, H]  |\Psi \rangle  \nonumber \\
&=&         \langle \Psi   |    \frac{1}{M}  \sum_i     {\bf p}_i   |\Psi \rangle     =    \frac{1}{M}   \langle \Psi   |   {\bf P} | \Psi \rangle    \;.   \label{asin}
\end{eqnarray}
The second equation is
\begin{equation}     \frac{d}{dt}  \langle \Psi   |  {\bf P}  |\Psi \rangle  = \frac{1}{i \hbar}   \sum_i  \langle \Psi   |  [{\bf p_i}, H]  |\Psi \rangle     \;,

\end{equation}
but
\begin{equation}   \frac{1}{i \hbar}   \sum_i  \langle \Psi   |  [{\bf p_i},  \sum_j  e_j \Phi({\bf r}_j)  ]    |\Psi \rangle   = -   \sum_i    \langle \Psi |     e_i  {\bf E}    |\Psi \rangle    = - \, Q\,  {\bf E}\;,
\end{equation}
where $Q= \sum_i e_i$ is the total charge of the body, whereas
\begin{eqnarray}        \frac{1}{i \hbar}   \sum_i  \langle \Psi   |  [{\bf p_i},  V_{int}]   |\Psi \rangle   &=&   -   \sum_i  \langle \Psi   |  [{\nabla_i},  V_{int}]   |\Psi \rangle  = -  \langle \Psi   |  [{\nabla_{\bf R}   },  V_{int}]   |\Psi \rangle   \nonumber  \\
  & = &
      -  \langle \Psi   |  (\nabla_{\bf R} V_{int} ) |\Psi \rangle  =0\;
\end{eqnarray}
(see (\ref{noR})).
Therefore  a narrow wave packet for the CM, centered at ${\bf R}_{CM}, {\bf P}_{CM}$,    obeys
\begin{equation}          \frac{d}{dt}  {\bf R}_{CM} =      \frac{1}{M}   {\bf P}_{CM}\;, \qquad     \frac{d}{dt}  {\bf P}_{CM} =  - \, Q\, {\bf E}\;,
\end{equation}
as expected.   If the electric field is due to a charge $Q_0$ at  ${\bf R}_0$ at a distance  from ${\bf R}_{CM}$  much greater than the body's size,  $L_0$,  one finds Coulomb's law,
\begin{equation}     \frac{d}{dt}  {\bf P}_{CM} =    {\bf F} =     -   \nabla_{{\bf R}_{CM}} \frac{ Q  Q_0}{|{\bf R}_{CM} -  {\bf R}_0|}\;
\end{equation}
in the first approximation  (in which the body can be regarded as  pointlike),   by repeating an argument similar to the one  used to get  (\ref{zeroth}).

