The Joy of Quantum Physics_Appendices_only

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The Delights ofthe Appendices to

The Joy of Quantum Physics

Michael A. Morrison

Homer L. Dodge Department of Physics & Astronomy

University of Oklahoma

Version 8.37: August 3, 2010

Date Printed: August 3, 2010

c2010 by Michael A. Morrison.Not to be distributed or copied without permission of the author.


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ContentsVersion 8.37: August 3, 2010

A. Quantum mechanics: Greatest Hits 1A.1. Of states, ensembles, and measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1A.2. Three great ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A.3. Four great postulatesand their consequences . . . . . . . . . . . . . . . . . . . . . . . . . . 4A.4. S tationary states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16A.5. Non-stationary states and the method of eigenfunction expansion . . . . . . . . . . . . . . . 20A.6. Generic problem-solving strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23A.7. Measurements in the microverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

B. Dimensional extermination 26B.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26B.2. Transformation procedure and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27B.3. Transforming the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31B.4. Conversion back to dimensional quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.5. Solving the dimensionless TISE for the Morse potential . . . . . . . . . . . . . . . . . . . . . 35B.6. Scale transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

B.7. F inal exhortations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

C. Constants and estimates 42

D. The Conversion Factory 47

E. SI units 50

F. Atomic units 53F.1. Introducing atomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53F.2. How to make atomic units work for you . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55F.3. Atomic unit replacement rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

G. Dirac notation 64G.1. Dirac shorthand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65G.2. Dirac notation and Hermiticity of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66G.3. Eigenfunction expansions in Dirac notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67G.4. Useful properties in Dirac notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68G.5. Projecting a state out of the time-independent Schrodinger equation . . . . . . . . . . . . . . 68

H. The mathematics of operators 70H.1. Operators in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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CONTENTS ii

I. Matrices and determinants 79I.1. Useful properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81I.2. Eigenvalues, eigenvectors, and matrix diagonalization . . . . . . . . . . . . . . . . . . . . . . . 81I.3. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

J. Making sense of spectral data 88

K. Angular-momentum coupling 94K.1. The total orbital angular momentum operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 95K.2. Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95K.3. The fundamental expansions of angular-momentum coupling . . . . . . . . . . . . . . . . . . 96K.4. Eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98K.5. Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

L. The prefix dictionary 102

M.The Greek alphabet 103

N. The Dirac delta function 104N.1. A Dirac-delta primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

N.2. A Dirac- delta users guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

O. Waves and the de Broglie-Einstein relations 109

P. The one-dimensional simple harmonic oscillator 113P.1. Hamiltonian eigenfunctions and stationary-state energies . . . . . . . . . . . . . . . . . . . . . 113P.2. Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117P.3. Momentum-space wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Q. The Seven Habits of Highly Effective Problem Solvers 120Q.1. Why you need to develop new habits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Q.2. Im plem entation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Q.3. The Principle of Brainstorming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Q.4. The Principle of Symmetry Seeking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Q.5. The Principle of Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Q.6. The Principle of Pattern Seeking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Q.7. The Principle of Back-to-Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Q.8. The Principle of Least Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Q.9. The Principle of Alert Awareness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Index 127

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Appendix A

Quantum mechanics: Greatest Hits


Before laying it before the public, it would be as well, perhaps,that I should refresh their memories as to the simpler facts

upon which this commentary is founded.The Man with the Watches, by Sir Arthur Conan Doyle

Quantum mechanics is a collection of postulates and consequences thereof, formulated in the language ofmathematics, which provides tools for the analysis, prediction, and understanding of observed phenomenain the microscopic domain. This appendix reviews the key ideas and postulates of quantum physics. It thenrecaps the high points of the two types of quantum states found in nature, stationary and non-stationarystates. These introductory sections set the stage for a birds eye tour of the major mathematical machineryof quantum mechanics and of strategies and tactics for conquering quantum problems. The last sectionbriefly discusses the effect of measurement on a quantum state. You can find additional information aboutDirac notation in Appendix G and about the mathematics of operators in Appendix H.

A.1 Of states, ensembles, and measurement

The most important notion in quantum physics is that of a state of a physical system. We associate aparticular quantum state with probabilistic information about the observables of the system. In classicaland in quantum physics, an observable is a physically measurable quantity. The mathematical tool we useto represent an observable in quantum mechanics is an operator.1

1Notation: Some authors use the term dynamical variable for what I call an observable. Others use observable andoperator interchangeably. To distinguish the two, Ill put a little hat on operators but not on observables. Thus p is theobservable linear momentum, while p is the corresponding operator. The sole exception to this rule is the energy E, for whichthe operator is the Hamiltonian H.


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Appendix 2

A.1.1 States and their wave functions

In introductory quantum mechanics, we represent a quantum state mathematically by a wave function. Inthe simplest representation, the wave function depends on the position variables appropriate to the systemand on time. The most general wave function represents a state in which the various observables of thesystem do not have specific values; rather, each observable can assume one of a number of possible allowedvalues, each with a non-zero probability. The process of measuring a particular observable actualizes oneof these possibilities, leaving the system in a state where the measured observable does have a definite value(see A.7).Ensembles. Since the information contained in the wave function is inherently probabilistic, this infor-mation does not describe individual systems. Rather it describes a quantum-mechanical ensemble (or,simply, an ensemble): a huge number of identical systems all of which are in the same quantum state(see A.1.2). Each system is called a member of the ensemble. According to the Born interpretation,the wave function contains information about the probability that a member of the ensemble will exhibit aparticular value of an observable upon measurement of that observable. So when we talk about a quantumstate, were talking about an ensemble, not an individual system.

Bound versus continuum states. States come in two varieties: bound and continuum. Physically, thedistinction is simple: a particle in a bound state is spatially localized for all time in one region of spacenear

the potential energy that binds it. By contrast, a particle in a continuum state suffers no such restriction.If the continuum state is non-stationary, then the region of localization (that is, the region where the particleis most likely and be found in a position measurement) can move throughout space as time passes. To clarifythe distinction, consider the hydrogen atom: one electron in the attractive Coulomb field of a proton. In abound state, the probability of finding the electron far from the nucleus diminishes (exponentially) to zeroas distance from the nucleus increases to infinity (Chap. 5). By contrast, in a continuum state, the electroncan escape the vicinity of the proton to be found at arbitrarily large distances from it. (In this case,the electron is ionized.) Continuum states are especially important because of their role in the dynamics ofcollisions. For this reason, a continuum state is often called a scattering state.

A.1.2 Measurement and the Born interpretation

In general, quantum mechanics cannot predict the value that a single member of an ensemble will exhibitin a measurement of an observable. So we must be very careful when we interpret experiments. In quantumphysics, measurement has a precise meaning that follows from the intrinsically probabilistic character ofinformation. By a measurement on a microscopic system, we mean an ensemble measurement: identicalmeasurements performed on a huge number of identical systems that have been prepared so that prior tomeasurement each is in the same quantum state.2

The notion of an ensemble measurement helps us understand the nature of information in a wave func-tion. From a wave function we can determine the fraction of members of the ensemble that exhibit eachallowed value of whatever observable were measuring. A familiar instance is position: according to the Borninterpretation, we calculate information about a particles position by squaring the modulus of its wavefunction. Thus the wave function is a probability amplitude for various possible outcomes of a positionmeasurement. The generalized Born interpretation extends this approach to all observables. In a par-ticular quantum state every observable is characterized by a probability distribution of allowed values; wedetermine the probability for each allowed value from quantities we calculate from the wave function. Themathematics we use to perform this minor miracle is the method of eigenfunction expansion (A.5).

2Details: To ensure that all members of the ensemble are in the same state, they must be prepared identically. Of course,identical preparation does not imply that these members will behave the same in a subsequent ensemble measurement. Thisspecial outcomeall members exhibiting the same value of an observableoccurs only if the state prior to measurement wasan eigenstate of that observablethat is, in that state, the observable is sharp.



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Appendix 3

A.2 Three great ideas

The intrinsic weirdness of quantum physics is prominently on display in the three great ideas which, earlyin the 20th century, marked the transition from classical to quantum physics. These ideas are the heartand soul of the physics of matter. All three violate both our intuition as human beings and the insight we

gained from studying classical physics. The break with classical physics is especially dramatic: To describe amicroscopic particle, we must abandon completely the notion that a particle follows a well-defined trajectory.More radically, we must abandon the idea that we can study a physical system without interacting with it.These principles force us to jettison the classical precepts of causality, objectivity, and determinism.

