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Quantum Mechanics Made Simple:

Lecture Notes

Weng Cho CHEW 1

December 6, 2016

(^1) The author is with U of Illinois, Urbana-Champaign.

• 1 Introduction Acknowledgements viii
  • 1.1 Introduction
  • 1.2 Quantum Mechanics is Bizarre
  • 1.3 The Wave Nature of a Particle—Wave-Particle Duality
• 2 Classical Mechanics and Some Mathematical Preliminaries
  • 2.1 Introduction
  • 2.2 Lagrangian Formulation
    • 2.2.1 The Classical Harmonic Oscillator
    • 2.2.2 Continuum Mechanics of a Cylinder
  • 2.3 Hamiltonian Formulation
  • 2.4 More on Hamiltonian
  • 2.5 Poisson Bracket
  • 2.6 Some Useful Knowledge of Matrix Algebra
    • 2.6.1 Identity, Hermitian, Symmetric, Inverse and Unitary Matrices
    • 2.6.2 Determinant
    • 2.6.3 Eigenvectors and Eigenvalues
    • 2.6.4 Trace of a Matrix
    • 2.6.5 Function of a Matrix
• 3 Quantum Mechanics—Some Preliminaries
  • 3.1 Introduction
  • 3.2 Probabilistic Interpretation of the Wave Function
  • 3.3 Time Evolution of the Hamiltonian Operator
  • 3.4 Simple Examples of Time Independent Schr¨odinger Equation
    • 3.4.1 Particle in a 1D Box
    • 3.4.2 Particle Scattering by a Barrier
    • 3.4.3 Particle in a Potential Well
  • 3.5 The Quantum Harmonic Oscillator–A Preview
• 4 Time-Dependent Schr¨odinger Equation ii Quantum Mechanics Made Simple
  • 4.1 Introduction
  • 4.2 Quantum States in the Time Domain
  • 4.3 Coherent State
  • 4.4 Measurement Hypothesis and Expectation Value
    • 4.4.1 Uncertainty Principle–A Simple Version
    • 4.4.2 Particle Current
  • A Introduction to Gaussian Wave Packet
  • B Derivation from the Wave Equation
  • C Physical Interpretation
  • D Stability of the Plane Wave Solution
• 5 More Mathematical Preliminaries
  • 5.1 A Function is a Vector
  • 5.2 Operators
    • 5.2.1 Matrix Representation of an Operator
    • 5.2.2 Bilinear Expansion of an Operator
    • 5.2.3 Trace of an Operator
    • 5.2.4 Unitary Operators
    • 5.2.5 Hermitian Operators
  • 5.3 Commutation of Operators
  • 5.4 Expectation Value and Eigenvalue of Operators
  • 5.5 Generalized Uncertainty Principle
  • A Identity Operator in a Continuum Space
  • B Changing Between Representations
    • B1 Momentum Operator
    • B2 Position Operator
    • B3 The Coordinate Basis Function
  • C Time Evolution of the Expectation Value of an Operator
    • C1 Comparison to classical equations of motion
  • D Periodic Boundary Condition
• 6 Approximate Methods in Quantum Mechanics
  • 6.1 Introduction
  • 6.2 Use of an Approximate Subspace
  • 6.3 Tight Binding Model
    • 6.3.1 Variational Method
  • 6.4 Time Dependent Perturbation Theory
  • A Time Independent Perturbation Theory
    • A1 First Order Perturbation
    • A2 Second Order Perturbation
    • A3 Higher Order Corrections
• 7 Quantum Mechanics in Crystals Contents iii
  • 7.1 Introduction
  • 7.2 Bloch-Floquet Waves
    • 7.2.1 Periodicity of E(k)
    • 7.2.2 Symmetry of E(k) with respect to k
  • 7.3 Bloch-Floquet Theorem for 3D
  • 7.4 Fermi-Dirac Distribution Function
    • 7.4.1 Semiconductor, Metal, and Insulator
    • 7.4.2 Why Do Electrons and Holes Conduct Electricity?
  • 7.5 Effective Mass Schr¨odinger Equation
  • 7.6 Heterojunctions and Quantum Wells
  • 7.7 Density of States (DOS)
    • 7.7.1 DOS in a Quantum Well
    • 7.7.2 Quantum Wires
• 8 Angular Momentum
  • 8.1 Introduction
    • 8.1.1 Electron Trapped in a Pill Box
    • 8.1.2 Electron Trapped in a Spherical Box
  • 8.2 Mathematics of Angular Momentum
    • 8.2.1 Transforming to Spherical Coordinates
  • 8.3 Field-Atom Interaction–A Simple Introduction
    • 8.3.1 Magnetic Field and Zeeman Effect
  • A Infinitesimal Translation
  • B Infinitesimal Rotation
  • C Derivation of Commutation Relations
• 9 Spin
