Solved problems in quantum mechanics 2nd edition

UNITEXT for Physics Leonardo Angelini Solved Problems in Quantum Mechanics Second Edition UNITEXT for Physics Series Editors Michele Cini, University of Rome Tor Vergata, Roma, Italy Attilio Ferrari, University of Turin, Turin, Italy Stefano Forte, University of Milan, Milan, Italy Guido Montagna, University of Pavia, Pavia, Italy Oreste Nicrosini, University of Pavia, Pavia, Italy Luca Peliti, University of Napoli, Naples, Italy Alberto Rotondi, Pavia, Italy Paolo Biscari, Politecnico di Milano, Milan, Italy Nicola Manini, University of Milan, Milan, Italy Morten Hjorth-Jensen, University of Oslo, Oslo, Norway www.pdfgrip.com UNITEXT for Physics series, formerly UNITEXT Collana di Fisica e Astronomia, publishes textbooks and monographs in Physics and Astronomy, mainly in English language, characterized of a didactic style and comprehensiveness The books published in UNITEXT for Physics series are addressed to graduate and advanced graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching More information about this series at http://www.springer.com/series/13351 www.pdfgrip.com Leonardo Angelini Solved Problems in Quantum Mechanics 123 www.pdfgrip.com Leonardo Angelini Bari University Bari, Italy ISSN 2198-7882 ISSN 2198-7890 (electronic) UNITEXT for Physics ISBN 978-3-030-18403-2 ISBN 978-3-030-18404-9 (eBook) https://doi.org/10.1007/978-3-030-18404-9 © Springer Nature Switzerland AG 2019 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Quest’opera è protetta dalla legge sul diritto d’autore e la sua riproduzione è ammessa solo ed esclusivamente nei limiti stabiliti dalla stessa Le fotocopie per uso personale possono essere effettuate nei limiti del 15% di ciascun volume dietro pagamento alla SIAE del compenso previsto dall’art 68 Le riproduzioni per uso non personale e/o oltre il limite del 15% potranno avvenire solo a seguito di specifica autorizzazione rilasciata da AIDRO, Corso di Porta Romana n.108, Milano 20122, e-mail segreteria@aidro.org e sito web www.aidro.org Tutti i diritti, in particolare quelli relativi alla traduzione, alla ristampa, all’utilizzo di illustrazioni e tabelle, alla citazione orale, alla trasmissione radiofonica o televisiva, alla registrazione su microfilm o in database, o alla riproduzione in qualsiasi altra forma (stampata o elettronica) rimangono riservati anche nel caso di utilizzo parziale La violazione delle norme comporta le sanzioni previste dalla legge L’utilizzo in questa pubblicazione di denominazioni generiche, nomi commerciali, marchi registrati ecc., anche se non specificatamente identificati, non implica che tali denominazioni o marchi non siano protetti dalle relative leggi e regolamenti This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland www.pdfgrip.com Preface This book is essentially devoted to students who wish to prepare for written examinations in a Quantum Mechanics course As a consequence, this collection can also be very useful for teachers who need to propose problems to their students, both in class and in examinations Like many other books of Quantum Mechanics Problems, one should not expect a particular novel effort The aim is to present problems that, in addition to exploring the student’s understanding of the subject and their ability to apply it concretely, are solvable in a limited time This purpose is unlikely to be combined with a search for originality Problems will therefore be found that are also present in other books from the Russian classics [1, 2], and, therefore, in the collection, extracted from them, cared for by Ter Haar [3, 4] Among other books of exercises that have been consulted are the Italian Passatore [5] and that most recently