 \subsubsection{Motion in a constant magnetic field}

Let us next  consider an external, uniform constant magneti field ${\bf B} $.   The vector potential can be taken as
\begin{equation}     {\bf A}({\bf r})    =   \frac{1}{2}   \, {\bf B} \times  {\bf r}  \;.     \label{vectorpot}
\end{equation}
The first Ehrenfest theorem  now  reads
\begin{equation}     \frac{d}{dt}  \langle \Psi   |  {\bf R}  |\Psi \rangle  =
  \langle \Psi   |    \frac{1}{M}  \sum_i   (  {\bf p}_i  -    \frac{e_i}{c}   {\bf A}({\bf r}_i) )   |\Psi \rangle   =
  \langle \Psi   |    \frac{1}{M}  \sum_i   \left(  {\bf p}_i  -    \frac{e_i}{2 c}  ( {\bf B} \times  {\bf r}_i  )   \right)   |\Psi \rangle
\end{equation}
The expectation value of the terms containing  $ {\bf r}_i $  vanish due to parity,
and one gets
\begin{equation}
   \frac{d}{dt}  \langle \Psi   |  {\bf R}  |\Psi \rangle  =  \frac{1}{M}   \langle \Psi   |   {\bf P} | \Psi \rangle \;;\qquad     \frac{d}{dt}   {\bf R}_{CM}  =     \frac{1}{M}  {\bf P}_{CM} \;.
\end{equation}
The second Ehrenfest equation  is
\begin{eqnarray}     \frac{d}{dt}  \langle \Psi   |  {\bf P}  |\Psi \rangle  &=&    \frac{1}{i \hbar}   \sum_i  \langle \Psi   |  [{\bf p_i}, H]  |\Psi \rangle     \nonumber \\
 &=&
 \frac{1}{i \hbar}   \sum_i  \langle \Psi   |  [{\bf p_i},              \sum_j    \frac{({\bf p}_j - \frac{e_j}{c} {\bf A}({\bf r}_j) )^2}{2m_j}  ]              |\Psi \rangle   \;.\end{eqnarray}
By using (\ref{vectorpot}),  one gets after some calculation,
\begin{equation}   \frac{1}{i \hbar}   \sum_i  [{\bf p_i},              \sum_j    \frac{({\bf p}_j - \frac{e_j}{c} {\bf A}({\bf r}_j) )^2}{2m_j}  ]  =     \sum_i   \frac{e_i}{ m_i c}  ({\bf p}_i -   \frac{e_i}{c} {\bf A}({\bf r}_i) ) \times  {\bf B}\;,
\end{equation}
where a vector notation is used throughout,   in order to avoid confusion between the space indices and  the particle indices, $i, j$.    We now recognize
\begin{equation}   {\bf  j}_i =  \frac{e_i}{ m_i}  ({\bf p}_i -   \frac{e_i}{c} {\bf A}({\bf r}_i) )
\end{equation}
as the familiar gauge-invariant (and conserved) electric current operator of the $i$-th constituent atom or molecule in an external magnetic field \footnote{See for instance Sec. 14.1.2 of \cite{KKGP}.}.   Writing now the total current as
\begin{equation}    {\bf J}({\bf R}_{CM}) =     \langle \Psi   |  \sum_i {\bf  j}_i      |\Psi \rangle   \;,
\end{equation}
one finds for the  CM  motion,
\begin{equation}         \frac{d}{dt}   {\bf P}_{CM}  =    \frac{1}{c}   {\bf J}   \times  {\bf B}\;,  \label{Lorentz}
\end{equation}
which is the Lorentz force.

  \subsubsection{A macroscopic particle with a magnetic moment   \label{SGClassic}}

Let us now consider a  magnetic field slowly varying in space.  More precisely, we assume that the  space variation of ${\bf B}({\bf r})$  within the body under consideration, is negligible.
We shall take
\begin{equation}       \nabla_{{\bf r}_i}  {\bf B}  =    0\;, \label{constB}
\end{equation}
as long as ${\bf r}_i$ is within the  support of the internal wave function  (which is a region of linear size $L_0$).
Note that under these circumstances the derivation of the Lorentz force  above continues to be valid:  the formula (\ref{Lorentz})   holds also for
non constant magnetic field    ${\bf B} $.

To analyze this problem,  we first modify the kinetic terms in $H$  in (\ref{macroEM})  as
\begin{equation}        \sum_i  \frac{({\bf p}_i - \frac{e_i}{c} {\bf A}({\bf r}_i) )^2}{2m_i}  \simeq \sum_i \left[    \frac{{\bf p}_i ^2}{2 m_i}   -
\frac{e_i}{4  m_i  c }   \{  {\bf p}_i \cdot  ({\bf B} \times {\bf r}_i) +    ({\bf B} \times {\bf r}_i) \cdot  {\bf p}_i \}  \right]  \label{this}
\end{equation}
where we dropped the terms proportional to the square of ${\bf A}$, and
  we have written  the vector potential

