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E-grāmata: Quantum Mechanics: Non-Relativistic and Relativistic Theory [Taylor & Francis e-book]

(University of Dschang, Cameroon)
  • Formāts: 16 pages, 1 Tables, black and white; 37 Line drawings, black and white; 37 Illustrations, black and white
  • Izdošanas datums: 02-Jun-2022
  • Izdevniecība: CRC Press
  • ISBN-13: 9781003273073
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  • Taylor & Francis e-book
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Quantum Mechanics: Non-Relativistic and Relativistic TheoryPalielināt
  • Formāts: 16 pages, 1 Tables, black and white; 37 Line drawings, black and white; 37 Illustrations, black and white
  • Izdošanas datums: 02-Jun-2022
  • Izdevniecība: CRC Press
  • ISBN-13: 9781003273073
Citas grāmatas par šo tēmu:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Rokasgrāmatas mājas lapa: https://www.taylorfrancis.com/books/9781003273073
This book presents an accessible treatment of non-relativistic and relativistic quantum mechanics. It is an ideal textbook for undergraduate and graduate physics students, and is also useful to researchers in theoretical physics, quantum mechanics, condensed matter, mathematical physics, quantum chemistry, and electronics.

This student-friendly and self-contained textbook covers the typical topics in a core undergraduate program, as well as more advanced, graduate-level topics with an elegant mathematical rigor, contemporary style, and rejuvenated approach. It balances theory and worked examples, which reinforces readers' understanding of fundamental concepts.

The analytical methods employed in this book describe physical situations with mathematical rigor and in-depth clarity, emphasizing the essential understanding of the subject matter without need for prior knowledge of classical mechanics, electromagnetic theory, atomic structure, or differential equations.

Key Features:

Remains accessible but incorporates a rigorous, updated mathematical treatment Laid out in a student-friendly structure Balances theory with its application through examples

Lukong Cornelius Fai is a professor of theoretical physics at the Department of Physics, Faculty of Sciences, University of Dschang, Cameroon. He is Head of Condensed Matter and Nanomaterials as well as the Mesoscopic and Multilayer Structures Laboratory. He was formerly a senior associate at the Abdus Salam International Centre for Theoretical Physics (ICTP), Italy. He holds a Master of Science in Physics and Mathematics (1991) as well as a Doctor of Science in Physics and Mathematics (1997) from Moldova State University. He is the author of over 170 scientific publications and five textbooks.
Preface xiii
About the Author xv
SECTION I Non-Relativistic Theory
1 Quantum Mechanics Basic Concepts
3(38)
1.1 Inadequacies of Classical Mechanics
3(2)
1.2 Wave Function
5(2)
1.3 Wave Function Statistical Interpretation
7(2)
1.4 Uncertainty of Two Types of Measurements
9(1)
1.5 Superposition Principle Generalized Formulation
10(1)
1.6 Operators of Physical Quantities
11(3)
1.6.1 Expectation Value (Observable) and Operator of a Physical Quantity
11(2)
1.6.2 Properties of Operators
13(1)
1.7 Linear Self-Adjoint (Hermitian) Operators
14(6)