published by Yung-Kuo Lim [6], which collects the work of 19 Chinese physicists The two volumes by Flügge [7] lie between a manual and a problem book, providing useful tips, though the presented problems are often too complex in relation to the purpose of this collection Many interesting problems are also found in Quantum Mechanics manuals In this case, the list could be very long I will only mention those who have devoted more space to problems: the classical manuals of Merzbacher [8] and Gasiorowicz [9], the volume devoted to Quantum Mechanics in the Theoretical Physics course by Landau and Lifchitz [10], the two volumes by Messiah [11] and the most recent works by Shankar [12], Gottfried-Yan [13], and Sakurai-Napolitano [14] One particular quote is due to Nardulli’s Italian text [15], both because of the abundance of problems it contains with or without solution, and the fact that many problems presented here have been proposed over the years to students of his course The category of problems that can be resolved in a reasonable time is not the only criterion for our choice No problem has been included that requires knowledge of mathematical methods that are sometimes absent from standard courses, such as, for example, Fuchsian differential equations When necessary, complementary mathematical formulas have been included in the appendix The most important characteristic of this book is that the solutions of many problems are presented with some detail, eliminating only the simplest steps This will certainly v www.pdfgrip.com vi Preface prove useful to the students Like in any other book, problems have been grouped into chapters In many cases, the inclusion of a particular problem in a particular chapter can be considered arbitrary: many exam problems pose cross-cutting issues across the entire program The obvious choice was to take into account the most distinctive questions For a time, this collection was entrusted to the network and used by teachers and students It is thanks to some of them that many of the errors initially present have been eliminated I thank Prof Stefano Forte for encouraging me to publish it in print after completing certain parts and reviewing the structure One last great thanks goes to my wife; the commitment needed to draft this text also resulted in a great deal of family burdens falling on her Finally, I apologize to the readers for the errors that surely escaped me; every indication and suggestion is certainly welcome Bari, Italy December 2018 Leonardo Angelini www.pdfgrip.com Contents Operators and Wave Functions 1.1 Spectrum of Compatible Variables 1.2 Constants of Motion 1.3 Number Operator 1.4 Momentum Expectation Value 1.5 Wave Function and the Hamiltonian 1.6 What Does a Wave Function Tell Us? 1.7 Spectrum of a Hamiltonian 1.8 Velocity Operator for a Charged Particle 1.9 Power-Law Potentials and Virial Theorem 1.10 Coulomb Potential and Virial Theorem 1.11 Virial Theorem for a Generic Potential 1.12 Feynman-Hellmann Theorem 1 2 4 10 One-Dimensional Systems 2.1 Free Particles and Parity 2.2 Potential Step 2.3 Particle Confined on a Segment (I) 2.4 Particle Confined on a Segment (II) 2.5 Particle Confined on a Segment (III) 2.6 Scattering by a Square-Well Potential 2.7 Particle Confined in a Square-Well (I) 2.8 Particle Confined in a Square-Well (II) 2.9 Potential Barrier 2.10 Particle Bound in a d Potential 2.11 Scattering by a d Potential 2.12 Particle Bound in a Double d Potential 2.13 Scattering by a Double d Potential 2.14 Collision Against a Wall in the Presence of a d Potential 2.15 Particle in the Potential VðxÞ_ Àcosh xÀ2 13 13 13 18 19 21 22 24 29 31 33 34 35 38 40 42 vii www.pdfgrip.com viii Contents 