Quantum mechanics; classical results, modern systems, and visualized examples 2nd edition

Mechanics Classical Results, Modern Systems, and Visualized Examples Second Edition Richard W.. Richard Wallace Quantum mechanics : classical results, modern systems, and visualized exam

Quantum Mechanics

Classical Results, Modern Systems, and Visualized Examples

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Mechanics

Classical Results, Modern Systems, and Visualized Examples

Second Edition

Richard W Robinett

Pennsylvania State University

1

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Robinett, Richard W (Richard Wallace)

Quantum mechanics : classical results, modern systems, and

visualized examples / Richard W Robinett.—2nd ed.

p cm.

ISBN-13: 978–0–19–853097–8 (alk paper)

ISBN-10: 0–19–853097–8 (alk paper)

1 Quantum theory I Title.

QC174.12.R6 2006

Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India

Printed in Great Britain

on acid-free paper by

Biddles Ltd, King’s Lynn, Norfolk

ISBN 0–19–853097–8 978–0–19–853097–8

10 9 8 7 6 5 4 3 2 1

Preface to the Second Edition

One of the hallmarks of science is the continual quest to refine and expand one’sunderstanding and vision of the universe, seeking not only new answers to oldquestions, but also proactively searching out new avenues of inquiry based onpast experience In much the same way, teachers of science (including textbookauthors) can and should explore the pedagogy of their disciplines in a scientificway, maintaining and streamlining what has been documented to work, but alsoimproving, updating, and expanding their educational materials in response tonew knowledge in their fields, in basic, applied, and educational research For thatreason, I am very pleased to have been given the opportunity to produce a SecondEdition of this textbook on quantum mechanics at the advanced undergraduatelevel

The First Edition of Quantum Mechanics had a number of novel features,

so it may be useful to first review some aspects of that work, in the context

of this Second Edition The descriptive subtitle of the text, Classical Results, Modern Systems, and Visualized Examples, was, and still is, intended to suggest a

number of the inter-related approaches to the teaching and learning of quantummechanics which have been adopted here

• Many of the expected familiar topics and examples (the Classical Results)

found in standard quantum texts are indeed present in both editions, but wealso continue to focus extensively on the classical–quantum connection as one

of the best ways to help students learn the subject Topics such as space probability distributions, time-dependent wave packet solutions, and thecorrespondence principle limit of large quantum numbers can all help studentsuse their existing intuition to make contact with new quantum ideas; classicalwave physics continues to be emphasized as well, with its own separate chapter,for the same reason Additional examples of quantum wave packet solutionshave been included in this new Edition, as well as a self-contained discussion

momentum-of the Wigner quasi-probability (phase-space) distribution, designed to helpmake contact with related ideas in statistical mechanics, classical mechanics,and even quantum optics

• An even larger number of examples of the application of quantum

mech-anics to Modern Systems is provided, including discussions of experimental

realizations of quantum phenomena which have only appeared since the FirstEdition Advances in such areas as materials science and laser trapping/cooling

vi PREFACE TO THE SECOND EDITION

have meant a large number of quantum systems which have historically beenonly considered as “textbook” examples have become physically realizable Forexample, the “quantum bouncer”, once discussed only in pedagogical journ-

als, has been explored experimentally in the Quantum states of neutrons in the Earth’s gravitational field.ÉThe production of atomic wave packets whichexhibit the classical periodicity of Keplerian orbitsÊis another example of a

Classical Result which has become a Modern System.

The ability to manipulate nature at the extremes of small distance and even atomic-level) and low temperatures (as with Bose–Einstein con-densates) implies that a knowledge of quantum mechanics is increasinglyimportant in modern physical science, and a number of new discussions of

(nano-applications have been added to both the text and to the Problems, including

ones on such topics as expanding/interfering Bose–Einstein condensates, thequantum Hall effect, and quantum wave packet revivals, all in the context offamiliar textbook level examples

• We continue to emphasize the use of Visualized Examples (with 200 figures

included) to reinforce students’ conceptual understanding of the basic ideasand to enhance their mathematical facility in solving problems This includesnot only pictorial representations of stationary state wavefunctions and time-dependent wave packets, but also real data The graphical representation ofsuch information often provides the map of the meeting ground of the some-times arcane formalism of a theorist, the observations of an experimentalist,and the rest of the scientific community; the ability to “follow such maps” is

an important part of a physics education

Motivated in this Edition (even more than before) by results appearing fromPhysics Education Research (PER), we still stress concepts which PER stud-ies have indicated can pose difficulties for many students, such as notions ofprobability, reading potential energy diagrams, and the time-development ofeigenstates and wave packets

As with any textbook revision, the opportunity to streamline the presentationand pedagogy, based on feedback from actual classroom use, is one of the mostimportant aspects of a new Edition, and I have taken this opportunity to removesome topics (moving them, however, to an accompanying Web site) and adding

new ones New sections on The Wigner Quasi-Probability Distribution (and many related problems), an Infinite Array of δ-functions: Periodic Potentials and the Dirac Comb, Time-Dependent Perturbation Theory, and Timescales in Bound State

É The title of a paper by V V Nesvizhevsky et al (2002) Nature 415, 297.

