1.1: An Approach to Quantum Mechanics The purpose of this tutorial is to introduce the basics of quantum mechanics using Dirac bracket notation while working in one dimension. Dirac notation is a succinct and powerful language for expressing quantum mechanical principles; restricting attention to one-dimensional examples reduces the possibility that mathematical complexity will stand in the way of understanding. 1.2: Atomic and Molecular Stability Quantum mechanics is not mathematically more difficult than classical physics, the real challenge it offers is of a conceptual nature. That’s because it says that understanding the stability of matter requires that it possess both particle and wave properties. The conceptual challenge is that particle and wave are contradictory concepts. 1.3: Atomic and Molecular Stability Sometimes practitioners of quantum mechanics misinterpret energy contributions when studying the details of atomic behavior. This can happen when classical concepts are allowed to intrude on the quantum realm where they are not valid. 1.4: Atomic and Molecular Stability John Slater pioneered the use of the virial theorem in interpreting the chemical bond in a benchmark paper published in the inaugural volume of the Journal of Chemical Physics. This early study indicated that electron kinetic energy played an important role in bond formation. Thirty years later Klaus Ruedenberg and his collaborators published a series of papers detailing the crucial role that kinetic energy plays in chemical bonding, thereby completing the project that Slater started. 1.5: Quantum Computation - A Short Course Quantum computers use superpositions, entanglement and interference to carry out calculations that are impossible with a classical computer. I will present mathematical analyses of some relatively simple and representative quantum circuits that are designed to carry out important contemporary processes such as parallel computation, teleportation, data-base searches, prime factorization, quantum encryption and quantum simulation. 1.6: Quantum Computation- A Short Course One of the most intriguing applications of entanglement is quantum teleportation, which uses entanglement and a classical communication channel to transfer a quantum state from one location to another. However, to truly understand teleportation it is necessary to distinguish it from cloning. So first we look at the quantum no-cloning principle followed by a one-page snapshot of teleportation. 1.7: Quantum Computation- A Short Course As can be seen in the previous tutorials, teleportation involves entanglement transfer. Alice projects her photons onto one of the entangled Bell states and Bob receives a photon state which using information provided via the classical communication channel can be transformed into the teleportee state. Given the importance of entanglement in quantum computing a more elaborate example of entanglement transfer is provided in the following tutorial. 1.8: Quantum Computation- A Short Course The third tutorial is a version of the first which shows how path information destroys interference and how the interference can be restored. This tutorial summarizes David Deutsch’s solution to an equivalent mathematical problem using a function introduced earlier. The double-slit apparatus, the Mach-Zehnder interferometer and Deutsch’s circuit are quantum computers which use superpositions and interference effects to cut the effort of answering the question by a factor of two. 1.9: Quantum Computation- A Short Course The sub-microscopic building blocks of the natural world (electrons, protons, neutrons and photons) do not behave like the macroscopic objects we encounter in daily life because they have both wave and particle characteristics. Nick Herbert called them quons since we always observe particles, but prior to measurement or observation quons behave like waves. This peculiar behavior is illustrated in the following tutorial. 1.100: Analysis of the Stern-Gerlach Experiment 1.101: Related Analysis of the Stern-Gerlach Experiment 1.103: Bloch Sphere 1.104: 88. Related Analysis of the Stern-Gerlach Experiment 1.105: Bill the Cat and the Superposition Principle 1.106: Schroedinger's Dog 1.107: The Bloch Sphere 1.108: Density Matrix, Bloch Vector and Entropy 1.109: State Vectors and State Operators- Superpositions, Mixed States, and Entanglement 1.10: Quantum Computation- A Short Course The last several tutorials were a bit off the theme of quantum computation. We get back on track with a look at data base searching the quantum way, or the best way to find a needle in a hay stack. This is followed by a demonstration of Simon’s algorithm, an illustration of quantum dense coding and an example of quantum error correction. 