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 Two 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 Harmonic Oscillator: Position and Momentum Harmonic Oscillator: Kinetic and Potential Energy Harmonic Oscillator: Expectation Value of x4 Harmonic Oscillator Ground State Finding the State of a Harmonic Oscillator (I) Finding the State of a Harmonic Oscillator (II) General Properties of Periodic Potentials The Dirac Comb The Kronig-Penney Model 44 45 46 47 47 49 50 52 55 and Three-Dimensional Systems Plane Harmonic Oscillator Spherical Harmonic Oscillator Reflection and Refraction in Dimensions Properties of the Eigenstates of J and Jz Measurements of Angular Momentum in a State with ‘ ¼ Angular Momentum of a Plane Wave Measurements of Angular Momentum (I) Measurements of Angular Momentum (II) Measurements of Angular Momentum (III) Dipole Moment and Selection Rules Quadrupole Moment Partial Wave Expansion of a Plane Wave Particle Inside of a Sphere Bound States of a Particle Inside of a Spherical Potential Well Particle in a Nucleus Particle in a Central Potential Charged Particle in a Magnetic Field Bound States of the Hydrogenlike Atom Expectation Values of r1n for n ¼ 1; 2; in the Hydrogenlike Atom Stationary States One-Dimensional Hydrogen Atom? A Misleading Similarity Determining the State of a Hydrogen Atom Hydrogen Atom in the Ground State Hydrogen Atom in an External Magnetic Field A Molecular Model 59 59 61 63 65 67 69 70 71 72 73 74 76 77 78 82 83 84 85 88 90 90 92 92 94 97 97 98 98 99 100 Spin 4.1 4.2 4.3 4.4 4.5 Total Spin of Two Electrons Eigenstates of a Spin Component (I) Eigenstates of a Spin Component (II) Determining a Spin State (I) Determining a Spin State (II) www.pdfgrip.com Contents ix 4.6 4.7 4.8 4.9 Determining a Spin State (III) Measurements in a Stern-Gerlach Apparatus Energy Eigenstates of a System of Interacting Fermions Spin Measurements on a Fermion 101 102 103 105 Time 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 Evolution Two-Level System (I) Two-Level System (II) Two-Level System (III) Two-Level System (IV) Time-Evolution of a Free Particle Particle Confined on a Segment (I) Particle Confined on a Segment (II) Particle Confined on a Segment (III) Harmonic Oscillator (I) Harmonic Oscillator (II) Harmonic Oscillator (III) Plane Rotator Rotator in Magnetic Field (I) Rotator in Magnetic Field (II) Fermion in a Magnetic Field (I) Fermion in a Magnetic Field (II) Fermion in a Magnetic Field (III) Fermion in a Magnetic Field (IV) Fermion in a Magnetic Field (V) Fermion in a Magnetic Field (VI) Measurements of a Hydrogen Atom 107 107 110 111 114 115 118 119 121 122 125 126 128 130 131 131 133 135 136 138 139 141 Time-Independent Perturbation Theory 6.1 Particle on a Segment: Square Perturbation 6.2 Particle on a Segment: Linear Perturbation 6.3 Particle on a Segment: Sinusoidal Perturbation 6.4 Particle on a Segment in the Presence of a Dirac-d Potential 6.5 Particle in a Square: Coupling the Degrees of Freedom 6.6 Particle on a Circumference in the Presence of Perturbation 6.7 Two Weakly Interacting Particles on a Circumference 6.8 Charged Rotator in an Electric Field 6.9 Plane Rotator: Corrections Due to Weight Force 6.10 Harmonic Oscillator: Anharmonic Correction 6.11 Harmonic Oscillator: Cubic Correction 6.12 Harmonic Oscillator: Relativistic Correction 6.13 Anisotropic Harmonic Oscillator 6.14 Charged Harmonic Oscillator in an Electric Field 6.15 Harmonic Oscillator: Second Harmonic Potential I 143 143 144 145 146 150 152 154 155 157 158 159 160 161 163 164 www.pdfgrip.com 11.4 Ground State of Helium 237 By inserting these results into the expression for E(Z ), we obtain E(Z ) = − e2 a0 27 Z − Z2 , which takes its minimum value for Z = Ze f f = 27 