Basics of Quantum Mechanics

Basics of Quantum Mechanics

Basics of Quantum Mechanics

By

Bharat Saluja

Basics of Quantum Mechanics

Bharat Saluja

ISBN - 9789361525957

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Preface

Welcome to the realm of quantum mechanics—a fundamental domain that unveils the intriguing behaviors of particles at the quantum level. This introductory text is meticulously crafted to engage and guide undergraduate students in the United States through the intricate tapestry of quantum phenomena. As you embark on this intellectual journey, prepare to witness the departure from classical mechanics and embrace the paradigm shift that defines the physics of the very small.

Quantum mechanics is not merely a theoretical abstraction but a living, breathing framework that governs the behavior of particles at the subatomic scale. With a solid foundation in classical physics, you are poised to explore the principles that underpin the quantum world—principles that challenge our intuitive understanding of reality. From the wave-particle duality to the enigmatic phenomena of superposition and entanglement, this book systematically navigates through the mathematical formalism and conceptual richness that define quantum mechanics.

Each chapter is meticulously crafted to build upon the preceding knowledge, leading you through the intricacies of wave functions, operators, and the Schrödinger equation. The quest for understanding extends to the realms of quantum states, observables, and the probabilistic nature of measurement—a departure from the deterministic universe of classical physics.

This text is designed to not only elucidate the theoretical foundations of quantum mechanics but also to equip you with the analytical tools essential for solving problems and making predictions in this quantum landscape. As you delve into this transformative realm, may this book serve as your intellectual guide, unraveling the mysteries of quantum mechanics and paving the way for a profound comprehension of the quantum world.

