Foundational principle in quantum physics
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
More formally, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a quantum system, such as position, "x", and momentum, "p". Such paired-variables are known as complementary variables or canonically conjugate variables.
First introduced in 1927 by German physicist Werner Heisenberg, the formal inequality relating the standard deviation of position "σx" and the standard deviation of momentum "σp" was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:
formula_1
where formula_2 is the reduced Planck constant.
The quintessentially quantum mechanical uncertainty principle comes in many forms other than position–momentum. The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time. The basic principle has been extended in numerous directions; it must be considered in many kinds of fundamental physical measurements.
Position–momentum.
It is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic scales that humans experience. Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, but the more abstract matrix mechanics picture formulates it in a way that generalizes more easily.
Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A nonzero function and its Fourier transform cannot both be sharply localized at the same time. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of the particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the de Broglie relation "p" 
 "ħk", where k is the wavenumber.
In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. However, the particular eigenstate of the observable A need not be an eigenstate of another observable B: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.
Visualization.
The uncertainty principle can be visualized using the position- and momentum-space wavefunctions for one spinless particle with mass in one dimension.
The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentum-space wavefunction is less localized so the possible momentum components the particle could have are more widespread. Conversely, the more localized the momentum-space wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly the less localized the position-space wavefunction, so the position coordinates the particle could occupy are more widespread. These wavefunctions are Fourier transforms of each other: mathematically, the uncertainty principle expresses the relationship between conjugate variables in the transform.
Wave mechanics interpretation.
According to the de Broglie hypothesis, every object in the universe is associated with a wave. Thus every object, from an elementary particle to atoms, molecules and on up to planets and beyond are subject to the uncertainty principle.
The time-independent wave function of a single-moded plane wave of wavenumber "k"0 or momentum "p"0 is
formula_3
The Born rule states that this should be interpreted as a probability density amplitude function in the sense that the probability of finding the particle between "a" and "b" is
formula_4
In the case of the single-mode plane wave, formula_5 is "1" if formula_6 and "0" otherwise. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet.
On the other hand, consider a wave function that is a sum of many waves, which we may write as
formula_7
where "A""n" represents the relative contribution of the mode "p""n" to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized. We may take this a step further to the continuum limit, where the wave function is an integral over all possible modes
formula_8
with formula_9 representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that formula_9 is the "Fourier transform" of formula_11 and that "x" and "p" are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta.
One way to quantify the precision of the position and momentum is the standard deviation "σ". Since formula_5 is a probability density function for position, we calculate its standard deviation.
The precision of the position is improved, i.e. reduced "σ""x", by using many plane waves, thereby weakening the precision of the momentum, i.e. increased "σ""p". Another way of stating this is that "σ""x" and "σ""p" have an inverse relationship or are at least bounded from below. This is the uncertainty principle, the exact limit of which is the Kennard bound.
Proof of the Kennard inequality using wave mechanics.
We are interested in the variances of position and momentum, defined as
formula_13
formula_14
Without loss of generality, we will assume that the means vanish, which just amounts to a shift of the origin of our coordinates. (A more general proof that does not make this assumption is given below.) This gives us the simpler form
formula_15
formula_16
The function formula_17 can be interpreted as a vector in a function space. We can define an inner product for a pair of functions "u"("x") and "v"("x") in this vector space:
formula_18
where the asterisk denotes the complex conjugate.
With this inner product defined, we note that the variance for position can be written as
formula_19
We can repeat this for momentum by interpreting the function formula_20 as a vector, but we can also take advantage of the fact that formula_11 and formula_9 are Fourier transforms of each other. We evaluate the inverse Fourier transform through integration by parts:
formula_23
where formula_24 in the integration by parts, the cancelled term vanishes because the wave function vanishes at both infinities and formula_25, and then use the Dirac delta function which is valid because formula_26 does not depend on "p" .