   as
\begin{equation}     {\bf A}({\bf r}_i)    =   \frac{1}{2}   \, {\bf B}  \times  {\bf r}_i  \;,  \qquad  (i=1,2,.\ldots, N)\;,     \label{vectorpotBis}
\end{equation}
even though now  ${\bf B}$ depends on the CM position ${\bf R}$.    To see how this step is justified,  consider
\begin{equation}   \nabla_{\bf r}  \times   \frac{1}{2}  ({\bf B}  \times {\bf r})  =     \frac{1}{2}   ({\bf r} \cdot \nabla)\, {\bf B}  + {\bf B}\;, \label{firstterm}
\end{equation}
but the first term is negligible,  as long as this expression is  used inside  the expectation value,
\begin{equation}    \langle   \Psi |  \cdots   | \Psi \rangle \;
\end{equation}
   (see (\ref{constB})).

   The right hand side of  (\ref{this}) can be seen to be equal to
\begin{equation}     \sum_i \left[    \frac{{\bf p}_i ^2}{2 m_i}   -
\frac{e_i}{2  m_i  c }    {\bf B}  \cdot  ( {\bf r}_i  \times  {\bf p}_i)   \right] =
  \sum_i \left[    \frac{{\bf p}_i ^2}{2 m_i}   -
\frac{e_i}{2  m_i  c }  {\bf B}  \cdot  {\bf \ell}_i \right]  \;,    \label{}
\end{equation}
where  ${\bf \ell}_i $  is the orbital angular momentum operator associated with the $i$-th constituent of the body.   We recognize
\begin{equation}    \frac{e_i    {\bf \ell}_i  }{2  m_i  c }
\end{equation}
as the magnetic moment produced by the orbital motion of the charged constituent (Bohr's magneton). By  denoting the total magnetic moment of the body as
\begin{equation}          \sum_i   \langle \Psi |    ( \mu_i +     \frac{e_i    {\bf \ell}_i  }{2  m_i  c } ) |\Psi \rangle  =   \mu\;,
\end{equation}
we find
\begin{equation}      \frac{d}{dt}   {\bf P}_{CM}  =    -     {\nabla_{{\bf R}_{CM}} }  \,  ( \mu \cdot {\bf B} )\;,  \label{NoSG}
\end{equation}
a familiar classical equation.

 \subsubsection{General consideration \label{general}}

One might think that the generalization of these results to include other effects, such as polarization,  charge conduction, etc., is quite a straightforward task.
However, a moment of  reflection convinces us that  to try to reproduce the whole richness of electromagnetic phenomena  of condensed matter  (see for instance, \cite{Landau8}) and classical electromagnetism,
starting from the Ehrenfest theorem for a macroscopic body with $N\sim 10^{26}$ atoms and molecules, is quite another matter.     The problem is that the reaction of the body to the external electromagnetic fields depends on the nature of the body, whether it is a conductor or a dielectic, or something else; in other words,  the full details of the internal Hamiltonian   and the internal  wave function $\psi({\tilde {\bf r}}_1,  {\tilde {\bf r}}_2, \ldots   {\tilde {\bf r}}_{N-1})$   describing the body, come into play.  In simple terms, what happens is that  each charged particle, electron, atomic nuclei, molecules, ions, react differently to the external electromagnetic field, and deform the internal structure of the body,
according to the way and strength with which they are bound in the body. All of the detailed spin-dependent effects also will  come into play.

Under such circumstance the analysis will quickly become very difficult;  the task requires    a systematic classification of different  physics
cases, and an elaborated work organization and preraration.

One has the feeling that what has been done in Sec.~\ref{constE} -  Sec.~\ref{SGClassic}  amounts to the  (though necessary!) very first few steps towards climbing up Mt. Everest.

