Table Of Content2™ Edition
QUANTUM
PHYSICS
H.C. VERMA
LAT KANPUR
2
SURYA PUBLICATIONS, GHAZIABAD
To the Readers
Iretassical physics describes the world to be deterministic where the entire future course is
cited, quantum physics pute up a theory based on the assamption that the world is
Pegtublltic where events take place on random chances. Yet there no contradiction between
Provapind classical physics follows smoothly from quantum physics as the size of the systers
jee is 2 beautiful subject having, a rigorous
B thematital foundation and at the same time has flavours of philosophy and many see
Eiferent shades of spiritualism in it. it invites lots of abstract imagination but connects the
vents of the material world to the fantasies of imagination.
study quantum physics.
‘The guiding principle in writing this book: had been to let the reader feel the excitement of
Tiietstanding the new way of looking at the nature and correlate with the events occurring
Mound. I presume this to be the first exposition of quantum physics for the reader after the Std
Sir keel ‘treauneut of topics like Dobr's model of hydrogen atom, pho:celectric effects
Tadionetivity etc. The treutuient is based on sound mathematical formulation put :he physical
nsight ig not allowed to get lost in mathematical complexities, Relatively less common
Inathematical tools, especially operator algelua and related concepts like eigenvalie equations
Tie, are developed in detail before making use of them. 1 kave made an effort to tell about the
Gee of the theory developed in applications to physical phenomena in order to create an
tachment towards the subject. That is why the ttle is Quran Physics and not just Quantum
mechanics. is
‘The structure of the chapters includes the Text, Solved Problems and Exercises. A section called
You learned in this chapter’ after the text in each chapter gives the summary of the key
Concepts discussed in that chapter, The exercise questions at the end of a chapter are generally
Shortand given 10 sharpen Ue understanding of the topics learned in the chapter and not to
foad the reader. In some of the chapters a section called ‘Postscript’ is appended which may be
treated as an extension of the ideas dealt in Uiat chapter. The Postscript material is generally
hot used in the later chapters and hence the completeness or continuity of the book is not
affected ifthe Postscripts are left out in the first reading.
‘Though the book ie written for any person having interest in Physics and looking for a learning
material on quantum physics, it can be used as a textbook for « 40-50 kour course on basics
quantum mechanics and applications.
‘The force behind writing this book had been my lively interactions with the students of Patna
University and IIT Kanpur to whom { have given coursts on quantum physics several times and
have enjoyee the subject immensely. However the sceds of this bao'c were prabahly sown by my
teachers Dr A P Shuida and Dr YR Waghmare at IIT Kenpur who taught me quantum physics
in such a way that created a passion in me for this subject. The script of this book has been
thoroughly read by Dr Devi Prusad Verma, Rtd Profeusor of Mathcmetics, who apart from
suggesting editorial changes provided ime vital feedback how a matured but _nonphysicist
person would respond to this writing. 1 also acknowledge the effor:s of Mr Brajesh Pandey who
has gone through the text and solved all exercise problems to ensure the correctness of
enswers. Lam thankf:l to CDTE, UT Kanpur for providing financial assistance (o prepare the
manuscript
have tried to keep errors away from this Luck to the dest of my ability but 1 caunut ensure that
itis error free. I will be grateful to all who communicate to me about any such error they come
Across. [ will slways look forward for suggestions to enhance the usefulness of this book.
