Quantum Mechanics Made Simple: Lecture Notes and Exercises, Exams of Quantum Mechanics

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Quantum Mechanics Made Simple:

Lecture Notes

Weng Cho CHEW^1

October 5, 2012

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

summer.

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 field in physics that has affected our understanding of nano-meter length scale systems important for chemistry, materials, optics, and electronics. 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. Quantum mechanics can also explain the radiation of hot 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, and micro-electronics. But many more emerging technologies require the understanding of quantum mechanics; and hence, it is important that scientists and engineers understand quantum mechanics better. One area is nano-technologies due to the recent advent of nano- fabrication techniques. Consequently, nano-meter size systems are more common place. In electronics, as transistor devices become smaller, how the electrons move through 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. 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 un- derstanding of micro/nano-electromechanical sensor systems (M/NEMS). Moreover, the un- derstanding of spins is important in spintronics, another emerging technology where giant magneto-resistance, tunneling magneto-resistance, and spin transfer torque are being used. Quantum mechanics is also giving rise to the areas of quantum information, quantum

Introduction 3

of an electron. This experiment was done by Davisson and Germer in 1927. 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,^2 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^3

∂^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).^4 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. 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)

(^2) 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. (^3) 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. (^4) This is also called Dirac constant sometimes.

4 Quantum Mechanics Made Simple

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, as well as predicting the existence of electron orbitals inside an atom, and how electron would interact with other particles. 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 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

Chapter 2

Classical Mechanics

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)

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 analysist can think of that (2.1.2) or (2.1.3) can be solved by the

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

Classical Mechanics 9

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

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 = 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 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. 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)

10 Quantum Mechanics Made Simple

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.

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