A.2.1 Quantization

Observables that in classical physics are continuous may be quantized in the microscopic realm. For example,in classical physics the magnitude L of the orbital angular momentum of a negatively charged particle inorbit around a positively charged particle at the origin can assume any valueand it does, as the particleloses energy continuously and spirals into the origin. But the orbital angular momentum of an electronbound to a proton in a hydrogen atom can assume only discrete values (Chaps. 4-5). Similarly, the energyof the electron is quantized. The electron is forbidden to assume certain values of energy. Fortunately,quantization keeps the electrons in an atom from plummeting into the nucleus.

We refer to the collection of allowed values of any observable as the spectrum of that observable. Notall allowed values of an observable are necessarily quantized. For example, the energy of a free particleis continuous; this observable can assume any value E 0. Moreover, some observables are allowed bothdiscrete and continuous values. In this case, we refer to the discrete part and the continuous part of thespectrum. In quantum mechanics, the allowed values of an observable are the eigenvalues of the operatorthat represents the observable (see Appendix H). We determine the spectrum of an observable by solvingthe eigenvalue equation of the corresponding operator (Appendix I).

A.2.2 Uncertainty

The inherent fuzziness in nature is quantified by the quantum mechanical uncertainty, a quantity wecalculate from the wave function. The signature of fuzziness of an observable in a particular quantum stateis the following: in an ensemble measurement of the observable, various members of the ensemble returndifferent valuesall drawn, of course, from the list of eigenvalues of the corresponding operator. In such astate the uncertainty in the observable is positive. Only in the special case that the quantum state is aneigenstate of the observable will all members return the save value. In this case, the uncertainty is zero,the observable has a single, well-defined value, and we say that in this state the observable is sharp.3

A more general expression of this extraordinary property is the Heisenberg uncertainty principle(HUP) (A.3.12). The HUP is one instance of the remarkable general principle that nature prohibits simul-taneous precise measurement of certain pairs of observables. (The Heisenberg principle refers to positionand momentum.) In general, an uncertainty relation is a mathematical statement about the product ofthe uncertainties of two observables. By the definition of the uncertainty, this product must be non-negative

(see A.6). If the state is an eigenstate of an observable, the uncertainty for that observable is zero.But an uncertainty relation is not restricted to any particular state: rather, its a general statement

about the uncertainties in two observables in any quantum state. If the uncertainty product is positive(rather than zero), then the uncertainty relation articulates a fundamental limitation on our knowledge of theobservable. Quantum mechanics provides a Generalized Uncertainty Principle (GUP)a prescription

3A cautionary note: Its vital that you distinguish quantum mechanical uncertainty, which is a well-defined quantity wecalculate from wave functions, from experimental uncertainty. The latter arises from limitations inherent in the apparatuswe use to perform measurements and has nothing to do with the quantum mechanical uncertainty. Many physicists prefer theword indeterminacy to uncertainty.



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Appendix 4

for deriving the uncertainty relations for any two observables for any quantum state from the commutatorof the corresponding two operators (see A.3 and Appendix H).

Not all pairs of observables are constrained by uncertainty relations. We can determine whether such aconstraint applies to a pair of observables by evaluating their commutator. Operators whose commutatoris non-zero (operators that dont commute) are constrained by an uncertainty relation. Operators whosecommutator is zero (commuting operators) are not. We call two observables that are subject to an

uncertainty relation incompatible observables . We call all other pairs compatible observables : Rule: If two observables are incompatible, then in any quantum state its impossible in principle to measure

both observables at the same time to infinite precision.

A.2.3 Duality

The physical description of the macroscopic world elegantly and neatly divides phenomena into two irre-deemably distinct categories: waves and particles. But in the microverse, matter and its constituents donot behave according to our classical notions of either waves or particles. Microscopic thing-a-ma-jigs areessentially other. But to avoid silly locutions, we go ahead and call them particles.

Quantum particles behave like waves in some experiments and like particles in others. According to

the Principle of Complementarity, microscopic particles never manifest both wave-like and particle-likebehavior in the same measurement.

The mathematical embodiments of this wave-particle duality are the Einstein-de Broglie relations(Appendix O),

= h/p, and = E/h. (A.1)

Each equation relates a wave-like property (on the left-hand-side) to a particle-like property (on the right-hand-side).

In replacing classical notions like determinism by quantization, uncertainty, and duality, quantum me-chanics wreaks epistemological havoc. It transform the very nature of knowledge. In quantum physics,knowledge is inherently, irredeemably probabilistic, statistical, and limited.4

A.3 Four great postulatesand their consequences

In the absence of data we must abandon the analytic or scientific method of investigation,and must approach it in the synthetic fashion. In a work, instead of taking known eventsand deducing from them what has occurred, we must build up a fanciful explanation if itwill only be consistent with known events. We can then test this explanation by any freshfacts which may arise. If they all fit into their places, the probability is that we are uponthe right track, and with each fresh fact this probability increases in a geometricalprogression until the evidence becomes final and convincing.

The Man with the Watches, by Sir Arthur Conan Doyle

The ideas in

A.1 and

A.2, along with spin (Chap. 7) and indistinguishability (Chap. 8), constitute theconceptual core of quantum physics. To implement these ideas in the study of physical systems and thedesign of technological devices we must express them mathematically. The primary mathematical devices

4Read on: The question of the meaning and philosophical implications of quantum physics is one each individual must decide.A fine, balanced non-technical introduction to these murky matters is David Alberts Quantum Mechanics and Experience(Cambridge, MA: Harvard University Press). A more penetrating and provocative inquiry is James T. Cushings QuantumMechanics: Historical Contingency and the Copenhagen Hegemony (Chicago: University of Chicago Press, 1994). To whetyour appetite, heres a quote from Cushings introduction: It is astounding that there is a formulation of quantum mechanicsthat has no measurement problem and no difficulty with a classical limit, yet is so little known.



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Appendix 5

of quantum mechanics are the operators and wave functions which represent a microscopic system, itsobservables, and its states. To associate these mathematical devices with the core ideas of A.2 is the job ofthe postulates reviewed in this section.5

A.3.1 The wave function

Postulate 1 (The wave function)Every physically realizable state of a system is represented in quantum mechanics by a wave function .This function contains all accessible physical information about the system in that state.

An essential attribute of wave functions is that they satisfy the Principle of Superposition: If 1 and2 represent two physically realizable states of a system, then any linear combination,

= c11 + c22, a superposition of two wave functions. (A.1)

with arbitrary complex constants c1 and c2, represents a third physically realizable state. This acutely non-classical principle follows from the linearity of the time-dependent Schrodinger equation (TDSE), which all

wave functions satisfy (see Postulate IV).6

Requirements for physical admissibility. Not any old function of position and time can represent aquantum state. In order that a wave function be subject to interpretation as a probability amplitude (seePostulate II), it must satisfy four requirements of physical admissibility:

(1) The wave function must be a continuous function of position. This condition ensures, amongother things, that the probability density ||2 is everywhere continuous.

(2) The wave function must be a smoothly varying function of position. That is, first andsecond derivatives with respect to all position variables must be continuous. Together with the firstrequirement, this ensures continuity of the probability current density (see Complement ?? to Chap. 5).

(3) The wave function must be single-valued. If this condition is met, then the probability densitythe current density will be single-valued. Were this not the case, then we could associate different

position probabilities with a single point in spaceand even in quantum mechanics you cant do that.7

(4) The wave function must be normalizable. For example, every wave function (x, t) of a particlein one dimension (1D) must be such that the normalization integral is finite: 8

N

(x, t)(x, t) dx. normalization integral (1D) (A.2a)

5Commentary: The basic assumptions (postulates) are not to be understood as mathematical axioms from which every-thing can be derived without using further judgement and creativity. An axiomatic approach of this kind does not appear tobe p ossible in physics. The basic assumptions are to be considered as a concise way of formulating the quintessence of manyexperimental facts. from Quantum Mechanics: Foundations and Applications (Third edition) by Arno Bohm, (New York:Springer-Verlag, 1993).

