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QUANTUM MECHANICS Professor John W. Norbury Physics Department University of Wisconsin-Milwaukee P.O. Box 413 Milwaukee, WI 53201 November 20, 2000 Contents 1 WAVE FUNCTION 7 1.1 Probability Theory 8 1.1.1 Mean, Average, Expectation Value 8 1.1.2 Average of a Function 10 1.1.3 Mean, Median, Mode 10 1.1.4 Standard Deviation and Uncertainty 11 1.1.5 Probability Density 14 1.2 Postulates of Quantum Mechanics 14 1.3 Conservation of Probability (Continuity Equation) 19 1.3.1 Conservation of Charge 19 1.3.2 Conservation of Probability 22 1.4 Interpretation of the Wave Function 23 1.5 Expectation Value in Quantum Mechanics 24 1.6 Operators 24 1.7 Commutation Relations 27 1.8 Problems 32 1.9 Answers 33 2 DIFFERENTIAL EQUATIONS 35 2.1 Ordinary Differential Equations 35 2.1.1 Second Order, Homogeneous, Linear, Ordinary Differ- ential Equations with Constant Coefficients 36 2.1.2 Inhomogeneous Equation 39 2.2 Partial Differential Equations 42 2.3 Properties of Separable Solutions 44 2.3.1 General Solutions 44 2.3.2 Stationary States 44 2.3.3 Definite Total Energy 45 1 2 CONTENTS 2.3.4 Alternating Parity 46 2.3.5 Nodes 46 2.3.6 Complete Orthonormal Sets of Functions 46 2.3.7 Time-dependent Coefficients 49 2.4 Problems 50 2.5 Answers 51 3 INFINITE 1-DIMENSIONAL BOX 53 3.1 Energy Levels 54 3.2 Wave Function 57 3.3 Problems 63 3.4 Answers 64 4 POSTULATES OF QUANTUM MECHANICS 65 4.1 Mathematical Preliminaries 65 4.1.1 Hermitian Operators 65 4.1.2 Eigenvalue Equations 66 4.2 Postulate 4 67 4.3 Expansion Postulate 68 4.4 Measurement Postulate 69 4.5 Reduction Postulate 70 4.6 Summary of Postulates of Quantum Mechanics (Simple Version) 71 4.7 Problems 74 4.8 Answers 75 I 1-DIMENSIONAL PROBLEMS 77 5 Bound States 79 5.1 Boundary Conditions 80 5.2 Finite 1-dimensional Well 81 5.2.1 Regions I and III With Real Wave Number 82 5.2.2 Region II 83 5.2.3 Matching Boundary Conditions 84 5.2.4 Energy Levels 87 5.2.5 Strong and Weak Potentials 88 5.3 Power Series Solution of ODEs 89 5.3.1 Use of Recurrence Relation 91 5.4 Harmonic Oscillator 92 CONTENTS 3 5.5 Algebraic Solution for Harmonic Oscillator 100 5.5.1 Further Algebraic Results for Harmonic Oscillator . . 108 6 SCATTERING STATES 113 6.1 Free Particle 113 6.1.1 Group Velocity and Phase Velocity 117 6.2 Transmission and Reflection 119 6.2.1 Alternative Approach 120 6.3 Step Potential 121 6.4 Finite Potential Barrier 124 6.5 Quantum Description of a Colliding Particle 126 6.5.1 Expansion Coefficients 128 6.5.2 Time Dependence 129 6.5.3 Moving Particle 130 6.5.4 Wave Packet Uncertainty 131 7 FEW-BODY BOUND STATE PROBLEM 133 7.1 2-Body Problem 133 7.1.1 Classical 2-Body Problem 134 7.1.2 Quantum 2-Body Problem 137 7.2 3-Body Problem 139 II 3-DIMENSIONAL PROBLEMS 141 8 3-DIMENSIONAL SCHR ¨ ODINGER EQUATION 143 8.1 Angular Equations 144 8.2 Radial Equation 147 8.3 Bessel’s Differential Equation 148 8.3.1 Hankel Functions 150 9 HYDROGEN-LIKE ATOMS 153 9.1 Laguerre Associated Differential Equation 153 9.2 Degeneracy 157 10 ANGULAR MOMENTUM 159 10.1 Orbital Angular Momentum 159 10.1.1 Uncertainty Principle 162 10.2 Zeeman Effect 163 10.3 Algebraic Method 164 4 CONTENTS 10.4 Spin 165 10.4.1 Spin 1 2 166 10.4.2 Spin-Orbit Coupling 167 10.5 Addition of Angular Momentum 169 10.5.1 Wave Functions for Singlet and Triplet Spin States . . 