The term formula_27 is called the momentum operator in position space. Applying Plancherel's theorem, we see that the variance for momentum can be written as
formula_28
The Cauchy–Schwarz inequality asserts that
formula_29
The modulus squared of any complex number "z" can be expressed as
formula_30
we let formula_31 and formula_32 and substitute these into the equation above to get
formula_33
All that remains is to evaluate these inner products.
formula_34
Plugging this into the above inequalities, we get
formula_35
and taking the square root
formula_36
with equality if and only if "p" and "x" are linearly dependent. Note that the only "physics" involved in this proof was that formula_11 and formula_9 are wave functions for position and momentum, which are Fourier transforms of each other. A similar result would hold for "any" pair of conjugate variables.
Matrix mechanics interpretation.
In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators. When considering pairs of observables, an important quantity is the "commutator". For a pair of operators Â and formula_39, one defines their commutator as
formula_40
In the case of position and momentum, the commutator is the canonical commutation relation
formula_41
The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum eigenstates. Let formula_42 be a right eigenstate of position with a constant eigenvalue "x"0. By definition, this means that formula_43 Applying the commutator to formula_42 yields
formula_45
where Î is the identity operator.
Suppose, for the sake of proof by contradiction, that formula_42 is also a right eigenstate of momentum, with constant eigenvalue "p"0. If this were true, then one could write
formula_47
On the other hand, the above canonical commutation relation requires that
formula_48
This implies that no quantum state can simultaneously be both a position and a momentum eigenstate.
When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is "not" a momentum eigenstate, however, but rather it can be represented as a sum of multiple momentum basis eigenstates. In other words, the momentum must be less precise. This precision may be quantified by the standard deviations,
formula_49
formula_50
As in the wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle.
Quantum harmonic oscillator stationary states.
Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in terms of the creation and annihilation operators:
formula_51
formula_52
Using the standard rules for creation and annihilation operators on the energy eigenstates,
formula_53
formula_54
the variances may be computed directly,
formula_55
formula_56
The product of these standard deviations is then
formula_57
In particular, the above Kennard bound is saturated for the ground state "n"
0, for which the probability density is just the normal distribution.
Quantum harmonic oscillators with Gaussian initial condition.
In a quantum harmonic oscillator of characteristic angular frequency "ω", place a state that is offset from the bottom of the potential by some displacement "x"0 as
formula_58
where Ω describes the width of the initial state but need not be the same as "ω". Through integration over the , we can solve for the full time-dependent solution. After many cancelations, the probability densities reduce to
formula_59
formula_60
where we have used the notation formula_61 to denote a normal distribution of mean "μ" and variance "σ"2. Copying the variances above and applying trigonometric identities, we can write the product of the standard deviations as
formula_62
From the relations
formula_63
we can conclude the following (the right most equality holds only when Ω = "ω"):
formula_64
Coherent states.
A coherent state is a right eigenstate of the annihilation operator,
formula_65
which may be represented in terms of Fock states as
formula_66
(whenever the denominator is nonzero).
While this is a universal result, it depends upon the observable chosen and that the deviations formula_67 and formula_68 are computed for a particular state.
Identifying formula_69 and the characteristic time
formula_70
gives an energy–time relationship formula_71
Although formula_72 has the dimension of time, it is different from the time parameter "t" that enters the Schrödinger equation. This formula_72 can be interpreted as time for which the expectation value of the observable, formula_74 changes by an amount equal to one standard deviation.
Examples:
formula_78
The probability amplitude of this state has a time-dependent interference term:
formula_79
The oscillation period varies inversely with the energy difference: formula_80.
Each example has a different meaning for the time uncertainty, according to the observable and state used.
Quantum field theory.
Some formulations of quantum field theory uses temporary electron–positron pairs in its calculations called virtual particles. The mass-energy and lifetime of these particles are related by the energy–time uncertainty relation. The energy of a quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of "all histories" must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.
The energy–time uncertainty principle does not temporarily violate conservation of energy; it does not imply that energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time. The energy of the universe is not an exactly known parameter at all times. When events transpire at very short time intervals, there is