6Details: This statement of Postulate I pertains to the class of so-called pure states. An ensemble is in a pure state ifits members are in a maximally sharp quantum state, that is, a state in which all values of a complete set of commuting

operators are sharp (Chaps. 25). An ensemble whose preparation has not actualized a maximally sharp state is said to b e ina mixed state. We cannot represent a mixed state with a wave function, because in essence we dont know enough about thestate to write one down. The extension of Postulate I to mixed states avers the existence of a density matrix that contains allthe accessible information about the state. The physics of mixed states is governed by quantum statistical mechanics. Youllfind a clear introduction to these ideas in Chap. 1 of Density Matrix Theory and Applications, Second Edition, by Karl Blum(New York: Plenum, 1996).

7Commentary: We impose the requirement of single-valuedness on the wave function itself rather than on the probabilitydensity ||2 to ensure that the probability density for superposition states will b e single-valued: if 1 and 2 are single valued,then so must be the probability density for the linear combination c11 + c22.

8A cautionary note: The requirement of normalizability does not imply that itself must be finite everywhere. Singularitiesare allowed so long as the normalization integral remains finite.



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Appendix 6

Only if N is finite can we normalize the wave function, which we do by multiplying the functionby 1/

N. The normalization integral for the resulting normalized wave function equals 1: this is the

famous normalization condition, which must be obeyed by all quantum mechanical wave functions:

(x, t)(x, t) dx = 1 normalization condition (1D). (A.2b)

Imposing this condition ensures that the total probability for finding the particle somewhere in thedomain < x < + will be unity, as it must be if the particle exists.9

A.3.2 The interpretation of the wave function

Postulate 2 (The Born interpretation.)We interpret the wave function as a position probability amplitude. In a one-dimensional state representedby a wave function (x, t),

P(x, t) dx = |(x, t)|2 dx = (x, t) (x, t) dx, position probability density, (A.3)

is the probability that in an ensemble measurement of position at time t the particle will be detected in theinfinitesimal volume element dx at x. Thus the position probability density |(x, t)|2 is the probability perunit length for detection of the particle will be detected in dx at x in a measurement at time t.

The generalization of this postulate to single-particle systems in two (2D) and three dimensions (3D)is straightforward. In 3D, for example, the wave function depends on three spatial coordinates (x,y,z),which we can collectively denote by r. According to the Born interpretation, the probability densityP(x,y,z; t) = |(x,y,z,t|2 times the volume element d3v = dx dy dz is the probability per unit volumefor detecting the particle in volume element d3v at r at time t. Its generalization to many particles isaddressed in Chap. 9.

A.3.3 The integrated probability density

Actual position measurements concern finite regions of space. Still, however good our apparatus, it cantdetect particles in an infinitesimal region. So when we apply Postulate II to actual experiments, we alwaysconstruct an integrated probability density. If, for example, the resolution of our apparatus can detectthe particle in a finite region [a, b], then the relevant quantity is

P([a, b], t) =ba

P(x, t) dx =ba

|(x, t)|2 dx, integrated probability density. (A.4)

9Details: We relax this requirement a bit for a continuum stationary state. The wave function of such a state cannotbe normalized because it pervades all space. The normalizability requirement for such a state is that the modulus of thewave function must be finite for all values of its variables. To illustrate, consider a continuum stationary state of energyE = 2k2/(2m) and spatial function k(x). Instead of the usual normalization condition (A.2b), the condition for a continuumstationary state must be written in terms of the Dirac delta function (Appendix N), as

k (x)k(x) dx (k k),

where the constant of proportionality is arbitrary. This condition is called Dirac delta function normalization. To learnmore about continuum states and their interpretation see A.4 and 8.5 in Understanding Quantum Physics (UQP).



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Appendix 7

A.3.4 Expectation values and uncertainties

The probabilistic interpretation of the wave function provides the foundation for calculating quantities thatcharacterize an ensemble as a whole. Although quantum mechanics gives only probabilistic information aboutthe results of measurement, it gives precise information about statistical properties of the ensemble. The

two most important such properties are the expectation value and the uncertainty of an observable.10

In quantum mechanics, the average value of an observable (in a state) is its expectation value and thestandard deviation is its uncertainty. For position, say, the expectation value and the uncertainty fromthe interpretation of (x, t) as a position probability amplitude, as well now see.

The expectation value. The ensemble average or expectation value of position is given by theexpectation value of the position operator x with respect to the wave function:

x(t)

x P(x, t) dx =

(x, t)x (x, t) dx. average position at t (1D) (A.5)As its name implies, x(t) is the average position we would expect to get in an ensemble measurement ofthis observable at time t.

The uncertainty. In general, the results of an ensemble measurement will be dispersed about the expecta-tion value in a pattern characterized by the uncertainty (x)(t). Like the expectation value,l we calculatethe uncertainty from the wave function. For position in 1D, the uncertainty is11

(x)(t)

(x x)2 =

x2 x2. position uncertainty at t (1D) (A.6)

In writing expressions that contain expectation values and other quantum mechanical devices, we oftenresort to an elegant shorthand devised by the brilliant theoretician P. A. M. Dirac during the early years ofquantum theory (see Appendix G).

A.3.5 Probabilistic & statistical information about momentum

Armed with the Born interpretation and the definitions of the mean position and uncertainty, we can figureout all that nature allows about position. But buried in the wave function is all information about thestate: contains information about all observables of the system. Happily, quantum mechanics providesa powerful, general procedurethe method of eigenfunction expansionfor determining informationabout any observable. One way to view this method is as a translation procedure: we translate the wavefunction into the language of whatever observable were interested in, then apply an extension of the Borninterpretation to the result (Postulate II).

Before we review this procedure (see A.5), lets recall the familiar example of the linear momentum.We can translate the position probability amplitude (x, t) into a momentum probability amplitude(p) using the Fourier transform. This transform relates the initial wave function to an momentumamplitude function A(k) by

(x, 0) = 12

A(k) eikx dk. (A.7a)

We define a momentum probability amplitude (p) by interpreting k as the wavenumber k = p/ ofthe particle and writing

10Jargon: Somewhat imprecisely, at least from the viewpoint of a statistician, the uncertainty is also often called the standarddeviation or the dispersion.

11Notation: The argument (t) denotes the time dependence of (x)(t); it does not imply that x is a function of time. Ill usethis argument only when I want to emphasize this time dependence.



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Appendix 8

(p) 1

Ap

, momentum probability amplitude. (A.7b)

The Fourier transform now directly relates the position and momentum amplitudes:

(x, 0) =12

(p) eipx/ dp. (A.7c)

If we want to calculate the momentum amplitude from the initial wave function, we just use the inverserelation12

(p) =12

(x, 0) eipx/ dx. momentum amplitude (A.7d)

At subsequent times t > 0, the relationship between the position and momentum amplitudes follows fromthe time-dependent Schrodinger equation (TDSE) (see Postulate IV):

(x, t) =12

(p) ei [px(p)t] dp. (A.8)

The function (p) is the dispersion relation. We determine this function from the appropriate expressionfor the total energy. Thus for a free particle, the dispersion relation is

(p) =p2

2m=

k2

2m. (free particle dispersion relation) (A.9)

Using F as shorthand for a Fourier transform, we can encapsulate Eqs. (A.7) in the easily remembered form

(x, 0) = F1[(p)]

(p) = F[(x, 0)](A.10)

When we refer to (p) as a momentum probability amplitude, we mean that from it we canconstruct the (1D) momentum probability density at time t. The mathematical structure of this

momentum density is identical to that of the position density in Postulate II:P(p, t) dp = |(p)|2 dp = (p)(p) dp, momentum probability density (A.11)

Like all probability amplitudes, this function must be normalized,

(p)(p) dp = 1. (A.12)

Thus we have translated the wave function into a function of momentum and generalized the Born inter-pretation to this observable. We can similarly treat any other observable, be it discrete or continuous.

Armed with the momentum probability amplitude (p), we can calculate the expectation value anduncertainty of the linear momentum using relationships analogous to those for position. The momentumexpectation value for a system with momentum amplitude (p) is

p

(p)p (p) dp, (A.13a)where the action of the momentum operator p on a function of momentum is simply to multiply the functionby the variable p. (The action ofp on a function of position is quite different.) The momentum uncertainty

12Jargon: The new function (p) is sometimes called the wave function in momentum space or the momentum repre-sentation of the state. Note that p is the variable conjugate to x; as such, the two are related by the commutator relationx,p = i . Such relationships are characteristic of conjugate variables in quantum mechanics.PrintAppendices Version: 8.42 Printed: August 3, 2010


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Appendix 9

is defined by

p

p p2 = p2 p2. (A.13b)In practice, its usually easier to evaluate p and p directly from the wave function (x, t) using the

momentum operator (see Postulate III)

p = i ddx . (A.14)For example, an equivalent expression for the average momentum is

p = i

(x, t)

d

dx(x, t)

dx, momentum operator (1D). (A.15)

A.3.6 Statistical quantities for arbitrary observables

By extension of Eqs. (A.5) and (A.6) for position and Eqs. (A.15) and (A.13b) for momentum, we canevaluate the expectation value and uncertainty of an arbitrary observable Q:

Q =

(x, t)Q(x, t) dx, (A.16a)Q =

Q2 Q2. (A.16b)

As noted above, the quantum mechanical uncertainty is analogous to the statisticalstandard deviation of anobservable of a state of a macroscopic system. In quantum mechanics, however, the uncertainty implicitlyrefers to the standard deviation of individual results obtained in an ensemble measurement about theiraverage value. This average value is given by the expectation value Q.