1.7.1 Translation Operator
18(2)
1.8 Eigenfunction and Eigenvalue
20(3)
1.8.1 Conclusion
21(2)
1.9 Properties of Eigenfunctions of Hermitian Operators
23(7)
1.10 Theorem on the Commutation of Operators and Their Physical Application
30(4)
1.11 Heisenberg Uncertainty Relations for Arbitrary Observables
34(1)
1.12 Limiting Transition from Quantum Mechanics to Classical Mechanics
35(6)
2 Schrodinger Equation
41(28)
2.1 Stationary States
41(12)
2.1.1 Particle in an Infinite Deep Potential Well
44(4)
2.1.2 A Particle in an Infinitely High Potential Well
48(2)
2.1.3 Coordinate Representation Delta Potential
50(3)
2.2 Time-Dependent Operators
53(16)
2.2.1 Classical Equation of Motion
53(3)
2.2.2 Quantum-Mechanical Poisson Bracket and Quantum Correspondence Principle
56(1)
2.2.3 Quantum Mechanical Equation of Motion
57(2)
2.2.4 Postulates of Quantum Mechanics
59(1)
2.2.5 Velocity and Acceleration of a Charged Particle in an Electromagnetic Field
60(1)
2.2.6 Probability Density and Probability Current Density
61(2)
2.2.7 Current Density of a Charged Particle in an Electromagnetic Field
63(1)
2.2.8 Change with Time of a Wave Packet
64(5)
3 Momentum Operator
69(44)
3.1 Translation Operator
69(2)
3.2 Momentum Operator
71(2)
3.3 Heisenberg Uncertainty Relation
73(4)
3.4 Momentum Representation
77(9)
3.4.1 Momentum Representation of Particle in Triangular Potential Well
80(2)
3.4.2 Momentum Representation of Particle in Delta Potential Well
82(4)
3.4.3 Momentum Representation of an Operator in Matrix Form
86(1)
3.5 Particle Hamiltonian in a Potential Field
86(3)
3.5.1 Hamilton Function Operator and Ehrenfest Theorem
87(2)
3.6 Angular Momentum Operator
89(5)
3.6.1 Infinitesimal Rotation Operator
89(1)
3.6.2 Angular Momentum Operator
90(1)
3.6.3 Commutation Relations of Angular Momentum Operators
91(1)
3.6.4 Eigenvalue and Eigenfunction of z-Component Angular Momentum Operator
92(2)
3.7 Square of Angular Momentum Operator
94(5)
3.7.1 Square of Angular Momentum Operator Commutation Relations
94(2)
3.7.2 Square of Angular Momentum Operator Eigenvalue in the Dirac Representation
96(3)
3.8 Square of Angular Momentum Operator Eigenstates
99(14)
3.8.1 Legendre Polynomials
99(3)
3.8.1.1 Asymptotic Legendre Polynomials
102(2)
3.8.2 Angular Momentum Eigenstates
104(2)
3.8.3 Dirac Representation Eigenstates
106(1)
3.8.4 Matrix Representation and Finite Rotations Eigenstates
107(6)
4 Total Angular Momentum
113(34)
4.1 Infinitesimal Symmetry Transformation Generator
113(1)
4.2 Total Angular Momentum Justification
113(1)
4.3 Addition of Two Angular Momenta
114(10)
4.3.1 Clebsch-Gordan Coefficients
115(1)
4.3.1.1 Other Representation of Clebsch-Gordan Coefficients
116(1)
4.3.1.2 Clebsch-Gordan Coefficients Recursion Relations
117(1)
4.3.2 Triangular Rule
118(6)
4.4 Spherical Spinors
124(8)
4.4.1 Spinor Rotation
125(4)
4.4.2 Spin Density
129(3)
4.5 Spin of a System of Two Particles
132(4)
4.6 Rotation Operator
136(8)
4.6.1 Finite Rotation Operator About Some Angle Along Some Axis
136(1)
4.6.2 Finite Rotation Operator for Spinor One-Half
136(2)
4.6.3 Finite Rotation Operator for Spinor One-Half General Case
138(2)
4.6.4 Rotation Operator Matrix
140(3)
4.6.4.1 Spherical Harmonics Connection
143(1)
4.7 Irreducible Tensor Operators
144(3)
4.7.1 Wigner-Eckart Theorem