1.110: The Gram-Schmidt Procedure 1.11: Quantum Computation- A Short Course The four remaining tutorials deal with this clash between quantum mechanics and local realism, and the simulation of physical phenomena. The first two examine entangled spin systems and the third entangled photon systems. 1.12: Quantum Computation- A Short Course This fourth tutorial provides a terse mathematical summary for both spin and photon systems. These four tutorials all clearly show the disagreement between the predictions of quantum theory and those of local hidden-variable models. 1.13: Quantum Mechanics and the Fourier Transform Wave-particle duality as expressed by the de Broglie wave equation is the seminal concept of quantum mechanics. Wave and particle are physically incompatible concepts because waves are spatially delocalized, while particles are spatially localized. In spite of this incongruity we find in quantum theory that they are necessary companions in the analysis of atomic and molecular phenomena. Both concepts are required for a complete examination of experiments at the nanoscale level. 1.14: Quantum Mechanics and the Fourier Transform However, just as with the circular aperture (Airy pattern) a single slit also yields a diffraction pattern when illuminated. Both are examples of the superposition principle because the photons that arrive at the detection screen can get there from any points within the aperture or slit. So, in general, we calculate the diffraction pattern by a Fourier transform of the coordinate space geometry, slit or circle or something more complicated. 1.15: Quantum Mechanics and the Fourier Transform 1.16: Quantum Mechanics and the Fourier Transform Moving on to the hydrogen atom, we can use the spatial and momentum wavefunctions for the 1s, 2s and 3s energy states to again illustrate visually the uncertainty principle. 1.17: Quantum Mechanics and the Fourier Transform The purpose of this tutorial is to introduce the basics of quantum mechanics using Dirac bracket notation while working in one dimension. Dirac notation is a succinct and powerful language for expressing quantum mechanical principles; restricting attention to one-dimensional examples reduces the possibility that mathematical complexity will stand in the way of understanding. 1.18: Exploring the Origin of Schrödinger’s Equations The purpose of this tutorial is to explore the connections between Schrödinger’s equations (time-dependent and time-independent) and prior concepts in classical mechanics and quantum mechanics. For the sake of mathematical simplicity we will work in one spatial dimension. 1.19: Basic Quantum Mechanics in Coordinate, Momentum and Phase Space 1.20: The Repackaging of Quantum Weirdness This tutorial deals with the concept of the quantum mechanical operator and how it extracts information from the wavefunction. There are at least nine formulations of quantum mechanics. In this tutorial the position and momentum operators will be examined in the coordinate, momentum and phase space formulations of quantum mechanics. 1.21: Quantum Principles Illuminated with Polarized Light 1.22: Relationship Between the Coordinate and Momentum Representations 1.23: Very Brief Relationship Between the Coordinate and Momentum Representations 1.24: Getting Accustomed to the Superposition Principle It is impossible to simulate a quantum mechanical superposition with a mixture of M&M candies, or any other ensemble of macroscopic objects (1). Among the defining characteristics of a superposition is that it is not a mixture. Miller acknowledges that while his demonstration is not "strictly accurate", it is "quite effective and achieves a variety of educational goals." 1.25: The Dirac Delta Function Dirac delta function is a generalized function introduced by the physicist Paul Dirac to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. 1.26: Elements of Dirac Notation In the early days of quantum theory, P. A. M. (Paul Adrian Maurice) Dirac created a powerful and concise formalism for it which is now referred to as Dirac notation or bra-ket (bracket ⟨|⟩⟨|⟩ ) notation. 1.27: The Dirac Notation Applied to Variational Calculations The particle-in-a-box problem is exactly soluble and the solution is calculated below for the first 20 eigenstates. All calculations will be carried out in atomic units. 