  • 9.1 Introduction
  • 9.2 Spin Operators
  • 9.3 The Bloch Sphere
  • 9.4 Spinor
  • 9.5 Pauli Equation
    • 9.5.1 Splitting of Degenerate Energy Level
  • 9.6 Spintronics
• 10 Identical Particles
  • 10.1 Introduction
  • 10.2 Pauli Exclusion Principle
  • 10.3 Exchange Energy
  • 10.4 Extension to More Than Two Particles
  • 10.5 Counting the Number of Basis States
  • 10.6 Examples
  • 10.7 Thermal Distribution Functions
  • A Quantum Statistical Mechanics
    • A1 Distinguishable Particles iv Quantum Mechanics Made Simple
    • A2 Identical Fermions
    • A3 Identical Bosons
  • B Most Probable Configuration
    • B1 Distinguishable Particles
    • B2 Identical Fermions
    • B3 Identical Bosons
  • C The Meaning of α and β
• 11 Density Matrix
  • 11.1 Pure and Mixed States
  • 11.2 Density Operator
  • 11.3 Time Evolution of the Matrix Element of an Operator
  • 11.4 Two-Level Quantum Systems
    • 11.4.1 Interaction of Light with Two-Level Systems
• 12 Quantization of Classical Fields
  • 12.1 Introduction
  • 12.2 The Quantum Harmonic Oscillator Revisited
    • 12.2.1 Eigenfunction by the Ladder Approach
  • 12.3 Schr¨odinger Picture versus Heisenberg Picture
    • 12.3.1 Quantum Hamiltonian Mechanics
  • 12.4 Quantization of Waves on a Linear Atomic Chain–Phonons
    • 12.4.1 Quantization: A Direct Approach
    • 12.4.2 Quantization: Mode Decomposition Approach
  • 12.5 The Continuum Limit
    • 12.5.1 Quantization in the Continuum Limit
  • 12.6 Quantization of Electromagnetic Field
    • 12.6.1 Hamiltonian
    • 12.6.2 Multimode Case and Fock State
    • 12.6.3 One-Photon State
    • 12.6.4 Coherent State Revisited
  • A Thermal Light and Black-Body Radiation
  • B Vacuum Fluctuation and Casimir Force
• 13 Schr¨odinger Wave Fields
  • 13.1 Introduction
  • 13.2 Fock Space for Fermions
  • 13.3 Field Operators
  • 13.4 Similarity Transform
  • 13.5 Additive One-Particle Operator
    • 13.5.1 Three-Particle Case
  • 13.6 Additive Two-Particle Operator
  • 13.7 More on Field Operators
  • 13.8 Boson Wave Field
  • 13.9 Boson Field Operators Contents v
  • 13.10Additive One-Particle Operator
• 14 Interaction of Different Particles
  • 14.1 Introduction
  • 14.2 Interaction of Particles
  • 14.3 Time-Dependent Perturbation Theory
    • 14.3.1 Absorption
  • 14.4 Spontaneous Emission
  • 14.5 Stimulated Emission
  • 14.6 Multi-photon Case
  • 14.7 Total Spontaneous Emission Rate
  • 14.8 More on Atom-Field Interaction
    • 14.8.1 Interaction Picture
    • 14.8.2 E · r or A · p Interaction
• 15 Quantum Interpretation and Quantum Information
  • 15.1 Introduction
  • 15.2 Interpretation of Quantum Mechanics
  • 15.3 Entangled States
    • 15.3.1 EPR Paradox
  • 15.4 Bell’s Theorem
    • 15.4.1 Prediction by Quantum Mechanics
    • 15.4.2 Prediction by Hidden Variable Theory
  • 15.5 Quantum Cryptography
    • 15.5.1 A Simple Quantum Encryption Algorithm
    • 15.5.2 No-cloning Theorem
  • 15.6 Quantum Computing
    • 15.6.1 Quantum Bits (Qubits)
    • 15.6.2 Quantum Gates
    • 15.6.3 Quantum Computing Algorithms
  • 15.7 Quantum Teleportation
  • 15.8 A Final Word on Quantum Parallelism
  • A Derivation of the Clauser-Horne-Shimon-Holt Inequality

vi Quantum Mechanics Made Simple

viii Quantum Mechanics Made Simple

Acknowledgements

I like to thank Erhan Kudeki who encouraged me to teach this course, and for having many interesting discussions during its teaching. I acknowledge interesting discussions with Fuchun ZHANG, Guanhua CHEN, Jian WANG, and Hong GUO (McGill U) at Hong Kong U. I also like to thank many of my students and researchers who have helped type the notes and proofread them. They are Phil Atkins, Fatih Erden, Tian XIA, Palash Sarker, Jun HUANG, Qi DAI, Zuhui MA, Yumao WU, Min TANG, Yat-Hei LO, Bo ZHU, and Sheng SUN.