144(3)
5 One-Dimensional Motion General Principles
147(16)
5.1 One-Dimensional Motion General Principles
147(2)
5.2 Potential Well
149(2)
5.3 Particle in a One-Dimensional Finite Square Well Potential
151(6)
5.4 Potential Barrier
157(1)
5.5 Particle in a Square Potential Barrier
158(5)
6 Schrodinger Equation
163(44)
6.1 Linear Harmonic Equation
163(1)
6.2 Harmonic Oscillator Eigenstates and Eigenvalues
163(13)
6.2.1 Hermite Polynomial and Harmonic Oscillator Eigenfunction
166(1)
6.2.1.1 Hermite Polynomials
166(2)
6.2.1.2 Hermite Polynomials Integral Representation
168(1)
6.2.1.3 Harmonic Oscillator Eigenfunction and Normalization Condition
168(1)
6.2.1.4 Hermite Polynomials Orthogonality Condition
169(7)
6.3 Motion in a Central Field
176(9)
6.3.1 Radial Schrodinger Equation
176(2)
6.3.2 Radial Wave Function Qualitative Investigation
178(4)
6.3.3 Continuous Spectra Radial Wave Functions
182(1)
6.3.3.1 Jost Function
183(1)
6.3.4 Delta Potential Radial Solution
184(1)
6.4 Motion in a Coulombic Field
185(22)
6.4.1 Hydrogen Atom (Spherical Coordinates)
185(3)
6.4.2 Eigenvalue and Eigenfunction
188(2)
6.4.2.1 Hydrogen Atom's Wave Function
190(1)
6.4.2.2 Laguerre Polynomials Integral Representation
191(1)
6.4.2.3 Eigenvalue and Degeneracy
192(1)
6.4.3 Hydrogen Atom (Parabolic Coordinates)
193(1)
6.4.3.1 Energy Levels
194(2)
6.4.3.2 Wave Functions
196(1)
6.4.4 Spherical Oscillator (Spherical Coordinates)
197(4)
6.4.5 Particle in an Infinite Deep Spherical Symmetric Potential Well
201(1)
6.4.6 Kepler Problem in Two Dimensions
202(5)
7 Representation Theory
207(28)
7.1 Matrix Wave Functions and Operator Representation
207(1)
7.2 Properties of Matrices
208(2)
7.3 Rule on Matrix Operations
210(2)
7.4 Action of an Operator on a Wave Function
212(1)
7.5 Mean Value of an Operator
213(1)
7.6 Eigenstate and Eigenvalue Problem
213(2)
7.7 Unitary Transformation in State Vector Space
215(4)
7.7.1 Unitary Matrix
216(1)
7.7.2 Matrix Element of a Transformation Operator
217(1)
7.7.3 Invariance of the Trace of a Matrix Under Unitary Transformations
218(1)
7.8 Schrodinger and Heisenberg Representations
219(1)
7.9 Interaction Representation
220(1)
7.10 Energy Representation
221(14)
7.10.1 Evolution Operator
223(1)
7.10.2 Oscillator in the Energy Representation
224(1)
7.10.2.1 Matrix Element of the Oscillator Coordinate
224(3)
7.10.2.2 Hamiltonian Operator Eigenvalue
227(1)
7.10.2.3 Harmonic Oscillator Ground-State Eigenfunction
227(2)
7.10.2.4 Quantization of Operators
229(6)
8 Quantum Mechanics Approximate Methods
235(42)
8.1 Variational Principle
235(3)
8.1.1 Ritz Method
235(3)
8.2 Case of the Hydrogen Atom
238(1)
8.3 Perturbation Theory
239(7)
8.3.1 Stationary Perturbation Theory - Non-Degenerate Level Case
239(7)
8.4 Perturbation Theory - Case of a Degenerate Level
246(15)
8.4.1 The Stark Effect
249(1)
8.4.1.1 Hydrogen Atom
250(2)
8.4.2 Stark Effect (Spherical Coordinates)
252(6)
8.4.3 Stark Effect (Parabolic Coordinates)
258(3)
8.5 Time-Dependent Perturbation Theory
261(7)
8.5.1 Transition Probability
263(2)
8.5.2 Adiabatic Approximation
265(3)
8.6 Time-Independent Perturbation
268(1)
8.7 Time and Energy Uncertainty Relation
268(2)
8.8 Density of Final State
270(1)
8.8.1 Transition Rate
271(1)
8.9 Transition Probability-Continuous Spectrum
271(3)
