Quantum Mechanics Simplified: Concepts & Applications, Lecture notes of Quantum Mechanics

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Quantum Mechanics Made Simple:

Lecture Notes

Weng Cho CHEW^1

September 23, 2013

(^1) The author is with U of Illinois, Urbana-Champaign. He works part time at Hong Kong U this

summer.

• 1 Introduction Acknowledgements vii
  • 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
  • 2.3 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 Wavefunction
  • 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
• 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 *Identity Operator in a Continuum Space
  • 5.4 *Changing Between Representations
    • 5.4.1 Momentum Operator
    • 5.4.2 Position Operator
    • 5.4.3 The Coordinate Basis Function
  • 5.5 Commutation of Operators
  • 5.6 Expectation Value and Eigenvalue of Operators
  • 5.7 *Generalized Uncertainty Principle
  • 5.8 *Time Evolution of the Expectation Value of an Operator
  • 5.9 Periodic Boundary Condition
• 6 Approximate Methods in Quantum Mechanics
  • 6.1 Introduction
  • 6.2 Use of an Approximate Subspace
  • 6.3 *Time Independent Perturbation Theory
    • 6.3.1 First Order Perturbation
    • 6.3.2 Second Order Perturbation
    • 6.3.3 Higher Order Corrections
  • 6.4 Tight Binding Model
    • 6.4.1 Variational Method
  • 6.5 Time Dependent Perturbation Theory
• 7 Quantum Mechanics in Crystals
  • 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 Contents iii
    • 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 Fermi Level and Fermi Energy
    • 7.7.2 DOS in a Quantum Well
    • 7.7.3 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
• 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
• 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 iv Quantum Mechanics Made Simple
  • 12.1 Introduction
  • 12.2 The Quantum Harmonic Oscillator Revisited
    • 12.2.1 Eigenfunction by the Ladder Approach
  • 12.3 Quantization of Waves on a Linear Atomic Chain–Phonons
  • 12.4 Schr¨odinger Picture versus Heisenberg Picture
  • 12.5 The Continuum Limit
  • 12.6 Quantization of Electromagnetic Field
    • 12.6.1 Hamiltonian
    • 12.6.2 Field Operators
    • 12.6.3 Multimode Case and Fock State
    • 12.6.4 One-Photon State
    • 12.6.5 Coherent State Revisited
  • 12.7 Thermal Light
• 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
  • 13.10Additive One-Particle Operator
  • 13.11The Difference between Boson Field and Photon Field
• 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 Information and Quantum Interpretation Contents v
  • 15.1 Introduction
  • 15.2 Quantum Cryptography
    • 15.2.1 No-cloning Theorem
    • 15.2.2 Entangled States
    • 15.2.3 A Simple Quantum Encryption Algorithm
  • 15.3 Quantum Computing
    • 15.3.1 Quantum Bits (Qubits)
    • 15.3.2 Quantum Gates
    • 15.3.3 Quantum Computing Algorithms
  • 15.4 Quantum Teleportation
  • 15.5 Interpretation of Quantum Mechanics
  • 15.6 EPR Paradox
  • 15.7 Bell’s Theorem
    • 15.7.1 Prediction by Quantum Mechanics
    • 15.7.2 Prediction by Hidden Variable Theory
  • 15.8 A Final Word on Quantum Parallelism
• A Gaussian Wave Packet
  • A.1 Introduction
  • A.2 Derivation from the Wave Equation
  • A.3 Physical Interpretation
  • A.4 Stability of the Plane Wave Solution
• B Generators of Translator and Rotation
  • B.1 Infinitesimal Translation
  • B.2 Infinitesimal Rotation
  • B.3 Derivation of Commutation Relations
• C Quantum Statistical Mechanics
  • C.1 Introduction
    • C.1.1 Distinguishable Particles
    • C.1.2 Identical Fermions
    • C.1.3 Identical Bosons
  • C.2 Most Probable Configuration
    • C.2.1 Distinguishable Particles
    • C.2.2 Identical Fermions
    • C.2.3 Identical Bosons
  • C.3 The Meaning of α and β
• D Photon Polarization
  • D.1 Introduction
  • D.2 2D Quantum Harmonic Oscillator
  • D.3 Single–Photon State