Table of Contents

1. Introduction to Quantum Theory 1

1.1 History and Motivation for Quantum Mechanics 1

1.2 Failings of Classical Physics 2

1.3 Blackbody Radiation 3

1.4 Photoelectric Effect 7

1.5 Photoelectric Effect 8

1.6 Compton Scattering 8

1.7 Particle Wave Duality 9

Conclusion 11

References 11

2. Mathematical Foundations 12

2.1 Complex Numbers 12

2.2 Linear Algebra 12

2.3 Hilbert Spaces 14

2.4 Operators 14

2.5 Eigenvalue Equations 15

Conclusion 17

References 17

3. Schrödinger Equation 18

3.1 Derivation from Wave Mechanics 18

3.2 Probabilistic Interpretation 19

3.3 Eigenfunctions and Eigenvalues 21

3.4 Mathematical Properties 22

3.5 Time Dependence 23

Conclusion 24

References 25

4. Quantum Dynamics 26

4.1 Free Particle Solutions 26

4.2 Particle in a Box 27

4.3 Quantum Tunneling 28

4.4 Time-Energy Uncertainty Principle 29

4.5 Harmonic Oscillator 29

4.6 Tunneling 30

4.7 Numerical Approximation Methods 31

Conclusion 32

References 32

5. Three Dimensional Systems 34

5.1 Schrödinger Equation in Three Dimensions 34

5.2 Quantum Tunneling 35

5.3 Degeneracy 36

5.4 Hydrogen Atom Solutions 37

5.5 Angular Momentum 38

5.5 Spin 39

Conclusion 39

References 40

6. Mathematical Tools of Quantum

Mechanics 41

6.1 Dirac Notation 41

6.2 State Vectors and Bra-Ket Notation 41

6.3 Operators 42

6.4 Expectation Values 43

6.5 Commutators 44

Conclusion 45

References 45

7. Quantum Dynamics in

Three Dimensions 46

7.1 Adiabatic Approximations 46

7.2 Scattering States 47

7.3 Partial Wave Analysis 48

7.4 Symmetries 49

Conclusion 50

References 51

8. Spin Systems 52

8.1 Stern-Gerlach Experiment 52

8.2 Spin Operators 53

8.3 Addition of Angular Momentum 55

8.4 Identical Particles 55

8.5 Spin-Orbit Coupling 56

Conclusion 57

References 57

9. Approximation Methods 59

9.1 Variational Method 59

9.2 Perturbation Theory 60

9.3 WKB Approximation 61

9.4 Time-Dependent Perturbation Theory 62

9.5 Variational Coupled Cluster 62

Conclusion 63

References 64

10. Atomic Structure 65

10.1 Central Force Problems 65

10.2 Self Consistent Field Method 66

10.3 Term Symbols 67

10.4 Zeeman Effect 68

10.5 Hyperfine Structure 68

Conclusion 69

References 70

11. Molecular Structure 71

11.1 Born-Oppenheimer Approximation 71

11.2 Hydrogen Molecular Ion 72

11.3 Valence Bond Theory 73

11.4 Molecular Orbital Theory 75

Conclusion 77

References 77

12. Electronic Properties of Solids 79

12.1 Band Structure 79

12.2 Bloch’s Theorem 82

12.3 Tight Binding Model 84

12.4 Nearly Free Electron Model 86

12.5 Bandgap Materials 88

References 90

13. Structural Properties of Solids 91

13.1 Crystal Structure 91

13.2 X-ray Diffraction 92

13.3 Lattice Vibrations 94

13.4 Phonons 94

Conclusion 96

References 96

14. Nuclear Physics 97

14.1 Constituents and Properties 97

14.2 Nuclear Force and Models 98

14.3 Radioactive Decay 99

14.4 Nuclear Reactions 100

Conclusion 101

References 101

15. Particle Physics 102

15.1 Standard Model Fundamentals 102

15.2 Quarks 105

15.3 Gauge Symmetries 108

15.4 Grand Unification Theories 108

Conclusion 110

References 111

16. Frontiers of Quantum Mechanics 112

16.1 Quantum Information Science 112

16.2 Quantum Computing 117

16.3 Quantum Cryptography 120

16.4 Entanglement 123

16.5 Bell’s Inequality 125

Conclusion 127

References 128

Glossary 129

Index 131

Chapter-1

Introduction to

Quantum Theory

1.1 History and Motivation for Quantum Mechanics

The origins of quantum mechanics trace back to the early 20th century when rapid advances in the fields of radiation, atomic structure and wave-particle phenomena revealed seeming contradictions between established classical physics theories and experimental observations. Groundbreaking work across physics subdisciplines ultimately converged toward a revolutionary understanding of nature in which matter and energy exhibit inherently probabilistic and discrete behaviors on the smallest scales.

Early Motivations

Several pivotal empirical findings in the late 19th century set the stage for the quantum revolution. In 1859, Gustav Kirchhoff demonstrated that blackbody radiation, or heat emitted from an idealized enclosed cavity, had a frequency spectrum connected directly to temperature. However classical predictions vastly diverged from measurements, becoming known as the ultraviolet catastrophe. The photoelectric effect, discovered by Heinrich Hertz in 1887 and studied by Philipp Lenard, also defied classical wave models of light, whereby electron emission from metals induced by light depended directly on frequency rather than expected intensity.

Other Puzzles Emerge

Further surprising experimental results continued to mount that could not be explained by classical physics across mechanics, electromagnetism and thermodynamics. In 1897, J.J. Thomson measured the charge-to-mass ratio of cathode rays, revealing fundamental particles later called electrons that were over 1,000 times lighter than hydrogen, the smallest atom. Additional issues arose from stability models of Rutherford’s 1911 nuclear atom, spectral lines of ionized atoms, and Compton scattering of X-ray photons.

The most serious classical physics limitations emerged from the developing field of spectroscopy and atomic structure models in the early 20th century. When Niels Bohr incorporated quantized angular momentum in 1913 to stabilize his new atom model, he introduced discontinuous quantum jumps between orbits that violated classical electrodynamics. Expressing atomic spectral frequencies through quantum integer spacing succeeded where electromagnetism failed dramatically.

Quantum Theory Takes Shape

By the mid 1920s, the problems confronting classical physics grew into a crisis. In 1923, Compton definitively demonstrated the particle-like properties of X-ray photons through wavelength shifts during collisions with electrons. De Broglie soon thereafter proposed the radical concept of wave-particle duality applied to matter. Just one year later, Bose and Einstein introduced Bose-Einstein statistics by treating light quanta as indistinguishable particles. The field was ripe for a unifying theory merging the particle and wave perspectives.

Fig. 1.1 Quantum Mechanics

https://images.app.goo.gl/CeaaHWk3zN6i6gsr9

In 1925-26, multiple researchers derived a new conceptual framework now known as quantum mechanics. Heisenberg pioneered matrix mechanics, Schrödinger devised his eponymous wave equation, and Born statistically interpreted its solutions. Dirac, von Neumann and others integrated these approaches into the accepted quantum theory formalism. By treating dynamics statistically using wave functions and operators, quantum mechanics provided probabilistic explanations for discontinuous atomic behaviors and discrete allowed states. This new physical theory excellently described phenomena classical theories struggled to model, launching a scientific revolution.