1.28: Raising and Lowering; Creating and Annihilating The purpose of this tutorial is to illustrate uses of the creation (raising) and annihilation (lowering) operators in the complementary coordinate and matrix representations. These operators have routine utility in quantum mechanics in general, and are especially useful in the areas of quantum optics and quantum information. 1.29: Single Slit Diffraction and the Fourier Transform 1.30: From Coordinate Space to Momentum Space and Back 1.31: The Position and Momentum Commutation Relation in Coordinate and Momentum Space 1.32: Simulating the Aharonov-Bohm Effect The Aharonov–Bohm effect is a phenomenon by which an electron is affected by the vector potential, A, in regions in which both the magnetic field B, and electric field E are zero. The most commonly described case occurs when the wave function of an electron passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being zero in the region through which the particle passes. 1.33: Basic Matrix Mechanics 1.34: Rudimentary Matrix Mechanics A quon (an entity that exhibits both wave and particle aspects in the peculiar quantum manner - Nick Herbert, Quantum Reality, page 64) has a variety of properties each of which can take on two values. For example, it has the property of hardness and can be either hard or soft. It also has the property of color and can be either black or white, and the property of taste and be sweet or sour. 1.35: Matrix Mechanics The basic principles of quantum theory can be demonstrated very simply by exploring the properties of electron spin using Heisenberg's formulation of quantum mechanics which is usually referred to as matrix mechanics. The matrix formulation provides clear illustrations of the following essential quantum mechanical concepts: eigenvector, operator, eigenvalue, expectation value, the linear superposition, and the commutation relations. 1.36: Aspects of Diracʹs Relativistic Matrix Mechanics The relativistic equation for the energy of a free particle has positive and negative roots, where the positive root signifies the energy of a particle and the negative root the energy of its antiparticle. This interpretation was confirmed experimentally with the discovery of the anti‐electron (positron) in 1932 by Anderson. 1.37: The Double-Slit Experiment Thomas Young used the double-slit experiment to establish the wave nature of light. Richard Feynman used it to demonstrate the superposition principle as the paradigm of all quantum mechanical phenomena, illustrating wave-particle duality as stated above: between release and detection quons behave as waves. 1.38: Double‐Slit Experiment with Polarized Light 1.39: The Consequences of Path Information in a Mach-Zehnder Interferometer This tutorial deals with the effect of path information in a Mach-Zehnder interferometer (MZI). A related analysis involving the double-slit experiment is available in the preceding tutorial. We begin with an analysis of the operation of an equal arm MZI illuminated with diagonally polarized light. 1.40: Another look at the Consequences of Path Information in a Mach-Zehnder Interferometer This tutorial deals with the effect of path information and its so-called erasure in a Mach-Zehnder interferometer (MZI). A related analysis involving the double-slit experiment is available with the title "Which Path Information and the Quantum Eraser." 1.41: The Double‐Slit Experiment with Polarized Light Fresnel and Arago ʺusing an apparatus based on Youngʹs [double‐slit] experimentʺ observed that ʺtwo beams polarized in mutually perpendicular planes never yield fringes.ʺ The purpose of this tutorial is to examine this phenomenon from a quantum mechanical perspective. 1.42: The Quantum Eraser Paul Kwiat and an undergraduate research assistant published ʺA Do‐It‐Yourself Quantum Eraserʺ in the May 2007 issue of Scientific American. The purpose of this tutorial is to show the quantum math behind the laser demonstrations illustrated in this article. 1.43: Which Way Did It Go? ‐ The Quantum Eraser Paul Kwiat and Rachel Hillmer, an undergraduate research assistant, published ʺA Do‐It‐Yourself Quantum Eraserʺ based on the double‐slit experiment in the May 2007 issue of Scientific American. The purpose of this tutorial is to show the quantum math behind the laser demonstrations illustrated in this article. 