Chapter 1

Introduction

1.1 Introduction

Quantum mechanics (or quantum physics) is an important intellectual achievement of the 20th century. It is one of the more sophisticated fields in physics that has affected our understanding of nano-meter length scale systems important for chemistry, materials, optics, electronics, and quantum information. The existence of orbitals and energy levels in atoms can only be explained by quantum mechanics. Quantum mechanics can explain the behaviors of insulators, conductors, semi-conductors, and giant magneto-resistance. It can explain the quantization of light and its particle nature in addition to its wave nature (known as particle- wave duality). Quantum mechanics can also explain the radiation of hot body or black body, and its change of color with respect to temperature. It explains the presence of holes and the transport of holes and electrons in electronic devices. The first observation of something quantum was probably by Michael Faraday in 1838 on cathode rays: the fact that when electrons bombard a cathode, light is given out. The next quantum phenomenon was the black body radiation of Gustav Kirchhoff in 1859-1860. Ludwig Boltzmann conjectured that atomic energy levels should be discrete around 1877. Heinrich Hertz first observed the photo-electric effect in 1887 which was later explained by Einstein in 1905 using the particle nature of light. Examples of famous physicists that contribute to quantum mechanics were Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie, Max Born, Paul Dirac, Werner Heisenberg, Wolfgang Pauli, Erwin Schrdinger, Richard Feynman. Marx Planck proposed the Planck radiation law in 1900 that

I(ν) = 2 hν^3 c^2

ehν/kbT^ − 1

where h ≈ 6. 626 × 10 −^34 J · s is Planck constant, c is velocity of light, ν the frequency of light, kb = 1.38064852(79) × 10 −^23 J/K is Boltzmann constant, and T the temperature in Kelvin. The above radiation law is obtained by assuming that light forms quanta of energy each of which is given by hν when the quantum theory of light was not well understood. When the Planck constant becomes negligibly small, implying that the quantization of light

Introduction 3

mechanics is certainly giving rise to interest in quantum information, quantum communica- tion, quantum cryptography, and quantum computing. Moreover, quantum mechanics is also needed to understand the interaction of photons with materials in solar cells, as well as many topics in material science.

When two objects are placed close together, they experience a force called the Casimir force that can only be explained by quantum mechanics. (This force is there even when no electromagnetic field is present in the classical sense. Even at zero temperature, and “zero” field, a vacuum electromagnetic fluctuation exists, giving rise to attractive force between two neutral objects.) This is important for the understanding of micro/nano-electromechanical systems (M/NEMS). Moreover, the understanding of spins is important in spintronics, another emerging technology where giant magneto-resistance, tunneling magneto-resistance, and spin transfer torque are being used. It is obvious that the richness of quantum physics will greatly affect the future generation technologies in many aspects.

1.2 Quantum Mechanics is Bizarre

The development of quantum mechanics is a great intellectual achievement, but at the same time, it is bizarre. The reason is that quantum mechanics is quite different from classical physics. The development of quantum mechanics is likened to watching two players having a game of chess, but the observers have not a clue as to what the rules of the game are. By observations, and conjectures, finally the rules of the game are outlined. Often, equations are conjectured like conjurors pulling tricks out of a hat to match experimental observations. It is the interpretations of these equations that can be quite bizarre.

Quantum mechanics equations were postulated to explain experimental observations, but the deeper meanings of the equations often confused even the most gifted. Even though Einstein received the Nobel prize for his work on the photo-electric effect that confirmed that light energy is quantized, he himself was not totally at ease with the development of quantum mechanics as charted by the younger physicists. He was never comfortable with the probabilistic interpretation of quantum mechanics by Born and the Heisenberg uncertainty principle: “God doesn’t play dice,” was his statement assailing the probabilistic interpreta- tion. He proposed “hidden variables” to explain the random nature of many experimental observations. He was thought of as the “old fool” by the younger physicists during his time.