8.9.1 Harmonic Perturbation
272(2)
8.10 Transition in a Continuous Spectrum Due to a Constant Perturbation
274(3)
9 Many-Particle System
277(16)
9.1 System of Indistinguishable Particles
277(2)
9.2 Interacting System of Particles
279(1)
9.3 System of Two Electrons
280(13)
9.3.1 Exchange Interaction
285(1)
9.3.2 Two Electrons in an Infinite Square Potential Well - Heisenberg Exchange Interaction
285(8)
10 Approximate Method for the Helium Atom
293(6)
10.1 The State of the Helium Atom
293(3)
10.2 Self-Consistent Field Method
296(3)
11 Approximate Method for the Hydrogen Molecule
299(6)
11.1 Vibrational and Rotational Levels of Diatomic Molecules
303(2)
12 Scattering Theory
305(50)
12.1 Scattering Cross Section and Elastic Scattering Amplitude
305(6)
12.1.1 Relation Between the Laboratory and Center-of-Mass Systems
308(3)
12.2 Method of Partial Waves
311(6)
12.3 S-Scattering of Slow Particles
317(2)
12.4 Resonance Scattering
319(3)
12.5 The Unitary Scattering Conditions
322(4)
12.5.1 Optical Theorems
322(4)
12.6 Time-Reversal Symmetry
326(1)
12.6.1 Inversion Operator and Reciprocity Theorem
326(1)
12.7 Schrodinger Equation Green's Function
327(2)
12.8 Born Approximation
329(18)
12.8.1 Scattering of Fast Charged Particles on Atoms
329(1)
12.8.1.1 Scattering Amplitude in Momentum Representation
329(5)
12.8.2 Perturbation Theory Method Approach for Born Approximation
334(5)
12.8.2.1 Phase Shift
339(1)
12.8.2.2 Spherical Potential Well
340(2)
12.8.2.3 Coulomb Interaction and Rutherford's Formula
342(4)
12.8.2.4 Lippman Schwinger Equation, ID Delta Potential
346(1)
12.9 Elastic and Inelastic Collisions
347(3)
12.9.1 Fast and Slow Particle Total Cross Section
347(3)
12.10 Wentzel-Kramer-Brillouin (WKB) Method
350(2)
12.10.1 Motion in a Central Symmetric Field
350(2)
12.11 Scattering of Indistinguishable Particles
352(3)
13 Polaron Theory
355(28)
13.1 Lee-Low-Pines (LLP) Technique
355(12)
13.1.1 Lee-Low-Pines (LLP) Bulk Polaron
355(6)
13.1.2 Lee-Low-Pines (LLP) Surface and Slow Moving Polaron
361(5)
13.1.3 Lee-Low-Pines (LLP) Surface and Fast Moving Polaron
366(1)
13.2 Polaron in a Quantum Wire
367(4)
13.3 Polaronic Exciton and Haken Exciton
371(12)
SECTION II Relativistic Theory
14 Case of an Electron
383(16)
14.1 Spin Operators
383(8)
14.1.1 Spin and Spin Operator Commutation Relations
385(2)
14.1.2 Pauli Matrices
387(1)
14.1.3 Derivation of Pauli Matrices
388(3)
14.2 Spinors
391(8)
14.2.1 Lorentz Transformation and Spinor Transformation
392(2)
14.2.2 Arbitrary Spinor Transformation
394(5)
15 Klein-Gordon Equation
399(10)
15.1 Probability and Charge Densities
402(1)
15.2 Motion in an Electromagnetic Field
403(1)
15.3 Spinless Charge Particle in a Coulombic Field
404(2)
15.4 Non-Relativistic Limiting Equation
406(3)
16 Dirac Equation
409(4)
17 Probability and Current Densities
413(2)
18 Electron Spin in the Dirac Theory
415(4)
19 Free Electron State with Defined Momentum-Positronium Motion
419(6)
19.1 Stationary Dirac Equation
419(6)
19.1.1 Dirac Hypothesis-Hole Theory
423(2)
20 Dirac Equation
425(18)
20.1 Electron Motion in an External Electromagnetic Field
425(12)
20.1.1 Quasi-Relativistic Approximation-Pauli Equation
426(4)
20.1.2 Second-Order Relativistic Correction
430(1)
20.1.2.1 Spin-Orbital Interaction
430(1)