16 given by E(Z e f f ) = − e2 a0 27 Z e f f − Z e2f f = −77.5 eV This value is very close to the already mentioned experimental value −78.98 eV and provides a better approximation compared to the calculation in Perturbation Theory (see Problem 6.27) Both approximations calculate the energy of the ground state as the expectation value of the Hamiltonian in the unperturbed state, but the variational method modifies the value of Z in such a way as to minimize the difference from the exact value Notice, moreover, that Z e f f < Taking into account that wave functions are factored, we can consider the approximation as resulting from a model in which each of the electrons moves in the mean field of a nucleus with an effective charge Z e f f · e, lower than the real one due to the screen effect produced by the other electron www.pdfgrip.com Appendix Useful Formulas A.1 Frequently Used Integrals A.1.1 Gaussian Integrals Having defined +∞ I0 (α) = −∞ π , α d x e−αx = (A.1) we have +∞ I2n+1 (α) = I2n (α) = +∞ −∞ −∞ d x x 2n+1 e−αx = 0, d x x 2n e−αx = (−1)n ∂n ∂n I0 (α) = (−1)n n ∂α ∂α n π , α (A.2) for n = 1, 2, The results for the first values of n are I0 = π , α I2 = π , α3 I4 = π α5 (A.3) Another Gaussian integral of frequent use is I (α, β) = +∞ −∞ d x e−αx +βx = π β2 e 4α , α (A.4) which also allows you to calculate © Springer Nature Switzerland AG 2019 L Angelini, Solved Problems in Quantum Mechanics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-18404-9 www.pdfgrip.com 239 240 Appendix: Useful Formulas +∞ I (α, β) = d x x sin(βx) e−α 2 x = = +∞ eıβx − e−ıβx −α2 x e = 2ı +∞ eıβx + e−ıβx −α2 x dx = e ı −∞ dx x −∞ ∂ =− ∂β β β ∂ − β 22 +∞ e 4α d x e−(αx+ı 2α ) + e−(αx−ı 2α ) = =− ∂β −∞ √ ∂ − β 22 π =− e 4α = ∂β α √ π β − β 22 e 4α = 4α A.1.2 (A.5) Integrals of Exponential Functions and Powers +∞ In (0, ∞) = d x x n e−x = (−1)n = (−1)n dn dα n α α=1 dn dα n +∞ d x e−αx α=1 = n! = (A.6) Similarly, we find the most general result is b In (a, b) = d x x n e−βx = (−1)n a dn I0 (a, b) dβ n (A.7) For a = and b = ∞, we get I0 (0, ∞) = A.2 ; β I1 (0, ∞) = − d = 2; dβ β β I2 (0, ∞) = d2 = (A.8) dβ β β Continuity Equation The continuity equation in Quantum Mechanics is ∂ P(r, t) = −∇ · j(r, t), ∂t www.pdfgrip.com (A.9) Appendix: Useful Formulas 241 where the probability density P is given by P(r, t) = |ψ(r, t)|2 = ψ ∗ (r, t) ψ(r, t) (A.10) and the probability current density is defined as j(r, t) = A.3 2ım ψ ∗ (r, t)∇ψ(r, t) − ψ(r, t)∇ψ ∗ (r, t) = ım (ψ ∗ (r, t)∇ψ(r, t)) (A.11) Harmonic Oscillator A.3.1 Operator Treatment The eigenvalues of the harmonic oscillator Hamiltonian are E n = (n + ) ω , Hˆ |n = E n |n (A.12) In terms of raising and lowering (creation and destruction) operators, a† = mω x −i p, 2mω a= mω x +i p, 2mω (A.13) the position and momentum operators are given by x= 2mω (a + a † ), mω (a − a † ) i p= (A.14) a and a + act on the energy eigenkets as follows a|n = A.3.2 √ n |n − a + |n = √ n + |n + (A.15) Position Basis Treatment The eigenfunctions in the position basis are given by φn (x) = mω π ξ2 e− Hn (ξ ), √ n n! where www.pdfgrip.com ξ= mω x, (A.16) 242 Appendix: Useful Formulas where Hn is the Hermite polynomial defined by d n e−ξ dξ n Hn (ξ ) = (−1)n eξ (A.17) Hermite polynomials are orthogonal polynomials, +∞ −∞ dξ Hn (ξ )Hm (ξ ) = √ n π n! δn,m (A.18) and they satisfy the following recurrence relation: 2ξ Hn (ξ ) = Hn+1 (ξ ) + 2n Hn−1 (ξ ) (A.19) First Hermite polynomials H0 (x) = 1, H1 (x) = 2x, H2 (x) = 4x − 2, H3 (x) = 8x − 12x, H4 (x) = 16x − 48x + 12, H5 (x) = 32x − 160x + 120x (A.20) A.4 Spherical Coordinates The transition from Cartesian coordinates to spherical coordinates occurs through the transformation: x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ A.5 A.5.1 (A.21) (A.22) (A.23) Angular Momentum Operator Treatment The operators J , Jx , Jy , Jz satisfy the following commutation relations: [J , Jx ] = [J , Jy ] = [J , Jz ] = 0, [Jx , Jy ] = i Jz , [Jy , Jz ] = i Jx , www.pdfgrip.com [Jz , Jx ] = i Jy Appendix: Useful Formulas 243 The J and Jz common basis is denoted by | j, m : J | j, m = j ( j + 1) | j, m , Jz | j, m = m | j, m The operators J± = Jx ± i Jy (A.24) satisfy the following commutation relations with the operators J and Jz : [J , J± ] = 0, [Jz , J± ] = ±J± (A.25) J± act on an eigenket common to J and Jz , raising or lowering the azimuthal quantum number: J± | j, m = A.5.2 j ( j + 1) − m(m ± 1) | j, m ± (A.26) Spherical Harmonics Definition Y ,m (θ, φ) = (−1)m + ( − |m|)! m P (cos θ ) eımφ , 4π ( + |m|)! (A.27) where P m are the Legendre associate functions defined for |z| ≤ 1, P m (z) = (1 − z ) |m| d |m| P (z), dz |m| (A.28) which, for m = 0, give us the Legendre polynomials P (z), P (z) = d (1 − z ) ! dz (A.29) They are orthogonal polynomials: +1 −1 dz P (z)P (z) = δ +1 , (A.30) Particular values of Legendre polynomials and associate functions: P (±1) = (±1) , P m (±1) = for m = www.pdfgrip.com (A.31) 244 Appendix: Useful Formulas First Legendre polynomials: (3z − 1), (A.32) 1 (5z − 3z), P4 (z) = (35z − 30z + 3) (A.33) P0 (z) = 1, P1 (z) = z, P2 (z) = P3 (z) = Orthonormalization relationship Y ∗,m (θ, φ) Y d ,m (θ, φ) =δ , δm,m (A.34) −1,m Y −1,m (θ, φ), (A.35) Recurrence relationship cos θ Y ,m (θ, φ) =a ,m Y +1,m (θ, φ) +a where a ,m = ( + + m)( + − m) (2 + 1)(2 + 3) (A.36) Sum theorem If ( , ) and (θ , φ ) are two space directions and θ is the angle between them, a Legendre polynomial can be expressed in terms of spherical harmonics: + P (cos θ ) = 4π Y + m=− ,m ( , )∗ Y ,m (θ , φ ) (A.37) First Spherical Harmonics Y0,0 (θ, φ) = √ , 4π Y1,0 (θ, φ) = Y2,0 (θ, φ) = 3 cos θ, Y1,±1 (θ, φ) = ∓ sin θ e±ıφ , 4π 8π (A.38) (A.39) 15 (3 cos2 θ − 1), Y2,±1 (θ, φ) = ∓ sin θ cos θ e±ıφ , 16π 8π Y2,±2 (θ, φ) = 15 sin2 θ e±2ıφ 32π www.pdfgrip.com (A.40) Appendix: Useful Formulas 21 (5 cos3 θ − cos θ ) , Y3,±1 (θ, φ) = ∓ sin θ (5 cos2 θ − 1)e±ıφ , 16π 64π Y3,0 (θ, φ) = 105 35 sin2 θ cos θ e±2ıφ , Y3,±3 (θ, φ) = ∓ sin3 θ e±3ıφ 32π 64π Y3,±2 (θ, φ) = A.6 A.6.1 245 (A.41) Schrödinger Equation in Spherical Coordinates Radial Equation If the potential energy V (r ) is central, the Schrödinger equation is separable into spherical coordinates The eigenfunction common to the operators H, L and L z , with eigenvalues E, ( + 1) and m , respectively, can be written in the form ψ E, ,m (r, θ, φ) = R E, (r ) Y ,m (θ, φ) = U E, (r ) Y r ,m (θ, φ), (A.42) where U E, (r ) is the solution to the radial equation: − d U E, + 2m dr 2 ( + 1) U E, + V (r )U E, = E U E, , 2mr (A.43) with m reduced mass of the system U E, (r ) must satisfy the condition lim U E, (r ) = r →0 A.7 (A.44) Spherical Bessel Functions The Spherical Bessel functions are solutions to the Spherical Bessel equation z2 A.7.1 d2 d φ(z) + 2z φ(z) + z − ( + 1) φ(z) = dz dz (A.45) Spherical Bessel Functions of the First and Second Kinds Two linearly independent integrals of (A.45) are given by the spherical Bessel functions of the first and second kinds j and y = (−1) +1 j− −1 For the first integer values of , they are www.pdfgrip.com 246 Appendix: Useful Formulas sin z , z (A.46) sin z cos z , − z2 z (A.47) j0 (z) = j1 (z) = cos z sin z − −1 , z2 z z2 j2 (z) = j3 (z) = sin z − z 15 − z3 z and y2 (z) = − y3 (z) = − (A.50) cos z sin z , − z2 z (A.51) sin z cos z − +1 , z2 z z2 15 + z3 z (A.49) cos z , z y0 (z) = − y1 (z) = − cos z 15 −1 , z z (A.48) cos z − z (A.52) 15 sin z −1 z2 z (A.53) Their asymptotic behavior is given by j (z) ∼ cos z − z +1 π (A.54) y (z) ∼ sin z − z +1 π , (A.55) z→∞ and z→∞ while the behavior in the origin is given by j (z) ∼ z→0 z (2 + 1)!! and y (z) ∼ − z→∞ (2 − 1)!! z +1 www.pdfgrip.com (A.56) (A.57) Appendix: Useful Formulas A.7.2 247 Spherical Hankel Functions Other linearly independent solutions to the Spherical Bessel equation are the spherical Hankel functions of the first and second kinds defined by h (1) (z) = j (z) + ı y (z), (A.58) h (2) (z) = j (z) − ı y (z) (A.59) Their asymptotic behavior is given by h (1) (z) ∼ ı (z− e z h (2) (z) ∼ −ı (z− e z z→∞ z→∞ +1 π) +1 , (A.60) π) (A.61) When the argument is an imaginary number, the Hankel functions have an exponential asymptotic behavior: (−z−ı +1 π ) e , (A.62) h (1) (ı z) ∼ z→∞ ı z and h (2) (ı z) ∼ z→∞ A.8 (z+ı e ız +1 π) (A.63) Hydrogen Atom First Energy Eigenfunctions Having introduced a0 = μe2 , the Bohr radius, the first two energy eigenfunctions are −3 − r ψ1,0,0 = √ a0 e a0 , π r −3 − r e 2a0 , ψ2,0,0 = √ a0 2 − a 2π − 23 r − r e 2a0 cos θ, ψ2,1,0 = √ a0 a0 2π −3 r − r ψ2,1,±1 = √ a0 e 2a0 sin θ e±ıϕ a0 2π www.pdfgrip.com (A.64) (A.65) (A.66) (A.67) 248 A.9 A.9.1 Appendix: Useful Formulas Spin Pauli Matrices σ1 = 01 10 , σ2 = −i i σi σ j = δi j + , σ3 = −1 i jk σk , (A.69) {σi , σ j } = σi σ j + σ j σi = 2δi j , [σi σ j ] = σi σ j − σ j σi = 2i A.9.2 (A.68) (A.70) i jk σk (A.71) Useful Relationships (A · σ ) (B · σ ) = (A · B) I + i (A × B) · σ , (A.72) where I is the identity matrix In particular, if A = B, (A · σ )2 = A2 I, eiθ ·σ = I cos θ + i(n · σ ) sin θ, where n = A.10 (A.73) θ θ (A.74) Time-Independent Perturbation Theory Let us consider the Hamiltonian H = H0 + H1 , where the solution to the H0 eigenvalue problem is supposed to be known: H0 |n = E n0 |n If E n0 is not a degenerate eigenvalue and the matrix elements m |H1 |n are small compared to E n0 , given the following expansions for the H eigenvalues E n and eigenkets |n : www.pdfgrip.com Appendix: Useful Formulas 249 E n = E n0 + E n1 + E n2 + · · · , |n = |n + |n + |n + · · · , we get E n1 = n |H1 |n , E n2 = m=n |n = m=n (A.75) | m |H1 |n |2 , E n0 − E m0 (A.76) m |H1 |n |m E n0 − E m0 (A.77) If E n0 is degenerate, the first-order corrections to the eigenvalues are given from the eigenvalues of the matrix representative of H1 in the E n0 eigenspace, obtained from det (H1 )m, j − E n1 δm, j = A.11 (A.78) Sudden Perturbation A sudden perturbation is an abrupt change of the Hamiltonian H0 → H = H0 + H , where H0 and H1 not depend on time A sudden perturbation does not modify the state vector Assuming that the system is initially in a state |n , an eigenket of H0 , the probability of measuring an energy E k , eigenvalue of the new Hamiltonian, that is, the probability of the transition |n → |k , is given by Pn→k = | k|n |2 (A.79) If it make sense to apply Perturbation Theory for non-degenerate eigenvalues, the probability of a transition to states k = n is Pn→k = k |H1 |n E k0 − E n0 www.pdfgrip.com (A.80) 250 A.12 Appendix: Useful Formulas Time-Dependent Perturbation Theory Let us consider a Hamiltonian H = H0 + H1 (t) for which the solution to the H0 eigenvalue problem is known: H0 |n = E n0 |n Suppose that H1 depends on time and the matrix elements m |H1 |n are small compared to E n0 We can write the system state vector in the form dn (t) e−i |ψ(t) = E n0 t |n (A.81) n Having called the probability of finding the system in the state vector | f as Pi→ f , provided that, at time t = 0, it is in state |i (0) , at first perturbative order, it results that t ı Pi→ f (t) = |d f (t)|2 = − dτ f |H1 (τ )|i eıω f i τ , (A.82) where ω f i = A.13 E 0f −E i0 and f = i Born Approximation Having called the wave vectors of the incident and deflected particle as k and k , respectively, the