Introductory Quantum Mechanics Textbook, Lecture notes of Quantum Mechanics

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Quantum mechanics is the centerpiece of modern physics. It

is the foundation of modern technologies benefiting everyday life

of everybody. For many decades, learning introductory quan-

tum mechanics was hampered by unintelligible paradoxes. How-

ever, the advanced quantum physics, quantum field theory, is free

from paradoxes. This is because in quantum field theory, both

electrons and radiation are treated as continuous fields, while

in some introductory quantum mechanics textbooks, electrons

are treated as fictitious material points. The split-personality of

electrons and radiation is the origin of paradoxes.

Starting from the modern all-fields view that both electrons

and radiation are continuous fields, this introductory quantum

mechanics textbook is logically consistent and covers all essen-

tial applications with no paradoxes. It is designed for freshman

and sophomore students majoring in physics, chemistry, solid

state electronics, materials science, and molecular biology. The

minimal mathematical prerequisite is the high-school advanced-

placement calculus or equivalent. It also provides a conceptual

bridge to quantum field theory. To improve intuitiveness, two-

colored graphs are used to show wavefunctions.

This textbook can be used for a one-semester course or a two-

semester course. For students already learned classical physics

and partial differential equations but not interested in advanced

topics, Chapters 2 through 7 fits a one-semester course. A more

complete course takes two semesters.

C. Julian Chen joined IBM T.J. Watson Research Center in

1985 as a Research Staff Member in the Department of Physical

Sciences, doing experimental and theoretical research on scan-

ning tunneling microscopy. He authored Introduction to Scan-

ning Tunneling Microscopy (Oxford University Press 1993, 2007,

2021), a standard reference book in nanoscience. In 2007, he

joined the Department of Applied Physics and Applied Mathe-

matics of Columbia University. For twelve consecutive years, he

teaches a graduate-level course Physics of Solar Energy, explain-

ing the quantum physics of solar cells and photosynthesis.

ii Preface

picometer resolution with negligible disturbance. Even the nodal structures inside the molecular wavefunctions are observed and mapped. Those ex- periments proved that wavefunctions are physical fields as designated by quantum field theory. No point-like electrons were observed.^2 Motivated by the experimental discovery and the need of my students to learn quantum mechanics without paradoxes, I tried to teach quan- tum mechanics based on the all-field concept for several years. This book manuscript is based on the lecture notes. My teaching experience indicated that the all-field concept has additional benefits. The difficult mathemati- cal prerequisites, such as Hilbert space, Lagrangian–Hamiltonian mechanics, and probability theory, can be avoided. Furthermore, by teaching partial differential equations during the early lectures in classical physics, espe- cially acoustics and electromagnetics, the mathematical prerequisite can be as low as high-school advanced-placement calculus. Therefore, it can be a freshman-sophomore course for students with less mathematical train- ing. Nevertheless, it provides many topics in quantum mechanics sufficient for applications in condensed-matter physics, materials science, solid-state electronics, all branches of chemistry, and molecular biology. A conceptual bridge to quantum field theory is also established. To improve intuitiveness, two-color graphics is extensively used to rep- resent wavefunctions. According to the Wigner theorem, if the Hamiltonian is time-reversal invariant, all wavefunctions are real. By displaying wave- functions with two colors, for example, red or blue as positive or negative, intuitiveness is greatly improved. Color graphics also improves the under- standing of many other physical concepts.

Following is a brief summary of the Chapters. Chapter 1, A Review of Classical Physics. In Section 1.1, Newtonian me- chanics is presented for an easy transition to quantum mechanics. Neverthe- less, the na¨ıvet´e of the material-points concept is emphasized to avoid mis- understandings. Section 1.2 presents wave phenomena, including standing waves, eigenvalues, eigenfunctions, nodal structures, orthogonality, and su- perposition. For students with no exposure to partial differential equations, this section is a teaching-by-example introduction. Section 1.3 presents Maxwell’s theory that light as an electromagnetic wave. The concept of po- larization is presented in detail to prepare for a correct explanation of the electron spin. In some textbooks, spin is presented as a property of an single electron designated as a material point, that causes gross misunderstand- ings. In fact, electron spin is equivalent to polarization of light. Finally, starting with a simple proof of the Euler formula eix^ = cos x + i sin x, the powerfulness of complex numbers in treating wave phenomena is presented with examples in planar geometry and electromagnetism.