This is in striking contrast to the case of the external gravitational  forces considered in Sec.~\ref{special1} and Sec.~\ref{general1}.  Due to the universal attractive nature, proportional to the mass,  of the gravitational forces, most of the details of the internal interactions and wave functions drop  out of the equations.   Each constituent, small or large, inside the body reacts and accelerates in an identical manner under  the external gravitational force.  This explains the fact that  a macroscopic body, e.g., a metal piece,  which contain in its interior the electron gas which exhibits quantum-mechanical collective behavior even at room temperatures,  or even stars and planets with $N\sim  10^{60}$ atoms and molecules,
still  obey  Newton's law for its CM motion, as it is known empirically and as it has been shown in Sec.~\ref{special1}, Sec.~\ref{general1}.

\section{Boundaries  between quantum and classical   \label{QCL}}

Throughout the discussion of this work,
there is clearly no sharp boundary between the quantum-mechanical and classical behavior of any given system as its temperature is varied.    A rough estimate of  the temperatures   $T\sim T_0$   under which the  microscopic, mesoscopic or macroscopic  system with a characteristic  (lowest-excitation-) frequency
$\nu$   is in its quantum-mechanical lowest  states,     is given by
\begin{equation}    T  \le      T_0\;,    \qquad  T_0  =     \frac{h \nu }{k_B}  =      \nu   \cdot  4.8   \cdot  10^{-11}   \, K  \;.   \label{temp}
\end{equation}
The characteristic frequency  $\nu$ depends on many factors of the system, the size,  the mass, $N$, the atomic,  molecular or collective excitation energies, etc.

\begin{table}[h]
  \centering
  \begin{tabular}{|c |c|c| c|c|c|}
\hline

   Systems      &               size      &     $  N  $   &    $\nu$ &  $T_0$      &            Ref.         \\
 \hline
  Macro pendulum (LIGO)      &     $10.8$ Kg     &  $\sim 10^{27} $    &    $100$     &   $ \sim 1.4  \cdot  10^{-6}  K$       &       \cite{LIGO}       \\
 \hline
  Nanomech.  resonator     &      $10^{-13}  \, {\rm cm}^3$       &  $ \sim 10^{11}   $  &   $\sim  2\cdot 10^7$    &      $1  \, mK$       &       \cite{LaHaye}       \\
 \hline
     Micromech. drum    &     $\sim  \mu $ m     &   $\sim 10^8 - 10^{10} $    &    $10^9$    &      $0.1 K$     &       \cite{Aaron}       \\
 \hline
   $C_{70} $     &     $ 70$  {\rm  \AA} &  $70$     &     $10^{14}$    &       $\sim 3000  K$           &      \cite{C70}    \\
 \hline
\end{tabular}
\caption{\small     A small sample of the experiments aimed at demonstrating the quantum mechanical behavior of a mesoscopic or macroscopic system at sufficiently low temperatures.   The size (the volume, linear dimension  or weight) and the number of atoms/molecules  cited are a crude estimate made from the lines of the articles by the present author:  any lack of precision is to be attributed to him, not to the authors of the works cited.
    }\label{classquant}
\end{table}

Some mesoscopic or macroscopic systems have been  studied recently for the purpose of revealing their quantum-mechanical behaviors
at low temperatures \cite{C70}, \cite{Leggett},  \cite{C60}-\cite{Brand}.
A small  sample of these experiments is summarized in  Table~\ref{classquant}, together with a few of the characteristics of the system considered and the relevant temperatures.
For  $N$ (the number of atoms/molecules)  spanning  over  many orders of magnitude,  the rule-of-thumb  estimate  (\ref{temp}) for the  typical temperature which devides
quantum-mechanical and classical behaviors of the system,   seems to be  basically  valid.

  In the case of  the LIGO  pendulum
of reduced mass $2.7$ Kg  \cite{LIGO},   the quoted  temperature of   $1.4  \,  \mu K$  refers to the  occupation numbers of about  $200$.  In the case of a nanomechanical resonator of \cite{LaHaye},  the quoted temperature   of $56\,mK$ refers to the   quantum excitation levels up to  $\sim 58$.    In a most
 interesting  case of  the work by O'Connel et. al. \cite{Aaron},  the achieved temperature is  roughly consistent with the  system being  in its quantum ground state (no phonon excitations), as verified experimentally by coupling the system to a squid  qubit.