A.3.7 Observables and operators

Postulate 3 (Operators)Every observable of a system is represented by an operator. The operator is the mathematical tool we useto extract information about the observable from wave functions. For an observable that is represented inclassical physics by the function Q(x, p), the corresponding operator is Q(x,p), where the operators x and pappear in Tbl. A.1.

Table A.1. A dictionary of operators for aone-dimensional system. Each operator acts,in general, on a wave function (x, t).

Observable Operator Instructions

position

x multiply by x

momentum p i

xtotal energy H T + V(x)kinetic energy T 2

2m

2

x2

potential energy V multiply by V(x, t)

parity invert x through the origin



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Appendix 10

Restrictions on operators. Just as we must restrict the class of functions that can serve as probabilityamplitudes (to those that are continuous, normalizable, and single-valued), we must restrict the class ofoperators that can represent observables:

(1) All quantum mechanical operators must be Hermitian.

(2) All quantum mechanical operators must be linear (or anti-linear).13

Dos and donts of operator manipulation. Operators are so important that we must keep before usthe rules for manipulating them. Here are the key ones (for more, see Appendix H):

(1) Derive and simplify operator expressions and equations by letting operators act on anarbitrary physically admissible function of all the relevant variables. For instance, a one-dimensional operator expressed in terms of x and its derivatives should act on an arbitrary func-tion f(x).

(2) Operators act on everything to their right unless their action is constrained by parenthesesor brackets.

(3) The product of two operators is a third operator.

(4) An operator product implies successive operation.

(5) The order in which the operators act is vital; never assume that two operators commute.

A.3.8 Hermiticity

In 1D an operator Q is Hermitian if, for arbitrary well-behaved wave functions 1(x, t) and 2(x, t),

1(x, t)Q2(x, t) dx =

Q1(x, t)2(x, t) dx. (A.17a)In Dirac notation, we write this definition as

1(t) |Q2(t) =

Q1(t) | 2(t) 1(t) |

Q | 2(t). (A.17b)

The double bar notation in the matrix element 1(t) | Q | 2(t) signifies that Q can act on either1(x, t) or on 2(x, t). The Hermiticity of an operator Qa property of an operator, not of states of asystemcan be expressed even more abstractly using the adjoint notation (Appendix H) as Q = Q.14Consequences of Hermiticity. Hermiticity is an exceptionally powerful property. From it follow some ofthe most important results of quantum mechanics. These results concern solutions of the eigenvalue equationof an Hermitian operator.

(1) The eigenvalues of a Hermitian operator are real. The eigenvalues of an operator are the onlyvalues that can be obtained in a measurement of the corresponding observable. Since all measurablevalues are real numbers, we must use only Hermitian operators.

(2) The eigenfunctions of a Hermitian operator are orthogonal.15 Two functions are orthogonalif their overlap integral is zero. Two bound-state eigenfunctions of a one-dimensional Hamiltonian, forexample, satisfy this property:

13Details: All operators that can be expressed in terms of the position and momentum operators are linear. Moreover, alloperators that represent continuous transformations of position coordinates are linear. The most important anti-linear operatoris the time reversal operator. This operator, which switches the signs of momentum (and spin) variables, is extensively usedin the quantum theory of scattering (see Chap. 1).

14Jargon: Mathematicians often call a Hermitian operator a self-adjoint operator.15Details: This is a slight overstatement. If a particular eigenvalue is degenerate, which means that two or more eigenfunctions

share this eigenvalue, then these degenerate eigenfunctions may not be orthogonal. But if they arent we can always constructan equal number of orthogonal eigenfunctions, each of which corresponds to the same eigenvalue, as linear combinations of thefunctions in the original set. A common procedure for doing so is the Gram-Schmidt orthogonalization procedure.



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Appendix 11

n | n =

n(x)n(x) dx = 0, (if n = n). (A.18a)

As always, these eigenfunctions must be normalized. A combined statement that the eigenfunctions of

H are orthogonal and normalized is the orthonormality condition16

n | n =

n(x)n(x) dx = n,n (A.18b)

where the Kronecker delta function is

n,n =

0 if n = n1 if n = n

Kronecker delta function (A.18c)

Orthonormality plays a key role in the method of eigenfunction expansion mentioned above. Weuse orthogonality to determine the coefficients of an expansion in the complete set of eigenfunctions ofa Hermitian operator.

(3) The eigenfunctions of a Hermitian operator constitute a complete set. Completeness

is the heart of the method of eigenfunction expansion, since it justifies our writing down theexpansion in the first place! Completeness of a set of functions means that we can expand an (almost)arbitrary function in the elements of the set. The function we want to expand must be mathematicallywell-behaved and must obey the same boundary conditions as the functions in which were expandingit. In quantum mechanics, this means that the function must satisfy the usual conditions of physicaladmissibility (continuous, smoothly varying, and so on), and must obey the same boundary conditionsas the elements of the set in which were going to expand it.17 This set is called the basis (or basisset) for the expansion. Well see an important example of this principle in A.5.

(4) The eigenfunctions of a Hermitian operator satisfy closure. Although Ive singled out thisproperty, its just another way to express completeness; closure is a property of eigenfunctions at twodifferent positions. For a complete set of eigenfunctions { E(x) }, this property reads

n E(x)E(x) = (x x). closure (A.19)16Details: In the language of linear algebra, the orthogonality integral for two functions is called their inner product or

scalar product. Another way to state the orthogonality condition is to say that the scalar product of two eigenfunctions withdifferent eigenvalues is zero. Another way to state the normalization condition is to say that the scalar product of a normalizedeigenfunction with itself is 1. The latter quantity is usually called the norm of the function, so the normalization conditionamounts to the easily remembered statement that the norm of a normalized eigenfunction is 1. Well use such language instudying electron spin in Chap. 6.

17Details: For example, we couldnt expand a function that went to infinity as x in a basis of eigenfunctions of the simpleharmonic oscillator Hamiltonian. In practice, this isnt an issue, because such a function wouldnt be physically admissible.



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Appendix 12

A.3.9 Linearity and the principle of superposition

An operator Q is linear if, for any functions 1 and 2 and complex number a and b (Appendix H),

Q(a1 + b2) = a

Q1 + b

Q2. a linear operator. (A.20)

Physically, this property of quantum mechanical operators is related to the Principle of Superposition,Eq. (A.1), p. 5. In fact, this principle holds only because the Hamiltonian, which determines wave functionsvia the TDSE, is a linear operator.

A.3.10 The commutator

We can determine whether two operators commute by evaluating their commutator. For operators Q1 andQ2, the commutator is a third operator, defined as Q1, Q2 Q1Q2 Q2Q1. commutator (A.21)If

Q1 and

Q2 commute, their commutator

Q1,

Q2

is zero; if they dont, its non-zero:

Q1, Q2 = 0, commuting operators, (A.22a) Q1, Q2 = 0, non-commuting operators. (A.22b)A key property of commuting operators is:

Rule: Operators that commute define a complete set of simultaneous eigenfunctions.

Warning: Unlike numbers and functions, operators do not, in general, commute.

A.3.11 Eigenstates and simultaneous eigenfunctions

To understand the above rule, we need to recall the definition of a simultaneous eigenfunction. Ourstarting point is the notion of an eigenstate . A system in state (x, t) is an eigenstate of an observable

Q if (x, t) is an eigenfunction of Q:Qq(x, t) = qq(x, t), system is in an eigenstate of Q. (A.23)

In this case, we often use the eigenvalue q as a subscript to label the wave function, as above.