171 10.5.2 Clebsch-Gordon Coefficients 172 10.6 Total Angular Momentum 172 10.6.1 LS and jj Coupling 173 11 SHELL MODELS 177 11.1 Atomic Shell Model 177 11.1.1 Degenerate Shell Model 177 11.1.2 Non-Degenerate Shell Model 178 11.1.3 Non-Degenerate Model with Surface Effects 178 11.1.4 Spectra 179 11.2 Hartree-Fock Self Consistent Field Method 180 11.3 Nuclear Shell Model 181 11.3.1 Nuclear Spin 181 11.4 Quark Shell Model 182 12 DIRAC NOTATION 183 12.1 Finite Vector Spaces 183 12.1.1 Real Vector Space 183 12.1.2 Complex Vector Space 185 12.1.3 Matrix Representation of Vectors 188 12.1.4 One-Forms 188 12.2 Infinite Vector Spaces 189 12.3 Operators and Matrices 191 12.3.1 Matrix Elements 191 12.3.2 Hermitian Conjugate 194 12.3.3 Hermitian Operators 195 12.3.4 Expectation Values and Transition Amplitudes 197 12.4 Postulates of Quantum Mechanics (Fancy Version) 198 12.5 Uncertainty Principle 198 13 TIME-INDEPENDENT PERTURBATION THEORY, HY- DROGEN ATOM, POSITRONIUM, STRUCTURE OF HADRONS201 13.1 Non-degenerate Perturbation Theory 204 13.2 Degenerate Perturbation Theory 208 CONTENTS 5 13.2.1 Two-fold Degeneracy 209 13.2.2 Another Approach 211 13.2.3 Higher Order Degeneracies 212 13.3 Fine Structure of Hydrogen 212 13.3.1 1-Body Relativistic Correction 212 13.3.2 Two-Body Relativistic Correction 216 13.3.3 Spin-Orbit Coupling 217 13.4 Zeeman effect 220 13.5 Stark effect 221 13.6 Hyperfine splitting 221 13.7 Lamb shift 221 13.8 Positronium and Muonium 221 13.9 Quark Model of Hadrons 221 14 VARIATIONAL PRINCIPLE, HELIUM ATOM, MOLECULES223 14.1 Variational Principle 223 14.2 Helium Atom 223 14.3 Molecules 223 15 WKB APPROXIMATION, NUCLEAR ALPHA DECAY 225 15.1 Generalized Wave Functions 225 15.2 Finite Potential Barrier 230 15.3 Gamow’s Theory of Alpha Decay 231 16 TIME-DEPENDENT PERTURBATION THEORY, LASERS235 16.1 Equivalent Schr¨odinger Equation 236 16.2 Dyson Equation 240 16.3 Constant Perturbation 241 16.4 Harmonic Perturbation 244 16.5 Photon Absorption 247 16.5.1 Radiation Bath 247 16.6 Photon Emission 249 16.7 Selection Rules 249 16.8 Lasers 250 17 SCATTERING, NUCLEAR REACTIONS 251 17.1 Cross Section 251 17.2 Scattering Amplitude 252 17.2.1 Calculation of c l 255 6 CONTENTS 17.3 Phase Shift 257 17.4 Integral Scattering Theory 259 17.4.1 Lippman-Schwinger Equation 259 17.4.2 Scattering Amplitude 261 17.4.3 Born Approximation 262 17.5 Nuclear Reactions 264 18 SOLIDS AND QUANTUM STATISTICS 265 18.1 Solids 265 18.2 Quantum Statistics 265 19 SUPERCONDUCTIVITY 267 20 ELEMENTARY PARTICLES 269 21 chapter 1 problems 271 21.1 Problems 271 21.2 Answers 272 21.3 Solutions 273 22 chapter 2 problems 281 22.1 Problems 281 22.2 Answers 282 22.3 Solutions 283 23 chapter 3 problems 287 23.1 Problems 287 23.2 Answers 288 23.3 Solutions 289 24 chapter 4 problems 291 24.1 Problems 291 24.2 Answers 292 24.3 Solutions 293 Chapter 1 WAVE FUNCTION Quantum Mechanics is such a radical and revolutionary physical theory that nowadays physics is divided into two main parts, namely Classical Physics versus Quantum Physics. Classical physics consists of any theory which does not incorporate quantum mechanics. Examples of classical theories are Newtonian mechanics (F = ma), classical electrodynamics (Maxwell’s equa- tions), fluid dynamics (Navier-Stokes equation), Special Relativity, General Relativity, etc. Yes, that’s right; Einstein’s theories of special and general relativity are regarded as classical theories because they don’t incorporate quantum mechanics. Classical physics is still an active area of research today and incorporates such topics as chaos [Gleick 1987] and turbulence in fluids. Physicists have succeeded in incorporating quantum mechanics into many classical theories and so we now have Quantum Electrodynamics (combi- nation of classical electrodynamics and quantum mechanics) and Quantum Field Theory (combination of special relativity and quantum mechanics) which are both quantum theories. (Unfortunately no one has yet succeeded in combining general relativity with quantum mechanics.) I am assuming that everyone has already taken a course in Modern Physics. (Some excellent textbooks are [Tipler 1992, Beiser 1987].) In such a course you will have studied such phenomena as black-body radi- ation, atomic spectroscopy, the photoelectric effect, the Compton effect, the Davisson-Germer experiment, and tunnelling phenomena all of which cannot be explained in the framework of classical physics. (For a review of these topics see references [Tipler 1992, Beiser 1987] and chapter 40 of Serway [Serway 1990] and chapter 1 of Gasiorowicz [Gasiorowicz 1996] and chapter 2 of Liboff [Liboff 1992].) 