  For Buckminsterfullerene molecule  $C_{70}$,   the frequency $\nu$ is estimated from the known molecular  excitation energies, and  $T_0$  given by Arndt et. al.
 \cite{C70}  is the temperature of the  $C_{70}$  beam  at which the interference fringes \`a la Young disappear.  The temperature is raised by
 increasing the laser power bombarding  the $C_{70}$'s  before they enter the double slit region.  It is  remarkable  that by varying continuously the laser power
 the authors  of \cite{C70}  have been  able to  interpolate  smoothly  between  the pure-quantum-state  $C_{70}$   (with interference fringes \`a la Young after the passage of  the double slit)  and   radiating, mixed-state ``classical" $C_{70}$, i.e., with no  interference fringes.

The relatively higher temperatures needed in the case of
$C_{70}$   as compared to the other macroscopic quantum-classical states in Table~\ref{classquant} reflects the fact that  here the excitations relevant are  the  $O(eV)$  {\it  molecular} excitations, whereas
in other examples  the characteristic frequency (and the body  temperature)  refers to the collective, phonon excitation modes, or even to  macroscopic oscillation modes.

\section{Summary and Discussion \label{Summary}}

In this work we have derived Newton's (force-) law from the Schr\"odinger equation \footnote{Newton's other law's,  that of inertia (the first law) and that of action-reaction
(the third law) are incorporated in quantum mechanics from the beginning.}.   Classical mechanics, initiated by Kepler, Copernicus and Galilei,  was completed by Newton in the form of the
mathematical principles and three fundamental laws \cite{Newton}. After three and a half centuries classical mechanics was replaced by quantum mechanics,
so as to be able to describe correctly the microscopic world of electrons, atoms and molecules. Now,  to describe appropriately the macroscopic world, we have come back all the way out,   to ``explain" Newton's equations from the more fundamental quantum mechanics.

Concretely,  we first  identified the basic requirements for the CM of a macroscopic body in the vacuum to possess a classical trajectory, i.e., a well-defined position and momentum at each instant of time.
The first is the Heisenberg uncertainty relation.  A macroscopic body of $N\sim 10^{51}$ atoms satisfies, for its center of mass (CM),   the same Heisenberg
lower limit    for the product of the canonical position and momentum,  $\Delta X_i  \cdot   \Delta P_j  \ge  \delta_{ij}    \hbar/2$ as for an atom.   Namely the quantum uncertainties do not pile up.
This means that for a macroscopic body, and for the measurements within the  appropriate range  of  macroscopic scale and precision,   Heisenberg's relations can be considered irrelevant.
The position and momentum can be measured simultaneously, to an ``arbitrary" precision,   $\Delta X \approx 0$, $\Delta P\approx 0$.

The second is the absence of the diffusion for free wave packets.  Here the essential factor is the mass.  While for an electron with mass $\sim 10^{-27} $ g,   the time needed for doubling its wave packet size  (starting from a $ 1   \, \mu m$)    is  $10^{-8}$ sec,    the corresponding time for a macroscopic
body of $1$ g,  with the  initial CM wave packet of the same size $ 1   \, \mu m$,  exceeds   the age of the universe.
This means that once its position and momentum are  experimentally measured with certain precision    $\Delta X$,   $\Delta P$,    they do not grow in time: a macroscopic body follows a
classical (well-defined)  path:   a trajectory.   An extended wave packet is a uniquely quantum concept,  which applies only to  microscopic systems.

 The third, and the crucial,  factor  is the nonzero temperature of the body, which makes it a radiating, metastable state.  It is a mixed state.   {\it  Decoherence}  caused by the entanglement with the ``lost"  (unobserved) photons  it emitted and which carry away the information,  means  that a split wave packet for a macroscopic body -
 a coherent superposition of two macroscopically distinct state vectors   -    is a meaningless notion.  It cannot be prepared experimentally.