In an eigenstate of Q, this observable is said to be sharp. This term has a very specific meaning: in anensemble measurement of Q, all members of the ensemble will give the same valuethe eigenvalue q. In aneigenstate of the observable Q, the uncertainty of Q is therefore zero: Q = 0.

The most familiar eigenstate is the energy eigenstate. A system is in an energy eigenstate if its wavefunction is an eigenfunction of the Hamiltonian. Another name for an energy eigenstate is a stationary

state (A.4).



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Appendix 13

A function q,r is said to be a simultaneous eigenfunction of two operators Q and R if it satisfies theeigenvalue equations of both operators,18

Qq,r = qq,r

Rq,r = rq,r .a simultaneous eigenfunction (A.24a)

The converse also holds: any two operators that share a complete set of simultaneous eigenfunctions neces-sarily commute. Well return to the remarkable consequences of this rule in A.7.

A.3.12 Commutators and uncertainty relations

As noted in A.2, if two operators commute, then we say that the corresponding observables are compatible .If they dont commute, the observables are incompatible . These colorful names allude to the possibilityof measuring both observables simultaneously to infinite precision (in principle). Incompatible observablesare subject to the constraints of an uncertainty relation. The most famous example of incompatible observ-ables are position and momentum. The commutator of the corresponding operators is

x,

p

= i . Theconsequence of the incompatibility of x and p are that any quantum state of any system is subject to theHeisenberg uncertainty principle

(x)(t) (px)(t) 12. Heisenberg Uncertainty Principle (A.25)

Rule: Uncertainty relations like Eq. (A.25) are general statements about systems; they are not limited toparticular states. In no state of any system is the uncertainty product (x)(t) (px)(t) at any time lessthan /2.

The Heisenberg uncertainty principle illustrates a more general result that pertains to any two quantummechanical operators, whether or not they commute. According to this Generalized Uncertainty Prin-ciple (GUP), we calculate the uncertainty product for operators

Q1 and

Q2 from their commutator, as

19

(Q1) (Q2) 1

2 i Q1, Q2 . Generalized Uncertainty Principle (A.26)A.3.13 Constants of the motion

In quantum mechanics as in classical physics, certain observables have a special status. These are the con-stants of the motion: physical quantities that remain unchanged as the system evolves with time. Knowledgeof the constants of the motion provides profound and powerful insights into the systemespecially into itssymmetry properties (see Chap. 1).

The meaning of constant of the motion in quantum physics is inherently statistical. An observable is aconstant of the motion if the expectation value of the observable for any state doesnt depend on time:

ddt

Q = ddt

(t) | Q | (t) = 0 for any . constant of the motion (A.27)18Details: A familiar example is a stationary state of a particle in a one-dimensional symmetric potential. Since the spatial

function of such a state has definite parity, the stationary-state wave function is an eigenfunction of the Hamiltonian and theparity operator.

19Read on: You can find a discussion of this important result in 11.4 in Morrison (1990) and remarks on its consequencesfor measurement in 13.1 of that book.



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Appendix 14

To determine whether an observable is a constant of the motion we call on the commutator of thecorresponding operator with the systems Hamiltonian. The connection between these quantities is thegeneral relationship

d

dtQ = i

H,

Q

+

Q

t

(A.28)

Unless the operator Q explicitly depends on t (like the interaction Hamiltonians for time-dependent fieldsin Chap. 18), the last term on the right-hand side of Eq. (A.28) is zero and, if the commutator

H, Q isalso zero, Eq. (A.27) immediately follows. But Q may also be a constant of the motion if the commutatoris non-zero, provided its expectation value with respect to every state of the system is zero.

A.3.14 The Correspondence principle

Since its initial development early in the 20th century, physicists have demanded that the equations ofquantum mechanics must go over continuously into their classical counterparts in the classical limit. This

is the essence of the Correspondence Principle: Rule: In the classical limit, the laws of quantum mechanics must reduce to those of Newtonian Mechanics.

Here the phrase the classical limit is a bit vague, although we can make it specific when we discussparticular systems. For example, in exploring the stationary states of a one-dimension simple harmonicoscillator potential, we find that as the quantum number n increases, the quantum mechanical probabilitydensity |n(x)|2 increasingly resembles the classical probability. Here the classical limit is the limit of largequantum numbers. In other systems, the classical limit refers to large spatial dimensions or to the limit inwhich quantities involving are negligible.20

What does the Correspondence Principle imply for the expectation values and uncertainties we useto characterize ensemble measurements? These quantities express the essentially non-classical nature ofquantum physics, its probabilistic character. In the classical limit, all evidence of this character mustdisappear:

Q(t) Q(t)Q (t) 0.

(in the classical limit) (A.29)

Ehrenfests Theorem. The Correspondence Principle has a subtle, fascinating implication for the classi-cal idea of a particle trajectory. In some sense, the position probability density, the squared modulus of thewave function, must reduce in the classical limit to the trajectory the particle would have were it describedby classical mechanics.21 The quantum mechanical articulation of this idea are the equations of EhrenfestsTheorem. These equations are written in terms of the expectation values of position and momentum fora given quantum state. To clarify the connection to classical equations youve seen before, Ill write theseequations for a particle in 3D:22

20Jargon: Most commonly, youll see the classical limit indicated in sentences like, Quantum equations of motion must

reduce to their classical counterparts in the limit 0. Such locutions are not to be taken literally! Plancks constant is aconstant, so strictly speaking, writing 0 is meaningless. Rather, this limit is a shorthand to indicate that terms involving are being neglected compared to other terms, which must be significantly larger for this limit to apply.

21Read on: Strictly speaking, this statement should be couched in terms of the ensemble of microscopic particles whosecommon state is described by the wave function. That is, in the classical limit the quantum mechanical position probabilitydistribution must reduce to the corresponding statistical probability distribution for an ensemble of classical particles, each ofwhich has a trajectory. For more on this topic, see Sec. 15.1 of Quantum Mechanics, by Leslie E. Ballentine (Englewood Cliffs,NJ: Prentice-Hall, Inc. 1990).

22Details: The second of these equations holds rigorously only if the corresponding classical expression V(r) is linearwith respect to position, as in the case of a simple harmonic oscillator. More generally and rigorously, the right-hand-side



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Appendix 15

d

dtr(t) = 1

mp(t) (A.30a)

d

dtp(t) = V(r). (A.30b)

A.3.15 The time-development of a quantum state

Postulate 4 (The Schrodinger equation.)In the absence of measurement, the time development of the wave functions of a system is governed by thetime-dependent Schrodinger equation (TDSE),

H = i t

. (A.31)

The Hamiltonian H = T +V fully represents the system. To specify a particular state of the system, we mustgive the initial condition: the value of the wave function at one time (usually t = 0).

The Hamiltonian, the operator that corresponds to the total energy of the system, is the sum of the kinetic

and potential energy operators. Provided no external magnetic fields act on the system, the kinetic energyoperator is just T = p2/(2m), and the Hamiltonian has the form23

H = T + V = p 22m

. Hamiltonian (A.32)

A.3.16 The probability current density

Discussions of the time development of a state often refer to a quantity which contains information comple-mentary to that of the probability density: the probability current density. This quantity plays a keyrole in understanding the magnetic moment of a microscopic system (see Complement ?? of Chap. 5). In1D, the probability current density is defined in terms of the wave function by

j(x, t) i2m

(x, t)

x(x, t) (x, t)

x(x, t)

. (A.33)

Conservation of probability. To illustrate the importance of the probability density, lets consider thereasonable requirement that the probability associated with a particle must be conserved. We can articulatethis important law in two ways: conservation at a point and conservation in a finite region. In 1D, we writethe law of pointwise conservation of position probability as

tP(x, t) +

xj(x, t) = 0. (A.34a)

Integrating this equation in a finite region a x b yields

ddt

Pa,b([a, b], t) = j(a, t) j(b, t). (A.34b)

By extending the region [a, b] to encompass the entire domain of x, we regain the reassuring result that thetotal probability of finding a particle somewhere in space doesnt change with time:

of Eq. (A.30b) is V(r). In most applications, the position uncertainty is small enough (compared to the range of thepotential energy) that the approximation in Eq. (A.30b) is excellent.

23Details: In the presence of an external magnetic field B with vector potential A, the momentum operator becomes [p +eA]2.(This alteration is required by classical electrodynamics and is not peculiar to quantum mechanics.) This changes the kineticenergy operator and hence the Hamiltonian, as described in Chaps. 13 and 18.