ÊSee Yeazell et al (1989).

PREFACE TO THE SECOND EDITION vii

Systems: Classical Period and Quantum Revival Times reflect suggestions from

various sources on (hopefully) useful new additions A number of new in-text

Examples and end-of-chapter Problems have been added for similar reasons, as well as an expanded set of Appendices, on dimensions and mathematical methods.

An exciting new feature of the Second Edition is the development of a WebsiteËin support of the textbook, for use by both students and instructors, linkedfrom the Oxford University PressÌ web page for this text Students will find

many additional (extended) homework problems in the form of Worksheets on

both formal and applied topics, such as “slow light”, femtosecond chemistry, and

quantum wave packet revivals Additional material in the form of Supplementary Chapters on such topics as neutrino oscillations, quantum Monte Carlo approx-

imation methods, supersymmetry in quantum mechanics, periodic orbit theory

of quantum billiards, and quantum chaos are available

For instructors, copies of a complete Solutions Manual for the textbook, as well as Worksheet Solutions, will be provided on a more secure portion of the site,

in addition to copies of the Transparencies for the figures in the text An 85-page Guide to the Pedagogical Literature on Quantum Mechanics is also available there,

surveying articles from The American Journal of Physics, The European Journal

of Physics, and The Journal of Chemical Education from their earliest issues,through the publication date of this text (with periodic updates planned.) In

addition, a quantum mechanics assessment test (the so-called Quantum anics Visualization Instrument or QMVI) is available at the Instructors site, along

Mech-with detailed information on its development and sample results from earliereducational studies Given my long-term interest in the science, as well as thepedagogy, of quantum mechanics, I trust that this site will continually grow inboth size and coverage as new and updated materials are added Information onaccessing the Instructors area of the Web site is available through the publisher

at the Oxford University Press Web site describing this text

I am very grateful to all those from whom I have had help in learning quantummechanics over the years, including faculty and fellow students in my under-graduate, graduate, and postdoctoral days, current faculty colleagues (here atPenn State and elsewhere), my own undergraduate students over the years, andnumerous authors of textbooks and both research and pedagogical articles, many

of whom I have never met, but to whom I owe much I would like to thank allthose who helped very directly in the production of the Second Edition of thistext, specifically including those who provided useful suggestions for improve-ment or who found corrections, namely, J Banavar, A Bernacchi, B Chasan,

Ë See robinett.phys.psu.edu/qm

Ì See www.oup.co.uk

viii PREFACE TO THE SECOND EDITION

J Edmonds, M Cole, C Patton, and J Yeazell I have truly enjoyed recent laborations with both M Belloni and M A Doncheski on pedagogical issuesrelated to quantum theory, and some of our recent work has found its way intothe Second Edition (including the cover) and I thank them for their insights, andpatience

col-No work done in a professional context can be separated from one’s personallife (nor should it be) and so I want to thank my family for all of their helpand understanding over my entire career, including during the production ofthis new Edition The First Edition of this text was thoroughly proof-read by mymother-in-law (Nancy Malone) who graciously tried to teach me the proper use

of the English language; her recent passing has saddened us all My own mother(Betty Robinett) has been, and continues to be, the single most important rolemodel in my life—both personal and professional—and I am deeply indebted

to her far more than I can ever convey Finally, to my wife (Sarah) and children(James and Katherine), I give thanks everyday for the richness and joy they bring

to my life

Richard Robinett December, 2005 State College, PA

1 A First Look at Quantum Physics 3

2 Classical Waves 34

2.4 Inverting the Fourier transform: the Diracδ-function 46

3 The Schrödinger Wave Equation 65

x CONTENTS

4 Interpreting the Schrödinger Equation 84

4.2 Probability Interpretation of the Schrödinger Wavefunction 91

4.6 Energy Eigenstates, Stationary States, and the Hamiltonian Operator 107

4.7.1 Transforming the Schrödinger Equation Into Momentum

5 The Infinite Well: Physical Aspects 134 5.1 The Infinite Well in Classical Mechanics: Classical Probability

5.2.1 Position-Space Wavefunctions for the Standard Infinite Well 137 5.2.2 Expectation Values and Momentum-Space Wavefunctions for

6 The Infinite Well: Formal Aspects 166

CONTENTS xi

7 Many Particles in the Infinite Well: The Role of Spin and Indistinguishability 192

8 Other One-Dimensional Potentials 210

8.1.4 Infinite Array ofδ-functions: Periodic Potentials and the Dirac

8.3.1 The Schrödinger Equation in Three Dimensions 230

9 The Harmonic Oscillator 239

xii CONTENTS

9.5 Unstable Equilibrium: Classical and Quantum Distributions 254

10 Alternative Methods of Solution and Approximation

12.6.2 Propagator and Wave Packets for the Harmonic Oscillator 353

CONTENTS xiii

12.7 Timescales in Bound State Systems: Classical Period and Quantum

13 Operator and Factorization Methods for the Schrödinger

xiv CONTENTS

18 Gravity and Electromagnetism in Quantum Mechanics 540

18.6.3 Hyperfine Splittings: Magnetic Dipole–Dipole Interactions 574

18.7.1 Measuring the Spinor Nature of the Neutron Wavefunction 576

CONTENTS xv

19.2.2 Wave Equation for Scattering and the Born Approximation 606

A Dimensions and MKS-type Units for Mechanics, Electricity and Magnetism, and Thermal Physics 641