 How high the body temperature must be, to lose the quantum mechanical characteristics,  depends on many factors, the type of material, the states, the  density, size, etc.,
 and varies vastly.   We have discussed briefly this question in Sec.~\ref{QCL},  drawing lessons from  several existing experimental studies,  which  show    that
 the  rough estimate of the boundary  temperature  is given by a simple formula,   (\ref{temp}). It captures essentially the nature (and  the energy)  of the lowest  excitation  modes, characteristic of the system considered, such as  an atomic or molecular excitation, phonon excitations in a solid, or simply a macroscopic oscillation of the body.  The formula   (\ref{temp})  was found basically valid for  mesoscopic-macroscopic bodies with number of atoms (or  of  molecules)  in a vast range spanning  from $N = 70$ to $N\sim  10^{27}$.

Once these basic  elements,  with various subtlties, are grasped (Sec.~\ref{basic}),    the derivation of Newton's equation for a macroscopic body is a straightforward application of Ehrenfest's theorem, as we have illustrated in the case of weak gravitational forces  (Sec.~\ref{special1} and Sec.~\ref{general1}),  a harmonic potential (Sec.~\ref{special2}), and under weak and almost space-independent external electromagnetic fields (Sec.~\ref{EM}).   As remarked in Sec.~\ref{general}, our work is only at the very initial stage of a
highly nontrivial task, involving  classical electromagnetism with macroscopic bodies and  more general discussion of electromagnetism of continuous media.

 Even though we have not yet  worked  it out in great  detail, it appears to be basically straightforward to
 generalize the results of the present work to  the case of many macro-body problems,  and  to establish
  the Hamilton equations and other aspects of classical mechanics.   Appendix~\ref{canonical} gives a sketch of the derivation
  of the canonical equations.

Our way of  understanding  how  the classical mechanics emerges  from quantum mechanics  for an isolated macroscopic body,  which (we hope) complements  and completes the previous works on the environment-induced decoherence  mechanism \cite{JoosBis}-\cite{Arndt1}, seems   to point to a  new perspective on the relation between quantum and classical physics.
  Namely, for a macroscopic body at finite temperatures  (hence a mixed state),  {\it  classical mechanics for its center-of-mass motion emerges as the lowest-order approximation  \footnote{The content of Sec.~\ref{basic} and Sec.~\ref{Ehrenfest}  defines the approximations needed  to derive the classical equations.  The nature of the approximations involved there suggests extremely small corrections in general;
  their magnitude will however depend on the kind of materials, mass,  and temperatures, on the presence of the environment and so on.  The study of ``quantum corrections to Newton's equations" is a subject of future studies.   }  of
  the quantum mechanical equations.}

 This seems  to be a  novel point of view,  as  compared to  the traditional  way of  thinking about this problem,  for instance,    that  classical physics emerges in the so-called semi-classical approximation of quantum mechanics  (i.e., in the large-action limit),    or that   the so-called coherent states  are the closest analogue of classical  objects  in quantum mechanics.   Other textbook examples include   the formal similarity between the commutation relations in quantum mechanics and the Poisson brackets in classical mechanics, and the fact  that  in semiclassical limit the Schr\"odinger equation for the (exponent of)  the wave function reduces to the
  Hamilton-Jacobi equation. Still further examples are   the  Bohr-Sommerfeld quantization conditions and  Bohr's correspondence principle.

   Though these well-known ideas  certainly constitute a necessary condition for the  quantum mechanics to be compatible with classical mechanics,
   and in that sense, are manifestations of  deep consistencies within the quantum theory,
    they mostly concern pure quantum states.  None of these represents  therefore the true  explanation of the emergence of  classical physics from quantum mechanics,
    which  hinges upon the  {\it  mixed-state, large mass}  nature of a macroscopic body {\it  at finite temperatures},      as shown

     in the present work.