20.1.2.2 Fine Structure Levels
431(2)
20.1.2.3 Fine Structure Effect
433(4)
20.2 Bound Electronic States in a Coulombic Field
437(6)
21 Motion in a Magnetic Field
443(24)
21.1 Landau Levels
443(3)
21.2 Spin Precession in a Magnetic Field
446(6)
21.3 Theory of the Zeeman Effect
452(11)
21.3.1 Russell-Saunders Coupling
454(6)
21.3.2 Weak Field Limiting Case - Zeeman Effect
460(1)
21.3.3 Strong Field for Exceedingly Small Spin-Orbit Interaction - Paschen-Back Effect
461(1)
21.3.4 Landau Case
462(1)
21.4 Atomic Paramagnetism and Diamagnetism
463(4)
SECTION III Appendix: Special Functions
22 Gamma Functions
467(8)
22.1 First Kind Euler Integral-Beta Function
467(1)
22.2 Gamma Function (Second Kind Euler Integral)
468(2)
22.3 Gamma Function Analytic Continuation
470(2)
22.4 Hankel Integral Representations
472(1)
22.5 Reflection or Complementary Formula
473(2)
23 Confluent Hypergeometric Functions
475(6)
23.1 Classical Gauss Confluent Hypergeometric Function
475(2)
23.2 Euler Integral Representation: Mellin-Barnes Integral Representation
477(1)
23.3 Confluent Hypergeometric Function - Kummer Function
478(3)
24 Cylindrical Functions
481(18)
24.1 Cylindrical Function of the First Kind
481(1)
24.2 Neumann Function
482(1)
24.3 Hankel Functions
483(1)
24.4 Modified Bessel Function
483(1)
24.5 Modified Bessel Function with Imaginary Argument
484(1)
24.6 Bessel Function of the First Kind Integral Formula
485(3)
24.7 Neumann Function Integral Formula
488(1)
24.8 Hankel Function Integral Formula
489(3)
24.9 Airy Function
492(7)
25 Orthogonal Polynomials
499(32)
25.1 Orthogonal Polynomials General Properties
499(1)
25.2 Transforming Confluent Hypergeometric Function into a Polynomial
500(2)
25.3 Jacobi Polynomials
502(2)
25.4 Jacobi Polynomial Generating Function
504(2)
25.5 Gegenbauer Polynomials
506(2)
25.6 Gegenbauer Polynomial Generating Function
508(1)
25.7 First Kind TschebychefF Polynomial
509(2)
25.8 Generating Function of the First Kind TschebychefF Polynomial
511(1)
25.9 TschebychefF Polynomial of the Second Kind
512(2)
25.10 Generating Function of the Second Kind Tschebycheff Polynomial
514(1)
25.11 Legendre Polynomials
515(1)
25.12 Legendre Polynomial Generating Function
515(2)
25.13 Legendre Polynomials Integral Representation
517(1)
25.14 Associated Legendre Polynomials
518(2)
25.15 Associated Legendre Polynomials Integral Representation
520(1)
25.16 Spherical Functions
520(4)
25.17 Laguerre Polynomials
524(1)
25.18 Associated Laguerre Polynomial Generating Function
525(3)
25.19 Hermite Polynomials
528(1)
25.20 Hermite Polynomial Generating Function
528(3)
References 531(2)
Index 533
Lukong Cornelius Fai is professor of theoretical physics at the Department of Physics, Faculty of Sciences, University of Dschang. He is Head of Condensed Matter and Nanomaterials as well as Mesoscopic and Multilayer Structures Laboratory. He was formerly a senior associate at the Abdus Salam International Centre for Theoretical Physics (ICTP), Italy. He holds a Masters of Science in Physics and Mathematics (June 1991) as well as a Doctor of Science in Physics and Mathematics (February 1997) from Moldova State University. He is an author of over a hundred and seventy scientific publications and five textbooks
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