(^2) C. J. Chen, Introduction to Scanning Tunneling Microscopy, Third Edition, Oxford University Press 2021. Chapter 8, Imaging Wavefunctions.

iii

Chapter 2, Wave and Quantum. From the beginning, the correct con- cept in quantum field theory that electron is a continuous physical field, like electromagnetic fields, is emphasized. The terms electron and photon mean that energy, mass, and electrical charge are quantized when inter- acting with other fields. There are no material points. The chapter starts with Einstein’s theory of quantization of light: when light is generated or converted into other forms, energy is quantized. In free space, light is an electromagnetic wave, not a spray of geometrical points. Similarly, accord- ing to quantum field theory, electron is always a field, not a material point. Accordingly, Compton effect is presented in terms of waves. Finally, black- body radiation is presected in details, including Einstein’s 1916 derivation, that shows the true mesaning of quantization. Chapter 3, The Static Schr¨odinger Equation. A simple and intuitive derivation of Schr¨odinger’s equation is presented based on the de Broglie postulate and the classical energy integral. Solutions of the harmonic oscil- lator and the hydrogen atom are presented in real variables. The solutions of the harmonic oscillator are presented using creation and annihilation opera- tors in real variables. It is intuitive and very simple. Quantization of bosons in the occupation-number representation, or the Fock space, is presented to provide a bridge to quantum field theory.

Chapter 4, Many-Electron Systems. Many-electron wavefunctions and the Slater determinants are introduced based on Pauli exclusion principle and the electron spin. The Hartree-Fock method and the density func- tional theory are presented. Quantization of fermions, formulated by the anti-commutative creation and annihilation operators in the Fock space, is presented. It is intuitive, and the mathematics is simple. It provides another bridge to quantum field theory.

Chapter 5, The Chemical Bond. Four types of chemical bonds are de- scribed: the van der Waals bond, the covalent bond, the ionic bond, and the hydrogen bond. A perturbation theory of the covalent bond is described in detail. As an example, the hydrogen molecular ion is treated in detail analytically. The homonucleus diatomic molecules are presented. Chapter 6, The Dynamic Schr¨odinger Equation. By presenting the wave- function as a two-component real field, the dynamic Schr¨odinger equation is derived from de Broglie’s postulate and the classical energy integral. The two-component real wavefunction is invariant under an SO(2) group, which is equivalent to SU(1). The complex wavefunction is introduced with a proof of gauge invariance. The operator algebra initially presented in Chapter 3 for the harmonic oscillator is extended to complex wavefunctions. The ap- plications to angular momentum and Pauli’s derivation of energy levels of hydrogen atom are presented. Finally, the wavepacket as a macroscopic particle and the Ehrenfest theorem are presented. Chapter 7, Perturbation Theories. Both static and dynamic perturba- tion theories are presented through examples. The interaction of radiation

Contents

Chapter 3

The Static Schr¨odinger Equation

The year 1905 was Albert Einstein’s annum mirabilis when he published four papers on Annalen der Physik that literally started the modern physics. Similarly, the year 1926 was Erwin Schr¨odinger’s annum mirabilis with six papers published on the same journal Annalen der Physik that defined non- relativistic quantum mechanics. Thus commented Paul Dirac in 1929: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known”. Those papers, belonging to the defining publications of modern science, are still worth reading. Here is a brief summary of the important ones: The first paper, Quantization as an Eigenvalue Problem, Part I, re- ceived by Annalen der Physik on January 27, 1926, defined a wavefunction ψ, which is “everywhere real, single-valued, finite, and continuously differ- entiable up to the second order”. A differential equation of ψ is presented as a variation of the Hamilton-Jacobi equation. By applying that equation to the hydrogen atom, the Rydberg formula was explained. The second paper, Quantization as an Eigenvalue Problem, Part II, pre- sented a parallelism of the relation of quantum mechanics and classical mechanics with the relation of wave optics and geometrical optics. He em- phasized that the wave nature of electrons is fundamental, and the particle view is a macroscopic approximation. He also presented an even simpler way to introduce the differential equation of the wavefunction based on de

Fig. 3.1. Austrian banknote with a portrait of Schr¨odinger. It is a rare honor for a scientist to have a portrait printed on a banknote. Note the large value.