1.2 Failings of Classical Physics

While classical theories achieved great success explaining dynamics for large everyday objects and certain electromagnetic phenomena, they critically struggled to describe central processes underlying experimental observations made in the late 19th and early 20th centuries involving radiation, atomic transitions, particle scattering and more. Key empirical findings that classical physics failed to explain include:

Blackbody Radiation

The Rayleigh-Jeans Law derived from classical electromagnetism diverged severely from experimentally measured blackbody spectral radiancy at higher frequencies, predicting near-infinite output in the ultraviolet that became known as the ultraviolet catastrophe.

Fig. 1.2 Blackbody Radiation

https://images.app.goo.gl/ghuZCPF1jPmSace69

Planck successfully modeled the spectrum by introducing quantization of the radiation field into individual oscillators at allowed discrete energy levels rather than a continuum - avoiding the short-wavelength divergence but requiring non-classical processes.

Photoelectric Effect

Einstein explained the experimental properties of photoelectric emission of electrons from metals struck with light using a discrete, particle-based model for light quanta that had energy directly proportional to their frequency, unlike the prediction of classical wave theories depending solely on intensity.

Compton Scattering

Scattering profiles of X-rays with electrons matched predictions from particle-based conservation laws and required assigning particle properties to photons that carried momentum proportionally to their wavelength while experiencing discrete billiard ball-like collisions - departing radically from classical expectations.

Fig. 1.3 Compton Scattering

https://images.app.goo.gl/xNQMSMuyx2ptEwLF9

Atomic Spectra

Discrete electron transition levels between quantized energy states successfully described distinct spectral emission and absorption lines in atomic gases, contradicting predictions of continuum dynamics from classical electromagnetism.

These and other empirical phenomena could only be explained by non-classical models introducing discrete states and discontinuous transitions, motivating development of entirely novel physics formalisms departing fundamentally from classical theories. Pioneering work across disciplines ultimately unified into the modern quantum mechanical framework.

1.3 Blackbody Radiation

Blackbody radiation refers to the thermal electromagnetic radiation emitted by an idealized physical body that absorbs and emits all frequencies of radiation. The term blackbody refers to the object’s ability to absorb all radiation that hits it, resulting in it appearing black. Studying blackbody radiation in the late 19th century helped spur the development of modern quantum theory.

The amount of radiation emitted in any particular frequency range should depend only on the temperature of the body, not on the body’s chemical or physical structure. The thermal radiation emitted covers a continuous spectrum of frequencies. However, at any given temperature, the intensity or power per unit area in a narrow frequency range is proportional to the frequency cubed. The proportionality constant depends on only the temperature.

The spectrum of blackbody radiation, described by Planck’s law, followed purely from thermodynamics arguments that did not involve details of atomistic processes. However, the result was in conflict with classical physics because at high frequencies, Planck’s law predicted more high-frequency radiation than possible from equipartition of energy among linear oscillators in thermal equilibrium. This became known as the ultraviolet catastrophe.

To resolve this problem, around 1900 Max Planck hypothesized that the emission of radiation came in discrete packets of energy proportional to the radiation frequency. Einstein later applied this quantization to the electromagnetic radiation itself, proposing that high-frequency radiation itself was quantized into particles which were later called photons. This explained the shape of the blackbody spectrum at high frequencies.

These discoveries initiated quantum mechanics, one of the most important theories in modern physics. We will now examine blackbody radiation and the associated discoveries in more detail.

Blackbody Radiation Spectrum

The spectrum of blackbody radiation, described by Planck’s law, is given by:

I(\nu,T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/kT} - 1}

Where:

I(\nu,T) = spectral radiance (W/m^2/sr/Hz) at frequency \nu and temperature T

\nu = frequency (Hz)

T = absolute temperature (K)

h = Planck’s constant = 6.626 x 10^-34 Joule-sec

k = Boltzmann constant = 1.381 x 10^-23 Joule/K

c = speed of light = 3 x 10^8 m/s

This equation gives the intensity of radiation at different frequencies \nu emitted by a blackbody at temperature T. It is derived based on thermodynamics principles and matches experimental measurements extremely well.

Some key features of the blackbody curve:

-At low frequencies (long wavelengths), the intensity increases as \nu^2. This is the Rayleigh-Jeans law, corresponding to equipartition of energy.

-At