B Physical Constants, Gaussian Integrals, and the Greek

E Special Functions 666

xvi CONTENTS

F Vectors, Matrices, and Group Theory 674

PART I

The Quantum Paradigm

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(1) The incorporation of a wave property description of matter into a consistentwave equation, via the Schrödinger equation;

(2) The statistical interpretation of the Schrödinger wavefunction in terms of

a probability density (in both position- and momentum-space);

(3) The study of single-particle solutions of the Schrödinger equation, for bothtime-independent energy eigenstates as well as time-dependent systems, formany model systems, in a variety of spatial dimensions, and finally;

(4) The influence of both quantum mechanical effects and the constraintsarising from the indistinguishability of particles (and how that depends

on their spin) on the properties of multiparticle systems, and the resultingimplications for the structure of different forms of matter

By way of example of our approach, we first note that Fig 1.1 illustrates

an example of a precision measurement of the wave properties of ultracoldneutrons, exhibiting a Fresnel diffraction pattern arising from scattering from

4 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

Scanning slit position

100 µm 500

1000

1500

Figure 1.1 Fresnel diffraction pattern obtained from scattering at a sharp edge, obtained using ultracold neutrons by Gähler and Zeilinger (1991).

a sharp edge, nicely explained by classical optical analogies We devote Chapter 2

to a discussion of classical wave physics and Chapter 3 to the description ofsuch wave effects for material particles, via the Schrödinger equation Figure 1.2demonstrates an interference pattern using electron beams, built up “electron byelectron,” with the obvious fringes resulting only from a large number of indi-vidual measurements The important statistical aspect of quantum mechanics,simply illustrated by this experiment, is discussed in Chapter 4 and beyond

It can be argued that much of the early success of quantum theory can be traced

to the fact that many exactly soluble quantum models are surprisingly ent with naturally occurring physical systems, such as the hydrogen atom andthe rotational/vibrational states of molecules and such systems are, of course,discussed here The standing wave patterns obtained from scanning tunnel-ing microscopy of “electron waves” in a circular corral geometry constructedfrom arrays of iron atoms on a copper surface, seen in Fig 1.3, reminds us ofthe continuing progress in such areas as materials science and atom trapping

coincid-in developcoincid-ing artificial systems (and devices) for which quantum mechanics

is applicable In that context, many exemplary quantum mechanical models,which have historically been considered as only textbook idealizations, have alsorecently found experimental realizations Examples include “designer” potentialwells approximating square and parabolic shapes made using molecular beamtechniques, as well as magnetic or optical traps The solution of the Schrödingerequation, in a wide variety of standard (and not-so-standard) one-, two-, andthree-dimensional applications, is therefore emphasized here, in Chapters 5, 8, 9,

1.1 HOW THIS BOOK APPROACHES QUANTUM MECHANICS 5

Figure 1.2 Interference patterns obtained by using an electron microscope showing the fringes being

“built up” from an increasingly large number of measurements of individual events From Merli, Missiroli, and Pozzi (1976) (Photo reproduced by permission of the American Institute of Physics.)

and 15–17 In parallel to these examples, more formal aspects of quantum theoryare outlined in Chapters 7, 10, 12, 13, and 14

The quantum in quantum mechanics is often associated with the discrete

energy levels observed in bound-state systems, most famously for atomic systemssuch as the hydrogen atom, which we discuss in Chapter 17, emphasizing that this

is the quantum version of the classical Kepler problem We also show, in Fig 1.4,experimental measurements leading to a map of the momentum-space probab-

ility density for the 1S state of hydrogen and the emphasis on momentum-space

methods suggested by this result is stressed throughout the text The influence ofadditional“real-life”effects, such as gravity and electromagnetism, on atomic andother systems are then discussed in Chapter 18 We note that the data in Fig 1.4

6 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

Figure 1.3 Standing wave patterns obtained using scanning tunneling microscopy from a circular “corral”

of radius ∼70 Å, constructed from 48 iron atoms on a copper surface (Photo courtesy of IBM Almaden.)