Harish Chandra Verma
Chapter 1
Chapter 2
Chapter 3
New Lessons from Young’s Double-slit Experiment
Je
1.1 The great wave-particle struy
1.2 Young’s double-slit experiment settled the debate
1.3 Photoelectric effect reopened the issue
1.4 Light is wave as well as particle
1.5 A new way to think
1.6 Relation between wave and particle parameters
1.7 You learned in this chapter
Sclved Problems
Exercises
Postscript : More experiments to demonstrate the
particle nature of light
P11 Blackbody radiation
P12 Discrete wavelengths in atomic spectra
P13 Compton effect
P14 Single photon interference experiments
Rules of Measurement-
Lessons from Polarized Light
2.1 Polarization of light
2.2 Polarization by a Nichol prism
2.3 Polarization with a weak source
2.4 You learned in this chapter
Exercise
Material Particles Also Need Wave Description,
3.1 You too electron
3.2. Davison and Germer Experiment,
3.3 Double-slit experiment with electrons
3.4 The Trishul like symbol y—wave function
3.5 State of a particle
3.6 The abstract state and the wave function
3.7 An unrealistic but important wave function—
Dirae delta function
3.8 Another unrealistic but important wave function
plane waves
3.9 Wave packet—a more realistic wave function
3.10 Relativistic and nonrelativistic cases,
3.11 You learned in this chapter
21
30
32
Solved Problems 33
Exercises 36
Chapter 4 Position and Linear Momentul of a Particle
1 Position distribution and expansion in position
eigenfunctions as
4.2. Papansion in terms of miomentum eigenfuunctions 4
£3 Amexample of expansion af wave function P
4.4 The factor of = in |p} -
4.5 ‘The wave function as a vector 1
4.6 You learned in this chapter is
Solved Problems 47
Exercises pe
Chapter 5 A Restriction on Experimentalists—
Heisenberg Uncertainty Principle
5.1 Position and momentum uncertainty 56
5.2 Exact definition of Ax and 4p. 37
5.3. Heisenberg uncertainty principle 60
5.4 Experiments which tried to beat
uncertainty principle but failed 61
5.5 The best 1 can do—Gaussian wave packet 64
3.6 Can I continue to use classical physies for my
kancha? 06
5.7 You learned in this chapter 0
Solved Problems 67
Exercises a
Chapter 6 Multiplication is Position And Differentiation is
Momentum— Operator Formulation I
a
6.1 The operator ~ih- for momentum B
6.2 The operator for position “
6.3 Operator Formulation a
6A Operators for some of the measurable quantities 7
6.5 You learned in this chapter mt
Exercises a
Chapter 7 The Mathematics of Operators—
Operator Formulation IT
7.1 Linear operators Be
7.2 Operator algebra Y
2
7.3 Hermitian operators
7.4 You learned in this chapter
88
Solved Problems 85
Exercises 89
Chapter 8 More on Operators—
Operator Formulation III
8.1 Eigenvalues of a Hermitian operator are real 1
8.2 Eigenfunctions of « Hermitian operator
corresponding to different eigenvalues are
orthogonal to each other 92
8.3 Degenerace eigenvalues 92
8.4 Simultaneous eigenfunctions of two commuting .
operators 9,
8.5 Basis for the state space in terms of the
cigenfunctions of an operator me
8.6 Some important operators a
8.7 Expansion of a wave function in terms of
eigenfunctions of an operator 99
8.8 Expectation value of a dynamical variable 99
8.9 You learned in this chapter 100
Solved Problems 101
Exercises 105
Chapter9 As time passes—
Schrédinger Equation
9.1 Schrodinger equation 107
9.2 Solving Schrodinger equation 109
9.3 A free particle
un
9.4 Stationary states 116
9.5 Operators commuting with Hamiltonian 17
9.6 Emfest Theorem us
9.7 Quantum mechanics to classical mechanics 120
9.8 You learned in this chapter 121
Solved Problems
Exercises
Chapter 10 Can Energy be Created or Destroyed —
Energy-Time Uncertainty
10.1 Energy-time uncertainty principle 126
10.2 Time evolution of a simple wave function 126
10.3 Natural linewidth of spectral lines 128
10.4 Principle of energy conservation 130
10.5 Nuclear fusion 131
10.6 Anather interpretation of energy-time uncertainty 132
10.7 You Leamed in this Chapter 133
Solved Problems 13
Exercises 3
Chapter 11 Probability Can Flow—
Probability Current Density
11.1 Current density in.a fluid flow 136
11.2 Probability current density 138
11.3 Beam of particies 139
11.4 You learned in this chapter 1G
Solved Problems 140
Exercises 142
Chapter 12 Bound States of a Particle —
Deep Square Well
12.1 Time independent Schridinger equation 143
12.2 Bound state potentials uy
123 Boundary conditions at a sudden potential jump 143
12.4 Infinite square weli potential
12.5 Density of States
12.6 You learned in this chapter
Solved Problems