Born approximation to the scattering amplitude for the potential V (r) is given by f B (k, k ) = − m 2π dr e−ik ·r V (r) eik·r , (A.83) where m is the system reduced mass If the potential is central, the expression simplifies: f B (q) = − 2m 2q ∞ dr sin(qr ) V (r ) r, (A.84) where, since we consider elastic scattering, q = |k − k | = 2k sin angle of deflection www.pdfgrip.com θ and θ is the Appendix: Useful Formulas A.14 251 WKB Approximation We consider a one-dimensional system of a particle with mass m and energy E subject to a potential V (x) and define p(x) as the classical momentum: p(x) = E − V (x) If the energy E is less than the potential V (x) for each point outside of a certain range [a, b], the eigenvalue E belongs to the discrete spectrum In the WKB approximation, the eigenvalues are given by the relationship a b d x p(x) = (n + )π with n = 0, 1, 2, (A.85) Equivalently, if we consider an entire classical oscillation between the two classical turning points a and b and back to a, this relation can be rewritten in the form of the Bohr-Sommerfeld quantization rule: d x p(x) = d x d p = 2π (n + ), D with n = 0, 1, 2, (A.86) where, in the first expression, the integral is extended to the complete classical trajectory and, in the second one, to the D domain delimited by it If the energy E is greater than the potential V (x) for each point outside of the interval [a, b], the eigenvalue E belongs to the continuous spectrum and we are in the presence of a potential barrier The probability of crossing the barrier in WKB approximation is given by b (A.87) T = e− a dy | p(y)| www.pdfgrip.com References Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Dover New York (1972) Flügge, S.: Practical Quantum Mechanics, volume I e II Springer, Berlin (1971) Gasiorowicz, S.: Quantum Physics Wiley, New York (2003) Gol’dman, I.I., Krivchenkov, V.D., Kogan, V.I., Galitskiy, V.M.: Selected Problems in Quantum Mechanics Infosearch London (1960) Gol’dman, I.I., Krivchenkov, V.D.: Problems in Quantum Mechanics Pergamon Press London (1961) Gottfried, K., Yan, T.-M.: Quantum Mechanics: Fundamentals, ii edn Springer, Berlin (2004) Kogan, V.I., Galitskiy, V.M.: Problems in Quantum Mechanics Prentice-Hall London (1963) Landau, L., Lifchitz, E.: Physique Theorique vol III (Mecanique Quantique) Mir Moscou (1966) Lim, Y.-K (ed.): Problems and Solutions on Quantum Mechanics World Scientific (1999) 10 Merzbacher, E.: Quantum Mechanics Wiley, New York (1970) 11 Messiah, A.: Mecanique Quantique, volume I and II Dunod Paris (1962) 12 Nardulli, G.: Meccanica Quantistica, volume I and II Franco Angeli Milano (2001) 13 Passatore, G.: Problemi di meccanica quantistica elementare Franco Angeli Milano, ii edition (1981) 14 Sakurai, J.J., Napolitano, J.: Modern Quantum Mechanics, ii edn Addison-Wesley (2010) 15 Shankar, R.: Principles of Quantum Mechanics, 2nd edn Plenum Press, New York (1994) 16 Ter Haar, D.: Selected problems in Quantum Mechanics Infosearch Ltd., London (1964) © Springer Nature Switzerland AG 2019 L Angelini, Solved Problems in Quantum Mechanics, UNITEXT for Physics, https://doi.org/10.1007/978-3-030-18404-9 www.pdfgrip.com 253 ... teaching More information about this series at http://www.springer.com/series/13351 www.pdfgrip.com Leonardo Angelini Solved Problems in Quantum Mechanics 123 www.pdfgrip.com Leonardo Angelini... Spin 4.1 4.2 4.3 4.4 4.5 Total Spin of Two Electrons Eigenstates of a Spin Component (I) Eigenstates of a Spin Component (II) Determining a Spin State (I) Determining a Spin... power expansion in the variables qi of the potential V = n n cn qi , we obtain [ pi , V] = cn [ pi , qin ] = n cn [qin pi − i nqin−1 − qin pi ] = cn [ pi qin − qin pi ] = n n cn nqin−1 = −i = −i