1.2 1.4

Figure 1.4 Electron probability density obtained by scattering with three different energy probes, compared with the theoretically calculated momentum-space probability density for the hydrogen-atom ground state, from Lohmann and Weigold (1981) The data are plotted again the scaled momentum in atomic units (a.u.),

q = a0p/
.

was obtained via scattering processes, and the importance of scattering methods

in quantum mechanics is emphasized in both one-dimension (Chapter 11) andthree-dimensions (Chapter 19) The fact that spin-1/2 particles must satisfy the

Pauli principle has profound implications for the way that matter can arrange

1.1 HOW THIS BOOK APPROACHES QUANTUM MECHANICS 7

Figure 1.5 Plots of the ionization energy (solid) and atomic polarizability (dashed) versus nuclear charge, showing the shell structure characterized by the noble gas atoms, arising from the filling of atomic energy levels as mandated by the Pauli principle for spin-1/2 electrons.

itself, as shown in the highly correlated values of physical parameters shown inFig 1.5 for atoms of increasing size and complexity While it is illustrated here in

a numerical way, this should also be reminiscent of the familiar periodic table ofthe elements, and the Pauli principle has similar implications for nuclear struc-ture We discuss the role of spin in multiparticle systems described by quantummechanics in Chapters 7, 14, and 17

We remind the reader that similar dramatic manifestations of quantum nomena (including all of the effects mentioned above) are still being discovered,

phe-as illustrated in Fig 1.6 In a justly famous experiment,É two highly localizedand well-separated samples of sodium atoms are cooled to sufficiently low tem-peratures so that they are in the ground states of their respective potential wells(produced by laser trapping.) The trapping potential is removed and the res-ulting coherent Bose–Einstein condensates are allowed to expand and overlap,exhibiting the quantum interference shown in Fig 1.6 (the solid curve, showing

ÉFrom the paper entitled Observation of interference between two Bose condensates by Andrews et al.

(1997).

8 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

200 Position (µm)

400 0

30

60

Figure 1.6 Data (from Andrews et al (1997)) illustrating the interference of two Bose condensates as

they expand and overlap (solid curve), compared to a single expanding Bose condensate (dotted curve).

regular absorption variations across the central overlap region), while no suchinterference is observed for a single expanding quantum sample (dotted data.)Many of the salient features of this experiment can be understood using relativelysimple ideas outlined in Chapters 3, 4, and 9

The ability to use the concepts and mathematical techniques of quantummechanics to confront the wide array of experimental realizations that havecome to characterize modern physical science will be one of the focuses of thistext Before proceeding, however, we reserve the remainder of this chapter forbrief reviews of some of the essential aspects of both relativity and standardresults from quantum theory

1.2 Essential Relativity

While we will consider nonrelativistic quantum mechanics almost exclusively,

it is useful to briefly review some of the rudiments of special relativity and the

fundamental role played by the speed of light, c.

For a free particle of rest mass m moving at speed v, the total energy (E), momentum (p), and kinetic energy (T ) can be written in the relativistically

correct forms

E = γ mc2, p = γ mv, and T ≡ E − mc2 = (γ − 1)mc2 (1.1)where

1.2 ESSENTIAL RELATIVITY 9

The nonrelativistic limit corresponds to v /c << 1, in which case we can use

the series expansion

which are the familiar nonrelativistic results for motion at speeds slow compared

to the speed of light

In quantum mechanics the momentum is a more natural variable than v, and

a useful relation can be obtained from Eqn (1.1), namely

This form stresses the fact that E, pc, and mc2 all have the same dimensions(namely energy), and we will often use these forms when convenient As anexample, the rest energies of various atomic particles will often be quoted inenergy units; for the electron and proton we have

m e c2= 0.511 MeV and m p c2 = 938.3 MeV (1.6)

Recall that the electron volt or eV is defined by

1 eV = the energy gained by a fundamental charge e

which has been accelerated through 1 V

“ a 2 eV electron ” should be taken to mean that the electron has a kinetic energy T = E − mc2≈ p2/2m ≈ 2 eV We will often write pc =
2(mc2)T in

this limit

10 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

At the other extreme, in the ultrarelativistic limit when E  mc2(or v
c),

which is also seen to be consistent with the energy–momentum relation for truly

massless particles (such as photons), namely E = pc.

We list below several typical quantum mechanical systems and the magnitudes of the energies involved:

order-of-• Electrons in atoms: For the inner shell electrons of an atom with nuclear

charge +Ze, the kinetic energy is of order T ≈ Z2 13.6 eV We can say,

somewhat arbitrarily, that relativistic effects become nonnegligible when T 

0.05 mc2(i.e a 5% effect) This condition is satisfied when Z  43, implyingthat the effects of relativity must certainly be considered for heavy atoms

• Deuteron: The simplest nuclear system is the bound state of a proton and

neutron where the typical kinetic energies are T ≈ 2 MeV; this is to be

com-pared with m p c2 ≈ m n c2 ≈ 939 MeV so that the deuteron can be considered

as a nonrelativistic system to first approximation

• Quarks in the proton and pion: The constituent quark model of

element-ary particles postulates that three quarks of effective mass roughly m q c2 ≈

350 MeV form the proton; this implies binding energies and kinetic energies

of the order of 1− 10 MeV which is consistent with “nonrelativity.” The pion,

on the other hand, is considered a bound state of two such quarks, but has

rest energy m π c2 ≈ 140 MeV, so that binding energies (and hence kineticenergies) of order several hundred MeV are required and relativistic effectsdominate.Ê

• Compact objects in astrophysics: The electrons in white dwarf stars and

neut-rons in neutron stars have kinetic energies T e ≈ 0.08 MeV and T n ≈ 140 MeVrespectively, so these objects are “barely” nonrelativistic

Just as the speed of light, c, sets the scale for when relativistic effects are important,

quantum physics also has an associated fundamental, dimensionful parameter,

Ê The pion is really a quark–antiquark system Bound states of heavier quarks and antiquarks, which are more slowly moving, can be more successfully described using nonrelativistic quantum mechanics.

