Quantum Mechanics 5th Edition Alastair I M Rae

Supporting Material

Quantum Mechanics5th Edition

Alastair I M Rae

Contents

1 Coloured Versions of Some Figures 3

2 Updates and Corrections 6Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Bibliography 14Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Complementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15The conceptual problems of quantum mechanics . . . . . . . . . . . . . . . . . 16Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2

1Coloured Versions of Some Figures

FIGURE 2.6An image of the (111) surface of silicon obtained by scanning tunnelling microscopy. Thebright peaks correspond to silicon atoms. The hexagonal symmetry is a characteristic fea-ture of this surface. (Supplied by P. A. Sloan and R. E. Palmer of the Nanoscale PhysicsResearch Laboratory in the University of Birmingham.)

3

Coloured Versions of Some Figures 4

0,0

1,0 1,1

2,0 2,1 2,2

FIGURE 3.3Representations of the shapes of the spherical harmonics with quantum numbers l, m, wherel≤ 2 and the z axis is vertical. In the case of m = 0, the dark and light regions are of oppositesign; when m 6= 0, the function is complex and its phase changes by 2mπ during a completecircuit of the z axis.

Coloured Versions of Some Figures 5

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2Updates and Corrections

This section contains an updated list of corrections to known misprints in the text. Themajority have been corrected in the 2010 reprint of the book, but those marked with ⊗were identified too late to include in the reprint.

If you detect any misprints or mistakes that are not listed here, please let us know bye-mailing [email protected].

Chapter 2

• Worked Example 2.1 In the solution on page 24, the magnitude of the proton mass iswrong. The correct expression for En is

En = π2× (1.05×10−34)2n2/

(8×1.7×10−27×4×10−30)= 2.0×10−12n2 J

• In the solution to worked Example 2.4 on page 34, a should be a2

Chapter 3

• On the second last line of page 51, p′ = p+2 should be p′ = p−2

• In the last paragraph of the solution to Worked Example 3.2 on page 56, right-handshould be left-hand

• The expression for r′ on the first line of page 57 should be r′ = (mω/~)1/2r

• The first term in square brackets in the final equation of Worked Example 3.3 shouldnot have a minus sign

6

Updates and Corrections 7

Chapter 4

• The final equation in Worked Example 4.2 on page 78 should read

35

exp[ik0x]+45

exp[2ik0x]

• In the solution to Worked Example 4.3 on page 81, the final expression for the inte-gral in part (iii) should be

a2(

13− 2

π2

)= 0.13a2

• Worked example 4.4: None of the minus signs should be shown and neither shouldthe ~2s in the first three lines, and some factors of ~2 and one of Px should not appear.The correct example is shown below.

Find an expression for the commutator bracket of the x coordinate of aparticle and its kinetic energy.Solution

[X , T ] =1

2m[X , P2]

=1

2m

[X P2

x − P2x X + X P2

y − P2y X + X P2

z − P2z X]

=1

2m[(PxX + i~)Px− Px(X Px− i~)Px]

= i~Px/m

where we have used (4.42).

• Worked Example 4.5 on page 89 should refer back to Worked Example 4.3. Whenthe corrections to the latter are included in the former, the correct expression is

∆x∆px = a(

13− 2

π2

)1/2

× π~2a

= 0.36π~2

= 0.57~>12~

• The sentence beginning on the line following (4.89) on page 99 should read

As expected, therefore, the probability distribution for the momentum ofan electron in the ground state of the hydrogen atom . . .


Chapter 5

• In the solution to Worked Example 5.2 on page 109, ~ in the final expression shouldbe ~2

• In the solution to Worked Example 5.4 on page 117, the final expression should read

Y1,±1 = L±Y1,0 =±~exp(±iφ)∂

∂θcosθ =∓~sinθexp(±iφ)

• Also on page 117, the bracketed term in the third expression in the last equationshould be

(L∗x + iL∗y).

Chapter 6

• Figure 6.1 The lower rightmost beam from the final SGZ should be βz not βx. Thecorrect figure is shown below.

SGZ

SGX

SGZaz

az

ax

bz

bx

bz

• Equation (6.22) on page 128 should read

c+ =α†xαz =

1√2[1 1][1

0

]=

1√2

c− =β†xαz =

1√2[1 −1

][10

]=

1√2

• In the solution to Worked Example 6.2 on page 130, sin(φ/2) should be sin(θ/2)

• Also on page 130, Dirac was born and educated in Bristol.