7 8 CHAPTER 1. WAVE FUNCTION The most dramatic feature of quantum mechanics is that it is a proba- bilistic theory. We shall explore this in much more detail later, however to get started we should review some of the basics of probability theory. 1.1 Probability Theory (This section follows the discussion of Griffiths [Griffiths 1995].) College instructors always have to turn in student grades at the end of each semester. In order to compare the class of the Fall semester to the class of the Spring semester one could stare at dozens of grades for awhile. It’s much better though to average all the grades and compare the averages. Suppose we have a class of 15 students who receive grades from 0 to 10. Suppose 3 students get 10, 2 students get 9, 4 students get 8, 5 students get 7, and 1 student gets 5. Let’s write this as N(15) = 0 N(10) = 3 N(4)=0 N(14) = 0 N(9)=2 N(3)=0 N(13) = 0 N(8)=4 N(2)=0 N(12) = 0 N(7)=5 N(1)=0 N(11) = 0 N(6)=0 N(0)=0 N(5)=1 where N(j) is the number of students receiving a grade of j. The histogram of this distribution is drawn in Figure 1.1. The total number of students, by the way, is given by N = ∞ j=0 N(j) (1.1) 1.1.1 Mean, Average, Expectation Value We want to calculate the average grade which we denote by the symbol ¯ j or j. The mean or average is given by the formula ¯ j ≡j = 1 N all j (1.2) where all j means add them all up separately as j = 1 15 (10+10+10+9+9+8+8+8+8+7+7+7+7+7+7+5) =8.0 (1.3) 1.1. PROBABILITY THEORY 9 Thus the mean or average grade is 8.0. Instead of writing many numbers over again in (1.3) we could write ¯ j = 1 15 [(10 ×3)+(9× 2)+(8× 4)+(7× 5)+(5×1)] (1.4) This suggests re-writing the formula for average as j≡ ¯ j = 1 N ∞ j=0 jN(j) (1.5) where N(j)=number of times the value j occurs. The reason we go from 0to∞ is because many of the N (j) are zero. Example N(3) = 0. No one scored 3. We can also write (1.4) as ¯ j = 10 × 3 15 + 9 × 2 15 + 8 × 4 15 + 7 × 5 15 + 5 × 1 15 (1.6) where for example 3 15 is the probability that a random student gets a grade of 10. Defining the probability as P (j) ≡ N(j) N (1.7) we have j≡ ¯ j = ∞ j=0 jP(j) (1.8) Any of the formulas (1.2), (1.5) or (1.8) will serve equally well for calculating the mean or average. However in quantum mechanics we will prefer using the last one (1.8) in terms of probability. Note that when talking about probabilities, they must all add up to 1 3 15 + 2 15 + 4 15 + 5 15 + 1 15 =1 . That is ∞ j=0 P (j) = 1 (1.9) Student grades are somewhat different to a series of actual measurements which is what we are more concerned with in quantum mechanics. If a bunch of students each go out and measure the length of a fence, then the j in (1.1) will represent each measurement. Or if one person measures the [...]... which will give us the answer more quickly Expand (1.15) as j 2 − 2j j + j σ2 = j 2 P (j) − 2 j = 2 P (j) jP (j) + j 2 P (j) where we take j and j 2 outside the sum because they are just numbers ( j = 8.0 and j 2 = 64.0 in above example) which have already been summed over Now jP (j) = j and P (j) = 1 Thus σ2 = j 2 − 2 j 2 + j 2 giving σ2 = j 2 − j 2 (1.16) Example 1.1.2 Repeat example 1.1.1 using equation... is just like the ordinary density ρ of water The total mass of water is M = ρdV where ρdV is the mass of water between volumes V and V + dV Our old discrete formulas get replaced with new continuous