1.3 QUANTUM PHYSICS:
AS A FUNDAMENTAL CONSTANT 11

namely Planck’s constant Its first applications came in the understanding of some

of the quantum aspects of the electromagnetic (EM) field and the particle nature

of EM radiation

• In his investigations of the black body spectrum emitted from heated objects(so-called cavity radiation), Planck found that he could only fit the observedintensity distribution if he made the (then radical) assumption that the EM

energy of a given frequency f was quantized and given by

The constant of proportionality, h, was derived from a “fit” to the experimental

data, and has been found to be

• Einstein assumed the energy quantization of Eqn (1.10) was a more eral characteristic of light, and proposed that EM radiation was composed of

gen-photonsËor “bundles” of discrete energy E γ = hf He used the photon concept

to explain the photoelectric effect, and predicted that the kinetic energy of

elec-trons emitted from the surface of metals after being irradiated should be givenby

1

where W is called the work function of the metal in question Subsequent

experiments were able to confirm this relation, as well as providing another,

complementary measurement of h (P1.5) which agreed with the value

obtained by Planck

• The relativistic connection between energy and momentum for a masslessparticle such as the photon could be used to show that it has a momentumgiven by

12 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

Figure 1.7 Geometry for Compton scattering The incident

photon scatters from an electron, initially at rest, at an angleθ γ.

u g

E g

E e⬘

E g⬘

whereλ is the wavelength Arthur Compton noted that the scattering of

X-rays by free electrons at rest could be considered as a collision process wherethe incident photon has an energy and momentum given by Eqn (1.14), asshown in Fig 1.7 Conservation of energy and momentum (P1.6) can then beapplied to show that the wavelength of the scattered photon,λ
, is given by the

Compton scattering formula

λ
− λ = h

whereθ γ is the angle between the incident and scattered photon directions;X-ray scattering experiments confirmed the validity of Eqn (1.15)

The connection of Planck’s constant to the properties of material particles, such

as electrons, came later:

• Using yet another experimental “fit” to spectroscopic data, in this case theBalmer–Ritz formula for the frequencies in the spectrum for hydrogen, Bohrused semiclassical arguments to deduce that the angular momentum of theelectron was quantized as

2π = n
with n = 1, 2, 3 (1.16)

• Motivated by the dual wave-particle nature exhibited by light, for example, inCompton scattering, de Broglie suggested that matter, specifically electrons,would exhibit wave properties He postulated that the relation

λdB= h

1.3 QUANTUM PHYSICS:
AS A FUNDAMENTAL CONSTANT 13

apply to material particles as well as to photons, thereby defining the de Broglie wavelength He could show that Eqn (1.17) reproduced the Bohr condition of

Eqn (1.16), and thus explain the hydrogen atom spectrum

Example 1.1 de Broglie Wavelength of a Truck?

Over the roughly 80 years since the Davisson–Germer experiment 4 directly demonstrated the wave nature of electrons by the observation of the diffraction of electron beams from nickel crystals, with a wavelength consistent5 with Eqn (1.17), the quantum mechanical

wave-particle duality of objects of increasing size and complexity has been observed.

Only 3 years after the prediction by de Broglie, Davisson and Germer accelerated electrons through voltages of orderV ∼ 50 V to speeds given by

nicely matched the atomic spacings in their sample (already determined by X-ray scattering experiments).

It is sometimes useful to compare the quantum mechanical wavelength of a particle to other physical dimensions, including its own size While many particles which play a crucial role

in determining the structure of matter have finite and measurable sizes, all ultrahigh energy scattering experiments involving electrons (which therefore probe ultra-small distance scales) are so far consistent with the electrons having no internal structure; various experiments can

be interpreted as putting upper limits on an electron “size” of order 10−10Å = 10 −5F orroughly 50, 000 times smaller than a proton or neutron This justifies the assumption of a

“point-like” electron.

Sixty years after the Davisson and Germer experiments with electrons, single- and

double-slit diffraction of slow neutrons was observed, giving the “most precise realization hitherto for matter waves.”6 In this case, the neutrons have a physical size measured (in other experiments)

to be of order 1 F = 10 −5Å and ultracold neutrons withλ = 15 − 30 Å were utilized,

so that the spatial extent of the particle is still orders-of-magnitude less than its quantum mechanical wave length In the last decade or so, however, advances in atom interferometry have led to the observation of interference or diffraction phenomena for small atoms (helium, He), larger atoms (atomic sodium, Na), diatomic molecules (sodium dimer or Na2), small clusters of molecules (of H, He 2 , and D 2 ), and most recently C 60 molecules (buckeyballs), all of atomic dimensions, and with increasingly small de Broglie wavelengths Representative data (and references) are collected below.