• On page 133, equation (6.44) should read

H ′ = f (r)(J2− L2− S2)+(µB/~)B0(Lz +2Sz)


⊗ In the caption to figure 6.2 on page 135, the word ‘squared’ should be inserted before’magnitudes’.

• In figure 6.3 on page 136, the right hand label on the lowest energy state should bej = 1/2 (not j = 0).

⊗ In table 6.2 on page 143, 2/√

3 should be√(2/3) in lines 2 and 3 of the table.

• On page 145, some factors of µB have been omitted from equation (6.73)

E = ε±2µBB

E =12

[−ε±µBB+

(9ε

2±2εµBB+µ2BB2)1/2

]E =

12

[−ε±µBB−

(9ε

2±2εµBB+µ2BB2)1/2

]• The final two paragraphs in the solution to Worked example 6.5 should read

If B is not zero, but µBB << ε, we can ignore the terms in B2 in (6.73)and expand the round brackets to get

E = ε±2µBB

E =12

[−ε±µBB+3ε

(1± µBB

9ε

)]= ε± 2

3µBB

E =12

[−ε±µBB−3ε

(1± µBB

9ε

)]=−2ε± 1

3µBB

which agrees with the weak-field values obtained earlier (6.53).In the strong-field limit, we can ignore the term in ε2 and expand the

square bracket assuming µBB >> ε to get

E =±2µBB+ ε

E =12

[−ε±µBB+µBB

(1± ε

µBB

)]= µBB or − ε

E =12

[−ε±µBB−µBB

(1± ε

µBB

)]=−µBB or − ε

These results are identical with our previous calculation.

Chapter 7

⊗ β is put equal to one after it has fulfilled its role in setting up equations (7.7) to (7.9).



|n〉= |0n〉+ ∑k 6=n

〈0k|H ′|0n〉E0n−E0k

|0k〉+higher order terms (7.18)

• In Worked Example 7.3 on pages 158 and 159, the expressions for the matrix ele-ments should read

H ′12,12 = a−4V0

∫ a

−a

∫ a

−acos2

(πx2a

)sin2

(πya

)xy dxdy = 0

H ′12,21 = H ′21,12 = a−4V0

∫ a

−a

∫ a

−acos(

πx2a

)sin(

πya

)sin(

πxa

)cos(

πy2a

)xy dxdy =

102481π4 V0

H ′21,21 = a−4V0

∫ a

−a

∫ a

−asin2

(πxa

)cos2

(πy2a

)xy dxdy = 0


E =~2

2me

[k2− 2πk

a+

2π2

a2

]± 1

2

[~2

2me

(4πk

a− 4π2

a2

)2

+V 20

]1/2

(7.57)

Chapter 8

• On page 177, equation (8.7) and the following line should be deleted because theyreproduce (8.6) and the last line on page 176.


∂a1

∂t=−1

2ωpa2

∂a2

∂t=

12

ωpa1

(8.11)

• In Worked Example 8.2 on pages 179 and 180, the expression in the last line of texton page 179 should read k′ = k− ia′x (not k′ = k+ ia′x).

• In the first line of the equation at the top of page 180 (2π)−1/2 should be (2π)−1

• The last few words on page 184 should read

If, however, we had used the second instead of the first term on the . . .


Chapter 9

• In Worked Example 9.3 on page 210, the equation on the second last line of the pageshould read

β = 2Ze2/4πε0


uklm =Ar[exp(−ikr)− exp(i(kr− lπ+2δlm))]Ylm(θ,φ) (9.52)

Chapter 11

• The second equation in (11.28) on page 251 should be

σ·(P+ eA)cu+−mc2u−+Vu− = Eu−

Chapter 12

• Table 12.1 on page 267 should read

Alice Bob Eve Bob(Eve inactive) (Eve active)

O S M O R a/r K O R/S R K′z α 0 z α a 0 x α β 1*z β 1 z β a 1 x β α 0*x α 0 z α r z β β

z β 1 x α r x α α

x α 0 x α a 0 x α α 0x β 1 z β r x β β

z β 1 x β r z β β

x β 1 z β r z β β

x α 0 x α a 0 x α α 0x α 0 z α r z α α

z α 0 z α a 0 x α β 1*z β 1 z β a 1 z β β 1


φ(1)ψ(2)→ψ(1)ψ(2)

φ(1)χ(2)→χ(1)χ(2) (12.7)


• In the second equation (12.15) on page 272, A(0) should be B(0)