Schr¨odinger came up with the bizarre “Schr¨odinger cat paradox” that showed the struggle that physicists had with quantum mechanics’ interpretation. But with today’s understanding of quantum mechanics, the paradox is a thing of yesteryear.

The latest twist to the interpretation in quantum mechanics is the parallel universe view that explains the multitude of outcomes of the prediction of quantum mechanics. All outcomes are possible, but with each outcome occurring in different universes that exist in parallel with respect to each other ...

4 Quantum Mechanics Made Simple

1.3 The Wave Nature of a Particle—Wave-Particle Du-

ality

The quantized nature of the energy of light was first proposed by Planck in 1900 to successfully explain the black body radiation. Einstein’s explanation of the photoelectric effect (1905) further asserts the quantized nature of light, or light as a photon with a packet of energy given by^1

E = ℏω (1.3.1)

where ℏ = h/(2π) is the reduced Planck constant or Dirac constant, and h is the Planck’s constant, given by

h ≈ 6. 626 × 10 −^34 Joule · second (1.3.2)

However, it is well known that light is a wave since it can be shown to interfere as waves in the Newton ring experiment as far back as 1717. Hence, light exhibits wave-particle duality. Electron was first identified as a particle by Thomson (1897). From the fact that a photon has energy E = ℏω and that it also has an energy related to its momentum p by E = pc where c is the velocity of light,^2 De Broglie (1924) hypothesized that the wavelength of an electron, when it behaves like a wave, is

λ =

h p

where p is the electron momentum.^3 The wave nature of an electron is revealed by the fact that when electrons pass through a crystal, they produce a diffraction pattern. That can only be explained by the wave nature of an electron. This experiment was done by Davisson and Germer in 1927.^4 When an electron manifests as a wave, it is described by

ψ(z) ∝ exp(ikz) (1.3.4)

where k = 2π/λ. Such a wave is a solution to^5

∂^2 ∂z^2

ψ = −k^2 ψ (1.3.5)

(^1) In the photoelectric effect, it was observed that electrons can be knocked off a piece of metal only if the light exceeded a certain frequency. Above that frequency, the electron gained some kinetic energy proportional to the excess frequency. Einstein then concluded that a packet of energy was associated with a photon that is proportional to its frequency. (^2) This follows from Einstein theory of relativity that says the E (^2) − (pc) (^2) = (mc (^2) ) (^2) where m is the rest mass of the particle. Photon has zero rest mass. When p = 0, this is the famous E = mc^2 formula. (^3) Typical electron wavelengths are of the order of nanometers. Compared to 400 nm of wavelength of blue light, they are much smaller. Energetic electrons can have even smaller wavelengths. Hence, electron waves can be used to make electron microscope whose resolution is much higher than optical microscope. (^4) Young’s double slit experiment was conducted in early 1800s to demonstrate the wave nature of photons. Due to the short wavelengths of electrons, it was not demonstrated it until 2002 by Jonsson. But it has been used as a thought experiment by Feynman in his lectures. (^5) The wavefunction can be thought of as a “halo” that an electron carries that determine its underlying physical properties and how it interact with other systems.

6 Quantum Mechanics Made Simple

Mind you, in the above, the frequency is not unique. We know that in classical physics, the potential V is not unique, and we can add a constant to it, and yet, the physics of the problem does not change. So, we can add a constant to both sides of the time-independent Schr¨odinger equation (1.3.11), and yet, the physics should not change. Then the total E on the right-hand side would change, and that would change the frequency we have arrived at in the time-dependent Schr¨odinger equation. We will explain how to resolve this dilemma later on. Just like potentials, in quantum mechanics, it is the difference of frequencies that matters in the final comparison with experiments, not the absolute frequencies. The setting during which Schr¨odinger equation was postulated was replete with knowledge of classical mechanics. It will be prudent to review some classical mechanics knowledge next.

Chapter 2

Classical Mechanics and Some

Mathematical Preliminaries

2.1 Introduction

Quantum mechanics cannot be derived from classical mechanics, but classical mechanics can inspire quantum mechanics. Quantum mechanics is richer and more sophisticated than classical mechanics. Quantum mechanics was developed during the period when physicists had rich knowledge of classical mechanics. In order to better understand how quantum mechanics was developed in this environment, it is better to understand some fundamental concepts in classical mechanics. Classical mechanics can be considered as a special case of quantum mechanics. We will review some classical mechanics concepts here. In classical mechanics, a particle moving in the presence of potential^1 V (q) will experience a force given by

F (q) = −

dV (q) dq

where q represents the coordinate or the position of the particle. Hence, the particle can be described by the equations of motion

dp dt

= F (q) = −

dV (q) dq

dq dt

= p/m (2.1.2)

For example, when a particle is attached to a spring and moves along a frictionless surface, the force the particle experiences is F (q) = −kq where k is the spring constant. Then the equations of motion of this particle are

dp dt

= ˙p = −kq,

dq dt

= ˙q = p/m (2.1.3)

(^1) The potential here refers to potential energy.