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Appendix 16

d

dtP([a, b], t) = d

dt

P(x, t) dx = 0. (A.34c)

A.4 Stationary states

Stationary states are the simplest states in quantum mechanics. They are also the starting point for moststrategies for studying non-stationary states (see A.5). The defining property of a stationary state is themathematical structure of its wave function. Its signature is a wave function that is a separable functionof space and time with an exponential time dependence:

E(x, t) = E(x) eiEt/. stationary state wave function (A.1)

Solutions of the TDSE of this form exist only if the potential energy term in the Hamiltonian H = T + Vdoes not depend on time. If so, then Eq. (A.1) follows from the TDSE by application of separation ofvariablesa technique well use repeatedly in this book. If not, see Chaps. 17 and 18.

A.4.1 The physics of stationary states

The physical properties of a stationary state follows from Eq. (A.1). The time dependence of E(x, t) appearsentirely in the complex phase factor ei Et/. Therefore the product of E(x, t) and its complex conjugatedoes not depend on time. Physically, therefore, the probability density of a stationary state is independentof time. (Thats why physicists call the state stationary.) Consequently the statistical properties of astateits expectation value and uncertaintyare also independent of time. More generally, we have thefollowing important rule:24

Rule: In a stationary state, all physical properties of the system are independent of time.

This rule pertains to all observablesnot just the energy. For any observable Q, Q and Q are independentof time in a stationary state.

The energy of stationary state has a very special property: it is sharpit has a well-defined value, andthat value is the particular eigenvalue of H that appears in the phase factor ei Et/:

E = E, and E = 0 (in a stationary state) (A.2)

The spatial function. The spatial function E(x) in the wave function E(x, t) fully describes thespatial properties of the state, such as position and momentum. Mathematically, E(x) is the eigenfunction ofthe Hamiltonian with eigenvalueE. We determine spatial functions by solving the Hamiltonian eigenvalueequation, the time-independent Schrodinger equation (TISE)

HE(x) = EE(x), time-independent Schrodinger equation . (A.3)

Rule: The TISE is is a linear, homogeneous, second-order ordinary differential equation.

(In more than 1D and/or for more than one particle, its a partial differential equation.) To specify itssolutions, therefore, we must specify two boundary conditions , two values of E(x). For a 1D system,these conditions are specified in the asymptotic limits x . That is, rather than specify the value ofthe function at two finite points, we specify its values in the limits of the domain < x < +.

24Commentary: Even the probability current density j(x, t) of a stationary state is independent of time. But its not necessarilyzero. Indeed, in Chap. 5 we explore stationary states of atomic hydrogen in which a non-zero (but time independent) probabilitycurrent density leads to the orbital magnetic moment of the atom.



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Appendix 17

An integral form of the TISE. In some methods for solving the TISEnotably the variationalmethod of Chap. 16its useful to transform this differential equation into an integral for the energy.This transformation is not restricted to stationary states. For an arbitrary quantum state with (normalized)wave function (x, t), the expectation value of the energy at t is, by definition,

E(t) =

(x, t) H(x, t) dx (A.4)For a stationary state, the energy is sharp with value E. So the expectation valuethe average value of theenergy in this stateis time independent and evaluates to E:

E =

n(x) Hn(x) dx = E. (for a stationary state) (A.5a)Using Dirac notation as a shorthand for integration we write25

E = |

H | = E. (for a stationary state) (A.5b)

A.4.2 Boundary conditions: bound versus continuum states

As noted in A.1, quantum states come in two varieties, bound and continuum. Amongst stationary states,the two varieties are distinguished by the asymptotic boundary conditions on their spatial functions,that is, the behavior ofE(x) in the limits x . For a 1D system, boundary conditions are

n(x) x

0, bound stationary state, (A.6a)

limx

|E(x)| < , continuum stationary state. (A.6b)

The continuum state boundary condition states that the magnitude of the spatial function must remainfinite for all values of r. Mathematicians say that such a function is bounded at infinity.26

Energy quantization. For a bound stationary state, the boundary conditions Eq. (A.6a) have profoundconsequences. Most important, the energy of a bound stationary state is quantized. Only for certain, discretevalues ofE do there exist solutions of the TISE that obey this boundary condition. Only these solutions arenormalizable, and hence only these solutions correspond to physically realizable states. Since each bound-state energy is quantized, we can index the energy and the spatial function by an integer. This integer iscalled the principal quantum number n. For the minimum value of n, the index for the ground state ,we choose n = 1, except when its not.27 In two and 3D, to construct a unique name for a Hamiltonianeigenfunction requires more than one quantum number.

25A cautionary note: Notice that the normalized spatial function E (x) appears in the integral form of the TISE for astationary state.

26Read on: We cant normalize the spatial function for a continuum stationary state. Nevertheless, we can extend ourinterpretation of the wave function as a position probability amplitude to interpret continuum spatial functions as amplitudesfor relative probabilities. In 1D, the transmission coefficient T(E) for scattering of a particle with incident kinetic energy Eby a potential well or barrier is defined as a ratio of continuum spatial functions (pure momentum state functions ei kx).Hence the arbitrary normalization of the spatial function divides out of T(E), which is the physically measurable quantity. Forfurther discussion, see 8.5 in Morrison (1990).

27A cautionary note: A prominent exception to this convention is the simple harmonic oscillator, where for reasons ofconvenience we start indexing at n = 0. See Appendix P.



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Appendix 18

A.4.3 Bound states in one dimension

For a single particle in 1D, the bound-state solutions of the TISEthe spatial functions n(x) which satisfythe bound-state boundary conditions (A.6a)have properties that aid us in problem solving and physicalthinking.28

(1) All bound-state energies are non-degenerate. To a non-degenerate eigenvalue En, there corre-

sponds one and only one mathematically distinct spatial function n(x). By contrast, each continuumstate energy E > 0 is two-fold degenerate. For each such energy, there are two linearly independentspatial functions that satisfy the TISE and the continuum boundary conditions (A.6b).29

(2) There exists at least one bound state in any attractive potential. An attractive potentialV(x) has the property that, if we choose the zero of energy at the maximum value of the potential,then the potential will be negative for part of the domain < x < +. For any such potentialnomatter how weakwe can find at least one solution of the TISE that obeys the bound-state boundaryconditions. We say that such a potential supports at least one bound state.30

(3) The bound states of a symmetric potential have definite parity. If a potential is symmetricunder inversion, then V(x) = V(x), and each of its bound states has definite parity. This prop-erty describes the behavior of the Hamiltonian eigenfunction n(x) under inversion (Chap. 1): : x x = x. Inverting the coordinate x transforms functions of x, in general into different func-tions. An arbitrary function f(x) will thus be transformed into a new function g(x):

f(x) g(x) = f(x). transformation under inversion (A.7a)

The equality in this equation is vital: it defines the transformed functiong(x) as the function whosevalue atx is equal to that of the untransformed functionf at the pointx that results from the inversion.The operator that effects this transformation of functions is the parity operator :

f(x) = g(x) = f(x). parity operator (A.7b)If the function f(x) has definite parity, the Eq. (A.7b) becomes an eigenvalue equation, with eigenvalues1. A function with eigenvalue +1 is an even function and one with 1 is an odd function:

f(x) = f(x) + even odd (A.7c) Rule: An even function is unaffected by inversion; an odd function changes sign under inversion. A

function f(x) whose transform under inversion, g(x), is neither even nor odd does not have definite parity.

Rule: In 1D, the bound-state spatial functions of a symmetric potential are all even or odd. 31

States whose spatial functions are even are said to have even parity; those whose spatial functions areodd have odd parity. The classification of the stationary states of a symmetric potential into even-and odd-parity states is a powerful tool for understanding and simplifying the analysis of all states,stationary and non-stationary.

28A cautionary note: These properties pertain to single-particle systems in 1D only. For generalization to other systems,see Chaps. 25.

29Jargon: A set consisting of two or more functions, say { f1, f2, . . . , f N } is linearly independent if the only set of constants{ c1, c2, . . . , cN } such that

Nj cj fj = 0 is the set of zeroes, c1 = c2 = = 0. (This condition must hold for all values of

the independent variable(s) of f.) If even one coefficient is non-zero, then the set is said to be linearly dependent. Forinstance, { sin , cos } is linearly independent, because neither function in the set is proportional to the other function. But{ sin , cos , ei } is linearly dependent, because ei = cos + i sin .