2−1/2 [d(2)c(3)− c(2)d(3)] , if i = 1

and 2−1/2 [c(2)c(3)−d(2)d(3)] , otherwise. (12.18)

• Equation (12.20) There is a small misprint in equation (12.20) on page 276, whichshould read

χ = αx(1)αx(2)αx(3)

= 2−3/2(α1 +β1)(α2 +β2)(α3 +β3)

= 2−3/2[α1α2α3 +α1α2β3 +α1β2α3 +α1β2β3

+β1α2α3 +β1α2β3 +β1β2α3 +β1β2β3]. (12.20)

⊗ Equation(12.24) should read

yk =N−12

m−1

∑l=0

r−1

∑j=0

exp(

2πiN

k( j+ lr))

y j+lr

=N−12

m−1

∑l=0

exp(

2πiN

klr) r−1

∑j=0

exp(

2πiN

jk)

y j

=N−12

[1− exp(2πik))1− exp

(2πikm

) ] r−1

∑j=0

exp(

2πiN

jk)

y j (12.24)

where the expression in square brackets equals the sum over l in the line above be-cause the latter is a geometric series. Unless k = pm, where p is an integer ≤ r, thisterm equals zero, because the numerator is zero and the denominator is finite. How-ever, if k = pm both are zero and their ratio equals m.

• The first summation in the second line of equation (12.26) on page 279 should beover the index j (not k)

• In the first line of equation (12.31) on page 281, 2n/2 inside the brackets should be2n.

• The last few lines at the foot of page 279, continuing to page 280 should read

The probability of obtaining a particular value of k following such ameasurement is |〈k|ψ〉|2 = 〈yk|yk〉, using (12.26). Referring to (12.28) wesee that the term in square brackets is zero unless k = pN, where p is an


integer, in which case it equals m, so that

〈yk|yk〉=m2

N2

r−1

∑j=0

r−1

∑j′=0

exp(

2πiN

( j− j′)k)〈y j|y j′〉

=m2rN2

=1r

(12.29)

remembering that N = mr and that the |y j〉 are orthonormal.

Chapter 13

• On page 289 V (X) should be V (x) in equations (13.4) and (13.6) and V (r) in equation(13.9).

• On page 297, P− in (13.7) and the preceding text should read P−−. In the first termof the RHS of the last line of equation (13.18), 3− should be 3+.

• Equation (13.17) In the last line P− should be P−−

P++(θ) = sin2(θ/2)P+−(θ) = cos2(θ/2)P−+(θ) = cos2(θ/2)P−−(θ) = sin2(θ/2)

(13.17)

• Equation (13:20) There is a small misprint in equation (13.20) on page 298, whichshould read

n(1+,2+)−n(1+,3+)+n(2−,3+)≥ 0. (13.20)

• Equation (13.21) The P++ should be replaced with P−−, i.e.,

P+−(θ12)−P+−(θ13)+P−−(θ23)≥ 0. (13.21)

• On page 298, n(2+,3−) should read n(2−,3+) in both equations (13.19) and (13.20).In equation (13.21) P++(θ23) should read P−−(θ23)

• On pages 310 onwards, references to equation (13.36) should be to equation (13.35).

• On page 317, the reference to figure 13.6 should be to figure 13.7.

3Bibliography

The early editions of Quantum Mechanics contained a Bibliography. Over the years, notonly have many of the references in it become out of date, but the amount of relevantpublished material has exploded to the point where there is a great danger that importantworks will be left out. Moreover, the availability of web searches and other data retrievingtools means that readers can much more easily build up their own databases of relevantmaterial. For these reasons, a bibliography has not been included in recent editions.

However, the original bibliography is published here along with some more recent titlesthat are labelled ‘addition. If you think a useful reference has been left out, please let usknow by e-mailing [email protected].

Background

A more extensive treatment of the experimental evidence for the need for quantum mechan-ics than is given in Chapter 1 is contained in many textbooks on atomic physics, of whichthe following are good examples.

• Eisberg, R. M., and Resnick, R., Quantum Physics of Atoms, Molecules, Solids, Nu-clei and Particles, Wiley, New York, 1974.

• Enge, H. A., Wehr, M. R., and Richards, J. A., Introduction to Atomic Physics,Addison-Wesley, Reading, Massachusetts, 1972.

• Richtmeyer, F. K., Kennard, E. H., and Cooper, J. H., Introduction to Modern Physics(6th edn.), McGraw-Hill, New York, 1969.