vi Quantum Mechanics Made Simple

viii Quantum Mechanics Made Simple

Chapter 1

Introduction

1.1 Introduction

Quantum mechanics 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. Quantum mechanics has played an important role in photonics, quantum electronics, nano- and micro-electronics, nano- and quantum optics, quantum computing, quantum communi- cation and crytography, solar and thermo-electricity, nano-electromechacnical systems, etc. Many emerging technologies require the understanding of quantum mechanics; and hence, it is important that scientists and engineers understand quantum mechanics better. In nano-technologies due to the recent advent of nano-fabrication techniques. Conse- quently, nano-meter size systems are more common place. In electronics, as transistor devices become smaller, how the electrons behave in the device is quite different from when the devices are bigger: nano-electronic transport is quite different from micro-electronic transport. The quantization of electromagnetic field is important in the area of nano-optics and quantum optics. It explains how photons interact with atomic systems or materials. It also allows the use of electromagnetic or optical field to carry quantum information. Quantum 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 is important for the understanding

Introduction 3

light is a wave since it can be shown to interfere as waves in the Newton ring experiment as far back as 1717. 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.^2 De Broglie hypothesized that the wavelength of an electron, when it behaves like a wave, is

λ =

h p

where h is the Planck’s constant, p is the electron momentum,^3 and

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

When an electron manifests as a wave, it is described by

ψ(z) ∝ exp(ikz) (1.3.3)

where k = 2π/λ. Such a wave is a solution to^4

∂^2 ∂z^2

ψ = −k^2 ψ (1.3.4)

A generalization of this to three dimensions yields

∇^2 ψ(r) = −k^2 ψ(r) (1.3.5)

We can define

p = ℏk (1.3.6)

where ℏ = h/(2π).^5 Consequently, we arrive at an equation

ℏ^2

2 m 0

∇^2 ψ(r) =

p^2 2 m 0

ψ(r) (1.3.7)

where

m 0 ≈ 9. 11 × 10 −^31 kg (1.3.8)

The expression p^2 /(2m 0 ) is the kinetic energy of an electron. Hence, the above can be considered an energy conservation equation.

(^2) 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. (^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) 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. (^5) This is also called Dirac constant sometimes.

4 Quantum Mechanics Made Simple

The Schr¨odinger equation is motivated by further energy balance that total energy is equal to the sum of potential energy and kinetic energy. Defining the potential energy to be V (r), the energy balance equation becomes [ −

ℏ^2

2 m 0

∇^2 + V (r)

]

ψ(r) = Eψ(r) (1.3.9)

where E is the total energy of the system. The above is the time-independent Schr¨odinger equation. The ad hoc manner at which the above equation is arrived at usually bothers a beginner in the field. However, it predicts many experimental outcomes. It particular, it predicts the energy levels and orbitals of a trapped electron in a hydrogen atom with resounding success. One can further modify the above equation in an ad hoc manner by noticing that other experimental finding shows that the energy of a photon is E = ℏω. Hence, if we let

iℏ

∂t

Ψ(r, t) = EΨ(r, t) (1.3.10)

then

Ψ(r, t) = e−iωtψ(r, t) (1.3.11)

Then we arrive at the time-dependent Schr¨odinger equation: [ −

ℏ^2

2 m 0

∇^2 + V (r)

]

ψ(r, t) = iℏ

∂t

ψ(r, t) (1.3.12)

Another disquieting fact about the above equation is that it is written in terms of complex functions and numbers. In our prior experience with classical laws, they can all be written in real functions and numbers. We will later learn the reason for this. 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 (1.3.10), 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.

6 Quantum Mechanics Made Simple

q

V(q)

q

Figure 2.1: The left side shows a potential well in which a particle can be trapped. The right side shows a particle attached to a spring. The particle is subject to the force due to the spring, but it can also be described by the force due to a potential well.

Given p and q at some initial time t 0 , one can integrate (2.1.2) or (2.1.3) to obtain p and q for all later time. A numerical analyst can think of that (2.1.2) or (2.1.3) can be solved by the finite difference method, where time-stepping can be used to find p and q for all later times. For instance, we can write the equations of motion more compactly as

du dt

= f (u) (2.1.4)

where u = [p, q]t, and f is a general vector function of u. It can be nonlinear or linear; in the event if it is linear, then f (u) = A · u. Using finite difference approximation, we can rewrite the above as u(t + ∆t) − u(t) = ∆. tf (u(t)), u(t + ∆t) = ∆. tf (u(t)) + u(t) (2.1.5)

The above can be used for time marching to derive the future values of u from past values. The above equations of motion are essentially derived using Newton’s law. However, there exist other methods of deriving these equations of motion. Notice that only two variables p and q are sufficient to describe the state of a particle.