4 See Davisson and Germer (1927).

5 Their exact words are “The equivalent wave-lengths of the electron beams may be calculated from the

diffraction data in the usual way These turn out to be in acceptable agreement with the values of h/mv of the undulatory mechanics.”

6 Zeilinger et al (1988).

14 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

(Continued)

(atomic units)

approx size (diameter)

λ dB

(in Å) Reference

electron (e−) 5 × 10 −4 <10−10Å 1–2 Davisson/Germer (1927)

sodium (Na 2 ) 2 · 23 = 46 ∼8 Å 0.1 Chapman et al (1995)

buckeyballs (C 60 ) 60 · 12 = 720 7 Å 0.025 Arndt et al (1999)

It is clear that objects of an increasing classical nature (like C 60 , with a large number

of internal degrees of freedom involved in the many bonds) are seen to exhibit quantum mechanical behavior The possibility that the quantum mechanical wavelength of an object can be much smaller than its physical size (hence the question in the title of this Example) has been amply demonstrated.

The de Broglie relation contains the seeds of the position–momentum uncertainty principle, namely

where x and p are the uncertainties in a measurement of x and p respectively.

Equation (1.19) puts fundamental limitations on one’s ability to simultaneouslymeasure the position and momentum of a particle; it also leads to the notion

of zero-point energy, a minimum unavoidable energy of a particle confined to a

localized region of space

The example of a particle in a one-dimensional box illustrates this most simply

A particle of mass m confined to a one-dimensional box of length L will satisfy the “standing wave” condition for de Broglie waves if n (λ n /2) = L (compare this to Eqn (1.35) below and explain any differences) with n = 1, 2, 3 This corresponds to quantized momenta p n = n
π/L and energies given by

Tmin ∝ pmin2

2m ∝
2

1.3 QUANTUM PHYSICS:
AS A FUNDAMENTAL CONSTANT 15

While such “back-of-the-envelope” calculations should be used with care, theycan often provide insight into the ground state of a quantum system

• Electrons in atoms? The electrons in atoms and molecules are confined

to a region of size x ∼ 1 Å The corresponding unavoidable spread in

momentum is therefore roughly

• Photons in atoms? On the other hand, the photons emitted in the radiative

decays of such atoms, cannot have been “stored” in the atom beforehand Tosee this, we note that a photon “bouncing around” in an atomic-sized boxwill have the same p ∼ p as in Eqn (1.22) Because massless photons are

necessarily relativistic, the corresponding kinetic energy is then given by

which is much larger than the 1–10 eV observed in typical transitions

• Alpha particles in nuclei? Radioactive nuclei emit α particles (m α ≈ 4m p)with kinetic energies of a few MeV The minimum momentum in a heavy

nucleus of radius R ≈ 5 F is roughly

which is consistent with observations

• Electrons from neutron β decay? Neutrons decay via the process n → pe ν e

where the electron (anti-)neutrino is often not directly detected, but the ted electron of a few MeV is easily observed A “pre-existing” electron in the

emit-neutron,“waiting around” to decay, would have a value of pc roughly five times

16 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

larger than in Eqn (1.25) (since the neutron is roughly five times smaller than

a heavy nucleus) implying relativistic electrons with kinetic energies of order

E0(e) ≈ pc ≈ 200 MeV; the decay electrons must therefore be created ex nihilo

at the time of decay Arguments such as these were among the first pieces ofevidence used to predict that a new force beyond those known classically, the

so-called weak interaction, was required to explain such decays.

For single particles, it is often clear when wave mechanical effects are ant For example, in electron scattering from crystal planes, the Bragg condition

import-for constructive interference can be written in the import-form n λ = D sin(φ) where D

is the interatomic spacing andφ is the scattering angle; clearly λ must be

com-parable to the other spatial dimensions in the problem for the wave properties ofmatter to be visible Many problems have some other natural length scale againstwhich to compare the de Broglie wavelength

Example 1.2 Systems of Particles: Classical or Quantum Mechanics?

At high temperatures and/or low densities, the behavior of a gas can be described by classical statistical mechanics; the atoms, to a good approximation, move along classical trajectories.

At low temperatures and/or high densities, quantum effects become important The classical approximation will break down when the de Broglie wavelength of a typical particle becomes

comparable to the average interparticle distance; if the number density is n, this distance is roughly d ∼ n −1/3 which can then be compared toλ = h/p For a system in thermal equilibrium, the thermal energy is E = p2/2M = k B T /2 where k Bis Boltzmann’s constant

and T is the temperature; this gives

λ = h

p =
2π

For air at typical room condition, one can estimate that M ∼ 28 u (for diatomic nitrogen,

N2) and T ∼ 300 K to find λ ∼ 0.45 Å; at one atmosphere of pressure, one has Patm ∼

105N/m2 giving7 n∼ 2.4 × 10 25 m−3or d = n −1/3 ∼ 35 Å Since d  λ the system

can be considered classically.