formulas, as follows: ∞ P (j) = 1 → j= 0 ∞ j = j= 0 ∞ f (j) = ∞ −∞ ρ(x)dx = 1 jP (j) → x = −∞ xρ(x)dx j= 0 2 (1.19) ∞ −∞ f (x)ρ(x)dx (j − j )2 P (j) → σ 2 ≡ (∆x)2 = = j2 − j (1.18) ∞ f (j) P (j) → f (x) = j= 0... in quantum mechanics 1.1.2 Average of a Function Suppose that instead of the average of the student grades, you wanted the average of the square of the grades That’s easy It’s just ¯2 ≡ j 2 = 1 j N 1 j = N ∞ 2 all 2 ∞ j N (j) = j= 0 j 2 P (j) (1.10) j= 0 Note that in general the average of the square is not the square of the average j2 = j 2 (1.11) In general for any function f of j we have ∞ f (j) =... precise) to use the symbol σ rather than ∆, otherwise we get confused with (1.13) (Nevertheless many quantum mechanics books use expectation value, uncertainty and ∆.) The average squared distance or variance is simple to define It is = 1 N = σ 2 ≡ ( j) 2 1 N ∞ = ( j) 2 all (j − j )2 all (j − j )2 P (j) (1.15) j= 0 1 Note: Some books use N 1 instead of N in (1.15) But if N 1 is used −1 −1 then equation (1.16)... on Newtonian 1.2 POSTULATES OF QUANTUM MECHANICS 15 mechanics, all you ever do is solve F = ma In a course on electromagnetism you spend all your time just solving Maxwell’s equations Thus these fundamental equations are the theory All the rest is just learning how to solve these fundamental equations in a wide variety of circumstances The fundamental equation of quantum mechanics is the Schr¨dinger... j2 − j (1.18) ∞ f (j) P (j) → f (x) = j= 0 σ 2 ≡ ( j) 2 = ∞ ∞ −∞ (1.20) (x − x )2 ρ(x)dx = x2 − x 2 (1.21) In discrete notation j is the measurement, but in continuous notation the measured variable is x (do Problem 1.3) 1.2 Postulates of Quantum Mechanics Most physical theories are based on just a couple of fundamental equations For instance, Newtonian mechanics is based on F = ma, classical electrodynamics... from the average than the other distribution The “distance” of a particular point from the average can be written j ≡ j − j (1.13) But for points with a value less than the average this distance will be negative Let’s get rid of the sign by talking about the squared distance ( j) 2 ≡ (j − j )2 (1.14) Then it doesn’t matter if a point is larger or smaller than the average Points an equal distance away... In quantum mechanics each state of a physical system is specified by only one variable, namely the wave function Ψ(x, t) which is a function of the two variables position x and time t Footnote: In classical mechanics the state of a system is specified by x(t) 1.2 POSTULATES OF QUANTUM MECHANICS 17 and p(t) or Γ(x, p) In 3-dimensions this is x(t) and p(t) or Γ(x, y, px , py ) or Γ(r, θ, pr , pθ ) In quantum. .. and E ≡ i¯ ∂t and x ≡ x ˆ h h ˆ The essence of quantum mechanics are these operators Because they are operators they satisfy (1.75) An alternative way of introducing quantum mechanics is to change the classical commutation relation [x, p]classical = 0 to [ˆ, p] = i¯ which can only x ˆ h be satisfied by x = x and p = −i¯ ∂x ˆ ˆ h∂ Thus to “derive” quantum mechanics we either postulate operator definitions... Ψdx because I don’t know what to put in for p under the integral But wait! We just saw that in quantum mechanics p is an operator given in (1.58) The proper way to write the expectation value is ∞ p ˆ ∂ Ψdx ∂x −∞ ∞ ∂Ψ = −i¯ h Ψ∗ dx ∂x −∞ = Ψ∗ −i¯ h (1.62) (1.63) ˆ In fact the expectation value of any operator Q in quantum mechanics is ˆ Ψ∗ QΨdx ˆ Q ≡ (1.64) which is a generalization of (1.19) or (1.50) . as σ 2 = j 2 − 2j j + j 2 P (j) = j 2 P (j) − 2 j jP (j) + j 2 P (j) where we take j and j 2 outside the sum because they are just numbers ( j . square of the grades. That’s easy. It’s just ¯ j 2 ≡ j 2 = 1 N all j 2 = 1 N ∞ j= 0 j 2 N (j) = ∞ j= 0 j 2 P (j) (1.10) Note that in general the average