90 The Static Schr¨odinger Equation

Broglie wave. Three further problems were treated: the harmonic oscillator, rigid rotor and non-rigid rotor, all related to molecular physics. The dynamic Schr¨odinger’s equation was introduced in the sixth paper, received by Annalen der Physik on June 23, 1926. By treating radiation as classical electromagnetic waves, its effect to atomic systems was resolved as a perturbation. Tunneling at an atomic scale is also resolved. We will introduce Schr¨odinger’s equations in three stages. In this Chap- ter, the static Schr¨odinger’s equation for a single electron is introduced. In Chapter 4, the many-electron version is introduced. In Chapter 6, the dynamic Schr¨odinger’s equation is introduced.

3.1 The static Schr¨odinger equation

The static Schr¨odinger equation can be derived by applying the de Broglie postulate to the classical energy integral, Eq. 1.11,

p^2 2 me

• V (r) = E. (3.1)

According to de Broglie, the electron is a field. A bound electron is similar to a standing wave ψ(r), satisfying the Helmhotz equation, Eq. 1.120,

∇^2 ψ(r) + k^2 ψ(r) = 0. (3.2)

The wave vector is then

k^2 = −

∇^2 ψ(r) ψ(r)

According to the de Broglie’s relation, Eq. 2.24,

p = ℏk. (3.4)

Combining Eq. 3.3 and Eq. 3.4, one finds

p^2 = −

ℏ^2 ∇^2 ψ(r) ψ(r)

Insert Eq. 3.5 into Eq. 3.1, multiply both sides by ψ(r), a differential equa- tion for wavefunction ψ(r) is obtained,

ℏ^2

2 me

∇^2 ψ(r) + V (r)ψ(r) = Eψ(r). (3.6)

This is the the static Schr¨odinger equation.

92 The Static Schr¨odinger Equation

Fig. 3.3. Energy levels in a one-dimensional potential well. The wavefunctions are labeled by quantum number n. The wavefunctions are similar to the vibration of string, see Fig. 1.10. The energy eigenvalue is proportional to n^2.

where n = 1, 2 , 3 ,... is an integer. The wavefunctions are

ψn(x) = C sin

( (^) nπx L

The constant C will be determined shortly. Figure 3.2 shows the wavefunc- tions. It is similar to the sound waves in a bugle, see Fig. 1.17. In classical mechanics, as in Eq. 3.1, energy can take any value. In quantum mechanics, because of condition Eq. 3.12, energy is quantized : it can only take discrete values determined by Eqs. 3.8 and 3.12:

En = n^2 E 1 =

n^2 π^2 ℏ^2 2 meL^2

Those allowed values of energy are called the energy eigenvalues, adapted from German, the proper values of energy. Because the Schr¨odinger equation is linear to the wavefunction, the con- stant C does not affect the determination of energy eigenvalues. According to Schr¨odinger, the square of the wavefunction is proportional to the charge density distribution of the electron as a field in space:

ρ(x) = −eψ^2 (x). (3.15)

Because the total charge of an electron over the space equals to one elemen- tary charge −e, the integral over the entire space must equal to 1. In the current situation, the electron is confined in a well of width L, ∫ (^) L

0

ψ^2 (x)dx = 1. (3.16)

3.2 Wavefunctions in a potential well 93

The average value of the square of sine function over any number of half periods is 1/2. Therefore, for all quantum numbers, the constant is

C =

L

The wavefunctions are

ψn(x) =

L

sin nπx L

It is straightforward to show that eigenstates with different quantum numbers n are orthogonal. In fact, using Eq. 3.18,

∫ (^) L

0

dx ψn(x) ψm(x) = δnm, (3.19)

which is zero when n = m, and is 1 when n = m. Furthermore, the set of wavefunctions is complete. Any function f (x) in the interval [0, L] can be expanded as a sum of those wavefunctions,

f (x) =

∑^ ∞

n=

bnψn(x), (3.20)

with coefficients

bn =

∫ L

0

f (x) ψn(x) dx. (3.21)