Before closing, it is perhaps  appropriate to discuss, to avoid any misunderstanding,  the nature of  the quantum-classical transition  characterized by a temperature, (\ref{temp}), as compared  to  the  so-called critical temperatures associated  with various kinds of  critical phenomena and {\it phase transitions}.     Although  some mention of various kinds of phase transitions was made in Sec.~\ref{Intro},   as a way of illustrating the fact that any matter behaves quantum mechanically at temperatures sufficiently close to $0$,
 and accordingly,  in order to pinpoint the role of the temperature as a primary factor  in our problem,  we hope that
the careful reader has not been misled to the idea that, or that we are claiming that,  these two questions are substantially related.

{\it  The quantum-classical  ``transition" discussed in this work concerns  a change in the way we describe the world; it does not deal with real discontinuities in physics.}    The finite temperatures of a body are just signs of the body being in an excited, metastable quantum state. Any such system radiates spontaneously, and the entanglement with the photons it emitted (and which carry away information)  renders the system a mixture.
Decoherence which results  is one of the main causes of the body to behave classically, as we saw.

The phase transitions and critical phenomena, instead,   describe a genuine change  in physics occurring at a precise (critical)  temperature. They describe a sudden collective  rearrangement of atoms and molecules in a large system\footnote{Strictly speaking, a true discontinuity or a mathematical singularity in theromodynamical quantities,  occur only in the systems with an infinite degrees of freedom, as is well known.}, let it be a liquid, gas, solid, BEC or a plasma.      Even keeping in mind that the whole of  statistical mechanics requires  quantum mechanics
for its consistency at its basis,     some phase transitions may still be understood  to various degrees  within the context of classical physics, such as solid-liquid, liquid-vapor transitions.
 In some others,
such as spontaneous magnetization,  superconductivity,  BEC,  and superfluidity  transitions,  instead,  quantum aspects of the systems appear so much prominent.
Nonetheless,   all of these phase transitions are  sometimes referred to as
classical phase transitions.

  In the so-called quantum phase transitions,
occurring at $T=0$,    which are a very active research field
 today, instead,    phase transitions occur  between two fully quantum-mechanical  ground-state phases of the system.

 These considerations illustrate  clearly that the question of the emergence of the classical physics from quantum mechanics examined in the present work
 is conceptually distinct from, and should not be confused with,    the interesting physics of various phase transitions, classical or quantum.

\section*{Acknowledgments}

An inspiration for the present work came from an experiment performed in Pisa some $\sim15$ years ago,  by C. Bemporad, C. Bradaschia and F. Fidecaro,
in a  celebrative event  for the public including school children.

 The setup consisted of two  $\sim3m$ tall  glass tubes with diameter $\sim 30$ cm, with their interior  kept in a good vacuum.   A few-centimeter steel  ball in the one and a light plastic sponge in the other, were let to fall simultaneously,   demonstrating that they indeed fall together (!),   evoking Galileo's (actual or imaginary)  experiment done from the top of the Leaning Tower around the year  $\sim1590$.
The device
 is conserved in the entrance hall of the VIRGO/EGO  gravitational-wave observatory at Cascina, near Pisa. The author is deeply indebted  to F. Fidecaro for rearranging the reactivation of the  apparatus  and for demonstrating  this glass-tube ``Galileian" experiment  anew  for him.
He  also  thanks
F. Fidecaro and G. Cella for discussions on the physics of the gravitational wave detections  at LIGO and VIRGO.
He thanks   P. Menotti and S. Bolognesi  for comments on the manuscript and for discussions.  A special thanks of the author is reserved for Hans Thomas Elze  for  many interesting
discussions,  suggestions, and for useful information about the literature.
   This work is supported by  the INFN special research initiative grant, ``GAST"  (Gauge and String Theories).

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