On the other hand, the conduction electrons in a metal (which for many purposes can be considered as a gas) have a de Broglie wavelength which is

M/m e ≈ 225 times larger than

for gas atoms, so thatλ e∼ 100 Å The larger densities of solid matter, however, imply much

smaller interparticle distances; with a few conduction electrons per atom, electron densities

of n e ≈ (1–10) × 1028 m−3are typical, so that d = n −1/3 e ≈ 2–5 Å giving λ  d in this

case.

Ï One uses the ideal gas law, P = nk B T

1.4 SEMICLASSICAL MODEL OF THE HYDROGEN ATOM 17

1.4 Semiclassical Model of the Hydrogen Atom

An essentially classical approach to the bound-state dynamics of the electron–proton (hydrogen atom) system can be extended by using the wave mechanicsidea of de Broglie to derive many of the most important features of the hydrogenspectrum, as in the Bohr model We note that other systems can be usefullydescribed with the same techniques, including (i) multiply ionized atoms whereall but one electron has been removed and (ii) states where an outer valenceelectron is in a highly excited state, and far from the ionic core so that it appears

hydrogenic, so-called Rydberg atom states.

The Coulomb force between the two particles can be written in the form

where we will conventionally write (in the units used in this book) the

funda-mental constant of electrostatics in the form K = 1/4π 0 This force can be

derived from the Coulomb potential, namely

V (r) = − Ke2

• Before proceeding, let us pause and make a few comments about the sionful constants that appear in this and other atomic and nuclear physicssystems involving electromagnetism The combination of constants whichdetermines the electrostatic force between two fundamental charges can bewritten in the form

andα is dimensionless and is called the fine-structure constant The

combina-tion
c has dimensions and numerical values given by

c ≈ 1973 eV Å ≈ 197.3 MeV F ≈ 0.1973 GeV F (1.31)which are useful for atomic/molecular, nuclear, and elementary particle physicsproblems, respectively Together, these give

Ke2 ≈ 14.4 eV Å ≈ 1.44 MeV F ≈ 2.31 × 10−28J· m (1.32)Despite focusing on nonrelativistic systems, we will often manipulate factors

of c to make use of these combinations Now back to the hydrogen atom.

18 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

Figure 1.8 Standing wave pattern in circular

orbits for deBroglie waves.

“tie” ends together

R

2pR

l

For circular orbits in which the electron (mass m) is assumed to orbit around

the (stationary and infinitely heavy) proton, Newton’s law implies that

m v

2

r = ma C = F(r) = Ke2

where we have used the appropriate centrifugal acceleration; this relation is

classically consistent with any value of r.

If we wish to incorporate the wave properties of the electron via the de Broglierelation

á la Bohr, and the orbital angular momentum must be quantized This additional

constraint, along with Eqn (1.33), gives

The corresponding speeds in the Bohr model are also quantized and given by

n = αc

which reminds us that the electron orbital motion (in hydrogen at least) is relativistic The period (τ) of the classical orbit and the corresponding frequency

non-1.4 SEMICLASSICAL MODEL OF THE HYDROGEN ATOM 19

2mc

2α2

1

n2

≈ − (0.51 × 106eV)

2

1137

21

ellip-Example 1.3 “Sieve” for Rydberg Atoms

While for small values of n, the typical sizes of atoms, as exemplified by Eqn (1.37), are

small, for large values of the quantum number (the Rydberg atom limit), the spatial extent

of the state can easily fall into the micron range or larger The large sizes of such Rydberg atoms make possible “slit” experiments in which the results do not depend on the quantum mechanical wave properties of the system but, rather, on their classical physical size In one such experiment,8beams of Rydberg atoms (in this case highly excited sodium atoms) were produced with specified quantum numbers in the range 23< n < 65 These were allowed to

drift toward an array of rectangular, micrometer size slits in gold foil; the average slit size was

2× 10 µm If one assumes that the Rydberg atoms have an effective classical radius given

by ka0n2, where k is a dimensionless constant, one can argue from Fig 1.9 that if the center

of the atom is more than d = l/2 − ka0n2from the center of the slit, the atom will not pass

through it (being ionized instead upon contact with the foil) The transmission probability, T ,

ÐSee Fabre et al (1983) for details.

20 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

Figure 1.10 Transmission probability versus principal quantum number square, n2, for the Rydberg

“sieve”; the data are taken from Fabre et al (1983).

is then determined solely by geometry, and is given by

of the atomic beam (beam perpendicular to foil), while the diamonds are for incidence at an

angle such that the effective width of the slit, l, is reduced by a factor of two The predictions

of the simple “hard-sphere” model described above are indicated by the straight lines for the two cases.