Classical Mechanics and Some Mathematical Preliminaries 9

Assuming that q(t 1 ) and q(t 2 ) are fixed, then the function q(t) and ˙q(t) that varies between t 1 and t 2 should minimize S, the action. In other words, a first order perturbation in q from the optimal answer that minimizes S should give rise to second order error in S. Hence, taking the first variation of (2.2.2), we have

δS = δ

∫ (^) t 2

t 1

L( ˙q, q)dt =

∫ (^) t 2

t 1

L( ˙q + δ q, q˙ + δq)dt −

∫ (^) t 2

t 1

L( ˙q, q)dt

∫ (^) t 2

t 1

δL( ˙q, q)dt =

∫ (^) t 2

t 1

δ q ˙

∂L

∂ q˙

• δq

∂L

∂q

dt = 0 (2.2.3)

In order to take the variation into the integrand, we have to assume that δL( ˙q, q) is taken with constant time. At constant time, ˙q and q are independent variables; hence, the partial derivatives in the next equality above follow. Using integration by parts on the first term, we have

δS = δq

∂L

∂ q˙

t 2

t 1

∫ (^) t 2

t 1

δq

d dt

∂L

∂ q˙

dt +

∫ (^) t 2

t 1

δq

∂L

∂q

dt

∫ (^) t 2

t 1

δq

[

d dt

∂L

∂ q˙

∂L

∂q

]

dt = 0 (2.2.4)

The first term vanishes because δq(t 1 ) = δq(t 2 ) = 0 because q(t 1 ) and q(t 2 ) are fixed. Since δq(t) is arbitrary between t 1 and t 2 , we must have

d dt

∂L

∂ q˙

∂L

∂q

The above is called the Lagrange equation, from which the equation of motion of a particle can be derived. The derivative of the Lagrangian with respect to the velocity ˙q is the momentum

p =

∂L

∂ q˙

The derivative of the Lagrangian with respect to the coordinate q is the force. Hence

F =

∂L

∂q

The above equation of motion is then

p˙ = F (2.2.8)

Equation (2.2.6) can be inverted to express ˙q as a function of p and q, namely

q˙ = f (p, q) (2.2.9)

The above two equations can be solved in tandem to find the time evolution of p and q.

10 Quantum Mechanics Made Simple

For example, the kinetic energy T of a particle is given by

T =

m q˙^2 (2.2.10)

Then from (2.2.1), and the fact that V is independent of ˙q,

p =

∂L

∂ q˙

∂T

∂ q˙ = m q˙ (2.2.11)

or

q˙ =

p m

Also, from (2.2.1), (2.2.7), and (2.2.8), we have

p˙ = −

∂V

∂q

The above pair, (2.2.12) and (2.2.13), form the equations of motion for this problem. The above can be generalized to multidimensional problems. For example, for a one particle system in three dimensions, qi has three degrees of freedom, and i = 1, 2 , 3. (The qi can represent x, y, z in Cartesian coordinates, but r, θ, φ in spherical coordinates.) But for N particles in three dimensions, there are 3N degrees of freedom, and i = 1,... , 3 N. The formulation can also be applied to particles constraint in motion. For instance, for N particles in three dimensions, qi may run from i = 1,... , 3 N − k, representing k constraints on the motion of the particles. This can happen, for example, if the particles are constraint to move in a manifold (surface), or a line (ring) embedded in a three dimensional space. Going through similar derivation, we arrive at the equation of motion

d dt

∂L

∂ q˙i

∂L

∂qi

In general, qi may not have a dimension of length, and it is called the generalized coordinate (also called conjugate coordinate). Also, ˙qi may not have a dimension of velocity, and it is called the generalized velocity. The derivative of the Lagrangian with respect to the generalized velocity is the generalized momentum (also called conjugate momentum), namely,

pi =

∂L

∂ q˙i

The generalized momentum may not have a dimension of momentum. Hence, the equation of motion (2.2.14) can be written as

p˙i =

∂L

∂qi

Equation (2.2.15) can be inverted to yield an equation for q˙i as a function of the other variables. This equation can be used in tandem (2.2.16) as time-marching equations of motion.