1.44: Which Path Information and the Quantum Eraser 1.45: Terse Analysis of Triple-slit Diffraction with a Quantum Eraser 1.46: Which Path Information and the Quantum Eraser (Brief) 1.47: Terse Analysis of Triple-slit Diffraction with a Quantum Eraser 1.48: Which‐way Markers and Post‐selection in the Double‐slit Experiment 1.49: A Stern‐Gerlach Quantum ʺEraserʺ 1.50: Using the Mach-Zehnder Interferometer to Illustrate the Impact of Which-way Information 1.51: Quantum Theory, Wave-Particle Duality and the Mach-Zehnder Interferometer 1.52: Analysis of a Temporal Double-slit Experiment 1.53: An Analysis of Three‐Path Interference 1.54: An Analysis of Three-Slit Interference Quantum mechanics teaches that if there is more than one path to a particular destination interference effects are likely. Using Feynmanʹs ʹsum over historiesʹ approach to quantum mechanics the probability of arrival at a location is the square of the magnitude of the sum of the probability amplitudes for each path to that location. E.g., in the triple‐slit diffraction experiment the probability of a photon leaving a source and arriving at x on the detection screen shows three‐path interference. 1.55: Using a Mach-Zehnder Interferometer to Illustrate Feynman's Sum Over Histories Approach to Quantum Mechanics 1.56: The Paradox of Recombined Beams 1.57: Evidence for Quantized Gravitational States of the Neutron 1.58: Quantized Gravitational States A Variational Approach 1.59: The Quantum Bouncer Doesnʹt Bounce, Unless... 1.60: Kinetic Energy Is Important in the Nanoscale World 1.61: Energy Expectation Values and the Origin of the Variation Principle 1.62: Examining the Wigner Distribution Using Dirac Notation 1.63: The Wigner Function for the Single Slit Diffraction Problem 1.64: Wigner Distribution for the Double Slit Experiment 1.65: Wigner Distribution for the Triple Slit Experiment 1.66: Wigner Distribution for the Quadruple Slit Experiment 1.67: Quantum Tunneling in Coordinate, Momentum and Phase Space 1.68: Another Look at the Wigner Function 1.69: The Wigner Distribution Function for the Harmonic Oscillator 1.70: Wigner Distribution for the Particle in a Box 1.71: The Wigner Distribution for a Particle in a One‐dimensional Box 1.72: Superposition vs. Mixture 1.73: Time‐dependent Wigner Function for Harmonic Oscillator Transitions 1.74: Momentum Operator in Coordinate Space 1.75: Momentum Wave Functions for the Particle in a Box 1.76: A Graphical Illustration of the Heisenberg Uncertainty Relationship 1.77: The Quantum Harmonic Oscillator 1.78: Coherent Superpositions for the Harmonic Oscillator 1.79: The Harmonic Oscillator and the Uncertainty Principle 1.80: Another view of the Harmonic Oscillator and the Uncertainty Principle 1.81: Hydrogen Atom and Helium Ion Spatial and Momentum Distribution Functions Illustrate the Uncertainty Principle 1.82: The Position‐Momentum Uncertainty Relation in the Hydrogen Atom 1.83: Demonstrating the Uncertainty Priniciple for Angular Momentum and Angular Position 1.84: A Brief Tutorial on Wavepackets 1.85: The Difference Between Fermions and Bosons 1.86: Quantum Corrals - Electrons within a Ring Here we report the construction and characterization of structures for confining electrons to this length scale. The walls of these "quantum corrals" are built from Fe atoms which are individually positioned on the Cu (111) surface by means of a scanning tunneling microscope (STM). These atomic structures confine surface state electrons laterally because of the strong scattering that occurs between surface state electrons and the Fe atoms. 1.87: Planck's Radiation Equation Fit to Experimental Data 1.88: Planck's Radiation Equation Fit to Experimental Data - Another Algorithm 1.89: Fitting Einstein's Heat Capacity Equation to Experimental Data for Silver 1.90: Einstein's Heat Capacity Equation Fit to Experimental Data - Another Algorithm 1.91: Fitting Debye's Heat Capacity Equation to Experimental Data for Silver 1.92: Debye's Heat Capacity Equation Fit to Experimental Data - Another Algorithm 1.93: Wave-particle Duality and the Uncertainty Principle 1.94: Wave-Particle Duality for Matter and Light 1.95: What Part of the Quantum Theory Don’t You Understand? 1.96: Quantum Potpourri - An Attempt to Demonstrate Two Fundamental Quantum Concepts- Wave-particle Duality and The Superposition Principle 1.97: Quantum Dynamics- One Step at a Time 1.98: Quantum Mechanical Pressure To provide a simple explanation for the origin of energy quantization, the PIB model shows that reducing the size of the box increases the kinetic energy dramatically. This “repulsive” character of quantum mechanical kinetic energy is the ultimate basis for the stability of matter. It also provides, as we see now, a quantum interpretation for gas pressure. To show this we will consider a particle in the ground state of a three-dimensional box. 1.99: Visualizing the Difference Between a Superposition and a Mixture