An important limit is suggested by Eqn (1.36) where we note that n >> 1 is

required to obtain macroscopically large values of the angular momentum; thefact that quantum systems approach (in an average sense which we will discuss in

later chapters) their classical counterparts in this limit is called the correspondence

1.5 DIMENSIONAL ANALYSIS 21

principle, and was used heavily by Bohr in his analyses For example, he noted

that the photons emitted in transitions between the quantized energy levels inEqn (1.42) satisfy the Balmer formula, written here in the form

2π
f γ = hf γ = E γ = E n − E n
= m(Ke2
22)2

1

A classical particle undergoing circular acceleration would emit radiation at

its orbital frequency, f , which from Eqn (1.40) is given by exactly the limit

above The connections and interpolations between the quantum mechanicaland classical descriptions of the physical world are stressed in this book

It is interesting to note in this context the role that wave mechanics andCoulomb’s law (via
and e) play in determining the densities of “ordinary”

solid matter.ÑThe mass of atoms is due mostly to their nuclear constituents (theprotons and neutrons), while their size is determined by the quantum properties

of their electrons For example, an order of magnitude estimate of the density ofatomic hydrogen can be obtained by assuming that there is one proton mass in

a cube of size 2a0≈ 1 Å on a side; this gives a density of roughly

(2a0)3 ≈ 1.6 × 103kg /m3 ∼ 1.6 gr/cm3 (1.46)which is in the right “ball-park.”

A number of such useful results were derived using such early quantummechanical ideas, but in order to utilize the full predictive power of modernquantum mechanics, we will incorporate important notions of classical wavephysics (Chapter 2) into the Schrödinger equation formulation (Chapter 3 andbeyond) of quantum physics

1.5 Dimensional Analysis

Most problems in physics are ultimately to be related to measureable quantities

in the “real world”, and therefore have answers that carry dimensions For purely

ÑThe seemingly commonplace observation that “matter held together by Coulomb forces is stable” is

a remarkable and rather subtle consequence of many aspects of quantum theory; for a nice discussion, see Lieb (1976) The corresponding classical “no-go” theorem of Earnshaw (see Jones 1980 for a brief

history) goes something like “No system of charged particles can be in stable static equilibrium.”

22 CHAPTER 1 A FIRST LOOK AT QUANTUM PHYSICS

mechanical problems, any constant or variable representing a physical property(call itX ) will have dimensions, which can be constructed from various powers

of fundamental units of mass (M ), length (L), and time (T ) We can formalize

this statement by using the notation

[X ] ≡ dimensions of X = M a L b T c (1.47)

where a, b, c are real, possibly fractional, exponents Familiar examples such as force (F ), pressure (P), and density ( ρ) correspond to

[F] = MLT−2, [P] = ML−1T−2, and [ρ] = ML−3 (1.48)

Specific conventions giving the units of physical observables (such as the MKS or

meter-kilogram-second system) rely on this observation, but it is more general

It can often be used to “solve” for the dependence of the physical quantity inquestion on the dimensionful parameters of the problem

Example 1.4 The Dimensions of the Quantum Harmonic Oscillator

The only dimensional inputs to the classic problem of a mass and spring system are the mass,

m, and spring constant, K, which have dimensions

The period of the oscillatory motion,τ, should presumably depend on these parameters, plus

additional dimensionless constants; we therefore expect thatτ ∝ m α K β, so we write

1.6 QUESTIONS AND PROBLEMS 23

(Continued)

For the classical oscillator, there is no combination of m and K, which naturally gives a

length scale, or preferred amplitude In the quantum version of this problem, however, we have another dimensionful constant at our disposal, namely Planck’s constant or
, which has dimensions

In contrast to the classical case, we can now also construct a fundamental length or amplitude

by writing A ∝ m α K β
γ and proceeding as above to find thatγ = 1/2 and α = β =

1.6 Questions and Problems

Q1.1 Everyone comes to any text with some idea of what they want to get out of it.

What one question about, or important aspect of, quantum mechanics interests you the most? Look in the index, and see if that topic is covered in this book If it

is, find the reference, and see what you will have to learn in order to understand

it If it is not, do a library or web search (or ask someone) until you learn what you really want to know.

Q1.2 Try to imagine a worldÉÉin which the fundamental constants of relativity and

quantum mechanics were on a more “human scale,” namely c = 10 m/s and

= 0.1 J · s For example, how would the famous “twin paradox” of relativity

ÉÈ Recall that the MKSA system of units actually uses the Ampere, that is, current, as the defining unit for EM.

ÉÉ For an entertaining version of this “what-if exercise,” see Mr Tompkins in Wonderland by George

Gamow (1946).

above The connections and interpolations between the quantum mechanicaland classical descriptions of the physical world are stressed in this book... derived using such early quantummechanical ideas, but in order to utilize the full predictive power of modernquantum mechanics, we will incorporate important notions of classical wavephysics (Chapter... quantized and given by

n = αc

which reminds us that the electron orbital motion (in hydrogen at least) is relativistic The period (τ) of the classical orbit and