30A cautionary note: This characterization doesnt apply to infinite potentials like the infinite square well or the simpleharmonic oscillator. For infinite potentials, which are useful models but cannot be realized in nature, we choose the zero ofenergy at the bottom of the potential. Infinite potentials support an infinite number of bound states and have no continuum.

31Details: The spatial functions for a non-symmetric potential dont have definite parity.



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Appendix 19

A.4.4 The stationary states of a free particle

The simplest quantum mechanical system is the free particle: a particle in the absence of any forceswhatsoever. Since a potential energy V = 0 cannot bind the particle, all its quantum states are continuumstates. In 1D, these states are called pure momentum states, because they are eigenstates of linear

momentum.32

The free particle in one dimension. The Hamiltonian for a free particle is just the kinetic energy

operator, H = T. We can write its eigenfunctions either in terms of the wavenumber k or the linear momen-tum p = k:

k(x) =12

eikx, or k(x) =12

eipx/. (A.8)

The factors 1/

2 in k(x) and 1/

2 in p(x) ensure that these functions satisfy the Dirac deltafunction normalization condition for continuum states,

k(x)k(x) dx = (k k). (A.9)

The wave function of a pure momentum state is the product of k(x) and the usual stationary-statetime factor eiEt/. We can write this wave function as a travelling wave using the de BroglieEinsteinrelations (A.1). For a free particle of mass m, we have

E = =2k2

2m.

= k(x, t) = k(x) eiEt/ = 12

e(ikxt).

(for a free particle) (A.10)

It is sometimes convenient to write k(x, t) in terms of p and E rather than k and . We introduce anadditional factor of 1/

to ensure that the normalization condition (A.9) still holds, to wit:

p(x, t) =

1

2 ei (pxEt)/

.

pure momentum state wave

function for a free particle. (A.11)

Each pure momentum state is two-fold degenerate, because both momenta p correspond to the same freeparticle energy, E = p2/(2m).

The free particle in 3D. For a free particle in 3D with linear momentum 33

p = k = pxex + pyey + pzez, (A.12)

we can separate variables x, y, and z in the TISE,

Hk(x) =

Tk(x) =

2

2m

2

x2+

2

y 2+

2

z 2

k(x). = Ek(x). (A.13)

The resulting spatial function k(x) is the product ofthree single-variable functions, one for each coordinate,each of the form of (A.8). Combining the exponential functions in this product, we get

32Commentary: The Heisenberg Uncertainty Principle implies that in a pure momentum state we know nothing about position,x = . Although an idealization, such states are useful in, for example, the description of collisions.

33Commentary: In 3D the momentum p and the wave vector k are vectors. So when we say p is sharp, we mean thatall three Cartesian components of the linear momentum are sharp, px = py = pz = 0. In 1D, of course, the momentumand wavenumber in these equations are scalars, p p and k k. The symbols ex, ey , and ez denote unit vectors along theCartesian axes.



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Appendix 21

This set is called the basis of the expansion. The expansion coefficients cn(0) are, in general, complexnumbers. They carry an argument 0 to remind us that they correspond to t = 0; were we expandingthe wave function at time t, wed write cn(t).

(2) Evaluate all non-zero coefficients cn(0) from the initial wave function35

cn

(0) =n |

(0)

=

n

(x)(x, 0) dx, for all n. (A.1b)

(3) Use the resulting coefficients and the eigenvalues En to write the wave function for t > 0:

(x, t) =n

cn(0)n(x) ei Ent/. (A.1c)

Thats all there is to it.36

Warning: As in any eigenfunction expansion, we must include all of the eigenfunctions except thosewhose coefficients are zero.

Once we have completed Steps 1 and 2, however, we may approximate the expansion (A.1c) by neglectingterms in this summation except those with the largest coefficients. This approximation is called truncation.

Example A.1 (Derivation of the expansion coefficients.)The key to this derivation is the orthogonality of eigenfunctions of any Hermitian operatorin this case, ofthe Hamiltonian, Eq. (H.6a), p. 72.

The most important step in deriving an equation for the nth coefficient cn(0) in the eigenfunction expansionEq. (A.1a) is to isolate this coefficient. That is, I must mathematically transform the sum on the right-handside into a single term. To do so, I multiply both sides by n(x), then integrate the equation with respect tothe independent variable, x:

n(x)(x, 0) =

n

cn(0)n(x)n(x) (A.2a)

n(x)(x, 0) dx =

ncn(0)

n(x)n(x) dx (A.2b)

Invoking orthogonality introduces a Kronecker delta function on the right-hand side, to wit:

n(x)(x, 0) dx =

n

cn(0)n,n. (A.2c)

When I now execute the sum, the Kronecker delta neatly isolates a single coefficient:

n(x)(x, 0) dx = cn(0). (A.2d)

Now I can drop the superfluous prime from the principle quantum number and regain Eq. (A.1b).

35A cautionary note: Before evaluating any of these coefficients, think carefully about the symmetry properties of the Hamilto-nian eigenfunctions and of the initial wave function; from such symmetry properties we can reason out that certain coefficientsare zero and thus avoid evaluating them altogether! In the 1D quantum mechanics of a symmetric potential, parity may rendermany coefficients equal to zero.

36Details: Except for artificial models like the infinite square well and the simple harmonic oscillator, potential energies V(x)support both bound and continuum states. So in general we must include bound and continuum spatial functions in the initialexpansion (A.1a). Since continuum solutions of the TISE exist for any E > 0, these solutions contribute an integral, and thegeneralized expansion of the initial wave function looks like

(x, 0) =

n

cn(0)n(x) +

0

E (x)(x, 0) dE.

A similar form generalizes Eq. (A.1c) for t 0. See Chap. 12 in Morrison (1990) for details.



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Appendix 22

A.5.2 Quantum states in the language of energy

Using Eqs. (A.1b) and (A.1c) we can mathematically represent any quantum state either by a function ofposition, the wave function (x, t), or by a list of numbers, the energy expansion coefficients { cn(0) }.Either representation has the same information content; they just represent this information differently.Applying the translation metaphor of A.3, we can view the expansion of (x, t) in the basis { n(x) } as atranslation of the information about the state from the language of position to the language of energy.

To explain what this means we call upon the Born interpretation. According to its generalized form,each number cn(t) is an energy probability amplitude. That is, the real number |cn(t)|2 is the probabilitythat in an energy measurement at t = 0 we will detect the value En: P(En, t) = |cn(t)|2. The set of energyprobability amplitudes is ideal for calculation and interpretation of energy-related properties of the state.

Normalization in the language of energy. To use this interpretation properly we must ensure normal-ization. This requirement holds regardless of how we choose to represent the stateby the wave function(x, t), the momentum amplitude (p), or the energy amplitudes { cn(0) }. For the wave function, thenormalization condition is the integral (A.2b), p. 6. For the momentum amplitude, its the integral (A.12),p. 8. But for the energy representation { cn(0) } its a simple sum:

n |cn(0)|2 = 1. normalization condition (A.3)

Physically, this equation states that the sum of probabilities for all possible energies of the system must be 1.This makes sense: upon measurement of its energy, each particle must exhibit one of the values the list ofallowed energies, the eigenvalues of the Hamiltonian.

The expectation value and uncertainty in the energy. The energy amplitudes greatly facilitatecalculation of statistical energy-related properties of a non-stationary state. For instance, the mean value ofthe energy at time t is the expectation value

E(t) = (t) | H | (t). (A.4a)We could, of course, evaluate this quantity directly from the wave function,

E(t) =

(x, t) 22m

d2

dx2+ V(x)(x, t) dx. (A.4b)

But this is arguably the least efficient way to perform this chore.

The easiest way to evaluate E(t) is to first translate the wave function into the language of energy [thatis, calculate the expansion coefficients a la Eq. (A.1b)]. Then we need only evaluate the sum

E(t) =n

|cn(t)|2 En. average energy (A.5)

Evaluation of the energy uncertainty is equally easy, since to calculate

E = E2 E2, (A.6a)we require only two expectation values, (A.5) and

E2(t) =n

|cn(t)|2 E2n. (A.6b)



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Appendix 23

A.6 Generic problem-solving strategies

A great deal of your first exposure to quantum physics was probably devoted to solving 1D model problemslike square wells and harmonic oscillators. Some of the techniques for solving such problems are specific tothose problems. Others, like power-series expansion of differential equations, involve dense mathematical

detail; these we invoke as needed. But a few qualitative problem-solving gambits combine simplicity withwide applicability. They are generic, in the sense that they apply to an enormous variety of problems. Thisshort section reviews such strategies.