This is a special case of the Fourier theorem, the proof can be found in any mathematics textbook with Fourier series. Nonetheless, if Eq. 3.20 is true, it is easy to prove that the expression of the coefficients, Eq. 3.21, is correct. In fact, because of the orthonormal relation Eq. 3.19,

∫ (^) L

0

ψn(x)dx

∑^ ∞

m=

bmψm(x) =

∑^ ∞

m=

δnmbm = bn. (3.22)

3.2.2 The Dirac notation

The notations of wavefunctions, Eq. 3.18, and the integrals, Eqs. 3.16 and 3.19, occurs very often in quantum mechanics. In the third edition of Prin- ciples of Quantum Mechanics, Dirac introduced the bra and ket notations, that greatly simplifies mathematical notations in quantum mechanics. In the real formulation of quantum mechanics, bra and ket are equivalent. A wavefunction can be denoted either as a bra or as a ket. For the case of electrons in a one-dimensional potential well, it is either

〈n| =

L

sin nπx L

3.3 The harmonic oscillator 95

one has δˆ = −δˆ†. (3.34)

The Schr¨odinger equation, Eq. 3.6, can be written as

Hˆ|ψ〉 = E|ψ〉 (3.35)

by defining an energy operator or a Hamiltonian,

Hˆ ≡ − ℏ

2 2 me

∇^2 + V (r). (3.36)

The energy operator is self-adjoint. The proof is left as an exercise.

3.3 The harmonic oscillator

In quantum mechanics, the harmonic oscillator is of fundamental impor- tance. It describes the oscillation of molecules and solids near its equilib- rium point. The electromagnetic wave can be decomposed into a number of simple harmonic oscillators. Using the quantization procedure presented here, the electromagnetic waves can be quantized. It is the basis of quantum electrodynamics, the complete theory of radiation and matter. From Eqs. 1.26 and 3.6, the Schr¨odinger equation for a one-dimensional harmonic oscillator is ( −

ℏ^2

2 m

d^2 dx^2

m 2 ω^2 x^2

ψ(x) = Eψ(x). (3.37)

By introducing a dimensionless coordinate q defined as

q ≡

mω ℏ

x, (3.38)

the Schr¨odinger equation Eq. 3.37 is simplified to

1 2

d^2 dq^2

• q^2

ℏωψ(q) = Eψ(q). (3.39)

3.3.1 Creation operator and annihilation operator

As we have presented in previous sections, the basic concept in quantum mechanics is the wavefunction, and it is governed by a partial differential equation, the Schr¨odinger equation. In some cases, the process of obtaining solutions of partial differential equations can be greatly simplified with dif- ferential operators using algebraic methods. This is especially true for the case of the harmonic oscillator.

96 The Static Schr¨odinger Equation

In order to find an algebraic solution of the harmonic oscillator, a pair of operators are introduced: an annihilation operator,

ˆa =

q +

d dq

and a creation operator ˆa†^ =

q −

d dq

From Eqs. 3.32 and 3.34, the creation operator is the adjoint of the annihi- lation operator. The meanings of these terms well be clarified soon. Those operators greatly simplify the solution of the harmonic oscillator problem, and are the basis of quantum field theory. By acting on any function f (q), a simple algebra shows that the two operators satisfy a commutation relation. One one hand, we have

ˆaˆa†f (q) =

q +

d dq

q −

d dq

f (q)

q^2 −

d^2 dq^2

d dq

q − q

d dq

f (q)

q^2 − d^2 dq^2

f (q).

On the other hand, we have

ˆa†ˆaf (q) =

q −

d dq

q +

d dq

f (q)

q^2 −

d^2 dq^2

d dq

q + q

d dq

f (q)

q^2 −

d^2 dq^2

f (q).

Here, the obvious identity is applied:

d dq

(q f (q)) = q d dq

f (q) + dq dq

f (q) = q d dq

f (q) + f (q). (3.44)

Combining Eqs. 3.42 and 3.43, we find the commutation relation

[ˆa, aˆ†] ≡ ˆaˆa†^ − ˆa†ˆa = 1. (3.45)

Through a simple algebra, the Schr¨odinger equation Eq. 3.39 becomes ( ˆa†ˆa +

ℏωψ(q) = Eψ(q). (3.46)

The verification is left as a Problem.