Exercise
Postscript : A particle in a 2-D box
P12.1 Hamiltonian for a particle in 2-D square box 159
P12.2 The energy eigenvalues and eigenfunctions 160
P12.3 Energy eigenstates in the k-space 161
P12.4 Density of states, 161
Chapter 13 Even Finite Well Will Do —
Finite Square Well Potential
13.1 Hamiltonian commutes with parity if V(x) is even 163
15.2 Eigenvalues and eigenfunctions of energy for a
finite square well potential ned
13.3 Special features abe
13.4 Physical situations 169
13.5 You learned in this chapter 10
Solved Problems ad
Exercise atte
Postscript : A quantum well
P13.1 What is a quantum Well? ue
P13.2 Density of states in a quantum well Wp,
chapter 14 Everywhere Forbidden, Still Bound—
Attractive Delta Function
14. A single delta function potential ae
14.2 Double delta function 180
14.3 Application to a hydrogen snulecular ion as
144 You learned in this chapter
Solved Problems 184
Exercises s6
Chapter 15 Any Thing in Stable Equilibrium can Oscillate —
Linear Harmonic Oscillator
15.1 Energy eigenvalues re
15.2. Energy eigenfunct ek
15.3 Ground state ie
15.4 Expectation values of x, 27, per 6
in stationary states 198
15.5 An cxample of time development 200
15.6 Solving the differential equation 202
18.7 You learned in this chapter 203
Solved Problems 208
Exercises 209
Chapter 16 Towards Unbound, Step by Step —
The Step Potential
16.1 The step potential aun
16.2 Stationary States with E< Vo 212
16.3 Stationary states with £>Vp 5
16.4. Motion of a wave packet 218
16.5 You learned in this chapter 218,
Solved Problems 218
Exercises 220
Chapter 17 Going Through a Hill—
Barrier Penetration
17.1 Stationary states in a barrier potential 201
A wave packet incident on a potential barrier 228
Transmission probabilily for barriers of
arbitrary shape
17.4 Alpha decay
17.8 Field emission
17.6 Tunnelling between two metal blocks
17,7 Nuclear fusion in the sun
17.8 You learned in this chapter
Solved Problems aah
Exercise 225.
Chapter 18 Let us Rotate —
Angular Momentum
18.1 Wave functions and operators in three dimensions 237
18.2 Orbital angular momentum 238
18.3 Eigenvalues and eigenfunctions of the angular
momentum operators
18.4 Generalized definition of angular momentum
18.5 Eigenvalues of J?, J, from commutation relations.
18.6 You learned in this chapter
Solved Problems
Exercises 255
Chapter 19 The Spinning Electron —
Spin Angular Momentum
19.1 Relation between magnetic moment and
angular momentum 258
19.2 Stern-Gerlach experiment 259
19.3 The g-value 261
19.4 The spin wave functions and operators 262
19.5 The matrix representation of spin wave functions
and operators 204
19.6 Pauli spin operators and matrices 266
19.7 You learned in this chapter 266
Solved Problems 267
Exercises 270
Chapter 20 Now in 3-D—
Central Potential Problems
20.1 The Hamiltonian 272
20.2 Stationary state eigenfunctions and radial equation 373
20.3 Rigid rotator 274
20.4 A free particle 275
20.5 A rigid spherical box 278
20.6 Finite square well potential ars
20.7 You learned in this chapter bees
Solved Problems 282
Exercises 288
Chapter 21. The Lightest One —
Hydrogen Atom
21.1 The Hamiltonian 286
21.2 Stationary state eigenfunctions and eigenvalues
287
21.3 Comparison with Bohr’s mode, 289
21.4 Radial probability densi 289
21.5 Angular distribution in p-states, p.Pj.Pe orbitals 291
21.6 Fine structure of hydrogen spectral lines 293
21.7 You learned in this chapter 293
Solved Problems 294
Beercises 297
Chapter 22 The Beauty of Being Identical—
Identical Particles
22.1 Identical particles in classical mechanics 299
22.2 Identical particles in quantum mechanics 299
Z 22.3 Symmetric and antisymmetric wave functions 300
224 Bosons and Fermions 301
22.5 Writing wave function of a system of three
indistinguishable particles 301
22.6 Spin and space wave function 302
22.7 You learned in this chapter 303
Solved Problems 303
Exercises a0
Chapter 23 We are distinguishable—
Maxwell-Boltzmann Statistics
23.1 Why statistical mechanics 305
23.2 Microstate, macrostate and configuration 308
23.3
Maxwell~Boltzmana distribution function, 307
23.4 Adilute gas
310
23.5 Molar heat capacity 313
23.6 Criteria for applying classical statistics aig
23.7 You learned in this chapter ang
Solved Problems 313
Exercises 318
Chapter 24 All Can Come Together—
Bose-Einstein Statistics
24.1 Quantum statistics 319
24.2 Occupation probability for
indistinguishable Bosons 320
24.3 A gas of non-interactive Bosons 323
24.4 Blackbody Radiation
303
24.5 Bose-Einstein condensation
337
24.6 You learned ir. this chapter 328