Continuity. One consequence of the requirements that a physically admissible wave function must becontinuous and smoothly varying is that the spatial function of a stationary state and its first spatial derivativemust be continuous at every point. We use this continuity requirement to connect pieces of a spatialfunction we obtain by solving the TISE in separate regions of space.

Simplifications due to symmetry. If the potential V(x) is symmetric under inversion, so thatV(x) = V(x), then we can use the parity of the Hamiltonian eigenfunctions to simplify or set to zeromany of the integrals we encounter in solving the TISE. The principles we use are

The integral over x from

to +

of an odd function is zero.

The integral over x from to + of an even function is twice the integral from 0 to+ of the function.

Curvature and energy. The curvature of a function f(x) is its second derivative d2f / dx2. Curvatureis therefore a measure of how rapidly the slope df / dx changes with changing x. A function with largercurvature turns over faster (per unit increment in x) than a function with smaller curvature. The sign ofthe second derivative determines whether the slope of f(x) increases or decreases with changes in x.37

According to the TISE, Eq. (A.3), p. 16, the curvature of a spatial function E(x) is given by

d2Edx2

= 2m2

E V(x)E(x). (A.1)

The curvature of E(x) is proportional to E(x). Hence the sign of the curvature depends on the sign of

the function, as well as on the sign of E V(x). In a classically allowed region (CA region), E > V(x),so d2E(x)/ dx

2 has the opposite sign ofE(x). A spatial function in a CA region is therefore oscillatory.But in a classically forbidden region (CF region), E < V(x). Here the curvature of E(x) has the samesign as the function, which is therefore an evanescent (or decaying) function. The boundary between aCA and an adjacent CF region, the value of x where V(x) = E is called the classical turning point. Atthis point, the curvature of E(x) is zero, and the function changes character from oscillatory to evanescent(or vice-versa).

Equation (A.1) also shows that the curvature of the spatial function is related to its energy. Since E andthe particles de Broglie wavelength are inversely related,

E =2k2

2m=

2(2)2

2m2, (A.2)

the curvature influences the wavelength. This connection makes sense: the faster a function turns over withchanging x (in a CA region), the smaller its wavelength.38 Several important consequences follow from theserelations.

37Details: In differential geometry, the curvature at a point on a curve is defined to be the norm of the first derivative of theunit tangent vector at the point. That is, the curvature is the ratio of the first derivative vector to the unit normal (the principalnormal) at the same point. The curvature is thus a quantitative, point-by-point measure of how much the curve bends. (A linehas zero curvature.) The reciprocal of the curvature is called the radius of curvature of the curve.

38Details: Considerations of wavelength must be modified in a region where the potential energy is changing with x, sincestrictly speaking a spatial function in such a region has no wavelength. To adapt to this situation we introduce the notion of alocal wavelength [see 9.2 in Morrison (1990) and Chap. 5 in this book]. The generic principles of this section then apply.



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Appendix 24

The ground state has the largest allowed wavelength and no nodes. The ground state isthe stationary state whose spatial function has the largest curvature consistent with the boundaryconditions n(x) 0 as x . Consequently the ground state function never crosses the x axis:it has no nodes.39 This is also the state with the smallest allowed energy, if we measure energy fromthe bottom of the potential well.40

With increasing energy, the bound states gain additional nodes. As we go up the energy

ladder of bound states, from the ground state (with energy E1) to the first excited state (E2) to thesecond excited state (E3) and so, the curvature of n(x) increases. Since all bound-state functionsmust satisfy the usual boundary conditions, each successive bound state has one additional node. Ifwe label the ground state with n = 1, the number of nodes for the stationary state with energy Enis n 1.

With increasing energy, the wavelengths of the bound states decrease. As the curvature ofan oscillatory function increases, its (local) wavelength decreases. So as we climb the energy ladder,each successive bound state has a smaller wavelength.

A.7 Measurements in the microverse

A.7.1 Measurement and the collapse of the wave function

The TDSE fully describes the time evolution of a system with Hamiltonian H so long as no one performs ameasurement on the system. But neither this nor any other postulate describes what happens to the statewhen a measurement is performed. Using the Born interpretation (and eigenfunction expansion) we canpredict the probabilities for various possible outcomes of an ensemble measurement but not the effect of themeasurement on the state. In this sense, the study of state evolution in quantum mechanics bifurcates intotwo topics:

If no one is performing a measurement, the state evolves according to the TDSE.

At the instant a measurement is performed, the state changes (instantaneously) into the eigenstate

that corresponds to the value of the observable obtained in the measurement.

The question of what happens to a quantum state upon measurement of an observable teeters on theprecipice brink of metaphysics. According to the interpretation of quantum mechanics originally devised byNiels Bohr and adapted to measurement by John von Neumann and others, the measurement itself transformsthe state into an eigenstate of the measured observable. Precisely how this miracle occurs, no one knows. Thisinstantaneous change of state as a result of measurement is called the collapse of the wave function. Thisway of understanding the effect of measurement is part of the Copenhagen interpretation of quantummechanics.

We can begin to come to grips with this remarkable notion by thinking explicitly in terms of an ensemble .Suppose we measure the energy of a system thats initially in a non-stationary statemade up of contributionsfrom the ground and first excited stationary states:

(x, 0) = c1(0)1(x) + c2(0)2(x). (A.1a)39Memory Jog: Recall that a node in a function f(x) is a finite value of x at which f(x) = 0. Since all bound states go to

zero in the asymptotic limits x , we dont count these as nodes.40A cautionary note: Be careful when you relate curvature to energy that youre aware of the zero of energy. For example,

consider a 1D symmetric finite square well that supports three bound states. If we measure energy from the bottom of thewell, then the bound-state energies are positive, with E3 > E2 > E1. But if, as is often the case, we put the zero of energyat the top of the well, then the energies are negative and, although this string of inequalities still holds, the magnitudes of thebound-state energies are related by |E3| < |E2| < |E1|. In either case, the wavelengths of the states inside the well (the CAregion) are related by 1 > 2 > 3, with the ground state 1(x) having the smallest curvature and the largest wavelength.



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Appendix 25

Then at time t > 0 (absent an intervening measurement) the wave function is

(x, t) = c1(0)1(x) ei E1t/ + c2(0)2(x) e

iE2t/. (A.1b)

Initially, the energy is uncertain: E(0) > 0, as it is at all subsequent timesuntil we measure the energy.Heres what happens then.

Immediately before the measurement. From t = 0 until the instant of the measurement, all systems inthe ensemble are in the same state, with wave function (A.1b). This state has no definite, fixed energy;rather, it embodies two possible (allowed) energies, because the two energy probability amplitudesc1(0) and c2(0) are nonzero.

At the instant of the measurement. The measurement transforms this single ensemble into twosubensembles. One subensemble consists of all particles that manifested energy E1 in the mea-surement; the other consists of particles that manifested E2. From the energy amplitudes we cancalculate the fraction of systems in the original ensemble in each subensemble: |c1(0)|2 are in theone with energy E1, and |c2(0)|2 have energy E2.

Immediately after the measurement. Two distinct subensembles exist. In each subensemble, the en-ergy is sharp (E = 0), in one with value E1, in the other with E2. Each subensemble is therefore inan energy eigenstate. The measurement has somehow caused the wave function to collapse into

one of the two stationary-state wave functions in the initial state (A.1a).



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Appendix B

Dimensional extermination in quantum mechanics


Errors disappear like magic.

Carton

B.1 IntroductionIn working with dimensions, a little thought and common sense go a very long way. The primary goal ofdimensional extermination is to rewrite relationships among physical quantities (equations and inequalities)in terms of dimensionless quantities.1 Because physical quantitiesmass, energy, length, time, and soforthhave dimensions, theyre called dimensional quantities. For instance, the left- and right-handsides of the de Broglie relationship, p = /, must both have dimension of momentum, M L T1 . On theright-hand side, the wavelength has dimensions of length, L , and has dimensions E T = M L2 T1 ,energy times time. So the dimensions of the left- and right-hand sides do indeed agree. 2

Physical Quantity Dimension Physical Quantity DimensionLength L Time TMass M Temperature Current I

Table B.1. Notation for dimensions of primary physical quantities.For additional quantities, see the the tables in B.8.

Similarly, the time-independent Schrodinger equation (TISE), HE = EE, equates quantities with di-mensions of energy. The

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