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2™ 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