2.2 Lagrangian Formulation

Another way to derive the equations of motion for classical mechanics is via the use of the Lagrangian and the principle of least action. A Lagrangian is usually defined as the difference between the kinetic energy and the potential energy, i.e.,

L( ˙q, q) = T − V (2.2.1)

where ˙q is the velocity. For a fixed t, q and ˙q are independent variables, since ˙q cannot be derived from q if it is only known at one given t. The equations of motion are derived from the principle of least action which says that q(t) that satisfies the equations of motion between two times t 1 and t 2 should minimize the action integral

S =

∫ (^) t 2

t 1

L( ˙q(t), q(t))dt (2.2.2)

Classical Mechanics and Some Mathematical Preliminaries 7

Assuming that q(t 1 ) and q(t 2 ) are fixed, then the function q(t) 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˙

dt +

∂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.

Classical Mechanics and Some Mathematical Preliminaries 9

2.3 Hamiltonian Formulation

For a multi-dimensional system, or a many particle system in multi-dimensions, the total time derivative of L is

dL dt

i

∂L

∂qi

q ˙i +

∂L

∂ q˙i

q ¨i

Since ∂L/∂qi = (^) ddt (∂L/∂ q˙i) from the Lagrange equation, we have

dL dt

i

[

d dt

∂L

∂ q˙i

q ˙i +

∂L

∂ q˙i

q ¨i

]

d dt

i

∂L

∂ q˙i

q ˙i

or

d dt

i

∂L

∂ q˙i

q ˙i − L

The quantity

H =

i

∂L

∂ q˙i q ˙i − L (2.3.4)

is known as the Hamiltonian of the system, and is a constant of motion, namely, dH/dt = 0. As shall be shown, the Hamiltonian represents the total energy of a system. It is a constant of motion because of the conservation of energy. The Hamiltonian of the system, (2.3.4), can also be written, after using (2.2.15), as

H =

i

q ˙ipi − L (2.3.5)

where pi = ∂L/∂ q˙i is the generalized momentum. The first term has a dimension of energy, and in Cartesian coordinates, for a simple particle motion, it is easily seen that it is twice the kinetic energy. Hence, the above indicates that the Hamiltonian

H = T + V (2.3.6)

The total variation of the Hamiltonian is

δH = δ

i

pi q˙i

− δL

i

( ˙qiδpi + piδ q˙i) −

i

∂L

∂qi

δqi +

∂L

∂ q˙i

δ q˙i

Using (2.2.15) and (2.2.16), we have

δH =

i

( ˙qiδpi + piδ q˙i) −

i

( ˙piδqi + piδ q˙i)

i

( ˙qiδpi − p˙iδqi) (2.3.8)

10 Quantum Mechanics Made Simple

From the above, since the first variation of the Hamiltonian depends only on δpi and δqi, we gather that the Hamiltonian is a function of pi and qi. Taking the first variation of the Hamiltonian with respect to these variables, we arrive at another expression for its first variation, namely,

δH =

i

∂H

∂pi

δpi +

∂H

∂qi

δqi

Comparing the above with (2.3.8), we gather that

q˙i =

∂H

∂pi

p˙i = −

∂H

∂qi

These are the equations of motion known as the Hamiltonian equations. The (2.3.4) is also known as the Legendre transformation. The original function L is a function of ˙qi, qi. Hence, δL depends on both δ q˙i and δqi. After the Legendre transformation, δH depends on the differential δpi and δqi as indicated by (2.3.8). This implies that H is a function of pi and qi. The equations of motion then can be written as in (2.3.10) and (2.3.11).

2.4 More on Hamiltonian

The Hamiltonian of a particle in classical mechanics is given by (2.3.6), and it is a function of pi and qi. For a non-relativistic particle in three dimensions, the kinetic energy

T = p · p 2 m

and the potential energy V is a function of q. Hence, the Hamiltonian can be expressed as

H =

p · p 2 m

• V (q) (2.4.2)

in three dimensions. When an electromagnetic field is present, the Hamiltonian for an electron can be derived by letting the generalized momentum

pi = m q˙i + eAi (2.4.3)

where e = −|e| is the electron charge and Ai is component of the vector potential A. Conse- quently, the Hamiltonian of an electron in the presence of an electromagnetic field is

H =

(p − eA) · (p − eA) 2 m

• eφ(q) (2.4.4)

The equation of motion of an electron in an electromagnetic field is governed by the Lorentz force law, which can be derived from the above Hamiltonian using the equations of motion provided by (2.3.10) and (2.3.11).