A general familiarity with mathematical techniques, particularly calculus and elementarymatrix algebra, up to a level typical of that of a first-year undergraduate physics course, isassumed; suitable textbooks covering this material are

• Arfken, G., Mathematical Methods for Physicists, Academic, New York, 1985.

• Boas, M. L., Mathematical Methods in the Physical Sciences (2nd ed.), Wiley, NewYork, 1984.

14

Bibliography 15

Complementary

Some of the textbooks on quantum mechanics which treat the subject at approximately thesame level as the present volume are listed below with short comments.

• Bohm, D., Quantum Theory, Prentice-Hall, New York, 1951. This thorough, dis-cursive test is one of the few at this level to attempt a detailed discussion of theconceptual problems of the subject. It is of course rather out of date by now.

• Dicke, R. H., and Wittke, J. R., Introduction to Quantum Mechanics, Addison-Wesley,Reading, Massachusetts, 1960. In many ways the approach of this book is rather sim-ilar to that adopted in the present work, but it relies on a more detailed understandingof formal classical mechanics and uses rather complex mathematical arguments attimes.

• Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectures in Physics(vol. III, Quantum Mechanics), Addison-Wesley, Reading, Massachusetts, 1965. Thisbook contains many physical insights and is written in an attractive informal style.However, from the start it uses an abstract, matrix formulation which some studentsfind difficult.

• French, A. P., and Taylor, E. F., An Introduction to Quantum Physics, Nelson, Mid-dlesex, 1978. This book describes the physical principles behind many quantum pro-cesses in considerable detail, and this discussion is illustrated by a large number ofexamples. However, detailed mathematical arguments are avoided, which somewhatlimits its scope.

• Mathews, P. T., Introduction to Quantum Mechanics (3rd ed.), McGraw-Hill, Lon-don, 1974. This book contains a rather condensed and mathematically based treat-ment of the basic ideas of quantum mechanics and their application to a number ofphysical problems.

• Sillito, R. M., Non-Relativistic Quantum Mechanics (2nd ed.), University Press, Ed-inburgh, 1967. A good reliable text, if a little over detailed and mathematical inplaces, by the person who taught the subject to the present author.

Advanced

The following is a small selection of the many books on quantum mechanics which treatthe subject at a more advanced level.

• Dirac, P. A. M., Quantum Mechanics (4th ed.), Oxford University Press, London,1967. This classic text describes the formal, rigorous version of quantum mechanicsthat was first developed by its author.

Bibliography 16

• Gillespie, D. T., A Quantum-Mechanics Primer, International Textbook Company,1970. This short book contains a clear, simplified description of formal theory and isan excellent introduction to the book by Dirac (see above).

• Landau, L., and Lifschitz, E. M., Quantum Mechanics, Non-Relativistic Theory (2nded.) (trans. J. B. Sykes and J. S. Bell), Addison-Wesley, Reading, Massachusetts,1965. A long book which discusses a number of advanced topics in some detail.

• Ziman, J. M., Elements of Advanced Quantum Theory, University Press, Cambridge,1969. A good introduction to more advanced topics, particularly quantum field the-ory and the many-body problem.

Addition

• Brandsen, B. H., Quantum Mechanics (2nd ed.) Pearson Education Ltd., Harlow,2000. A long book, with a similar coverage to the present volume, but with muchmore mathematical detail. A useful reference book.

The conceptual problems of quantum mechanics

The following constitute a selection of some publications on the topics discussed in Chap-ter 11. They all contain references to further published work.

• Bellinfante, F. J., Measurement and Time Reversal in Objective Quantum Theory,Pergamon, Oxford, 1978. This book argues forcibly for quantum mechanics beingan objective theory and for the importance of an ‘indelible record’ as part of anymeasurement.

• Clauser, J. F., and Shimony, A., ‘Bell’s theorem: experimental tests and implica-tions’, Reports on Progress in Physics, vol. 41, pp. 1881–1927, 1978. This reviewarticle contains a thorough discussion of Bell’s theorem and carefully analyses thevarious experiments that have been performed to test it.

• D’Espagnat, B., Conceptual Foundations of Quantum Mechanics, Benjamin, Mas-sachusetts, 1976. This is probably the nearest there is to an authoritative refer-ence work on this subject. It reviews and analyses the whole field, including non-separability, measurement theories, and the associated philosophical problems.

• Hughes, R. I. G., The Structure and Interpretation of Quantum Mechanics, HarvardUniversity Press, 1989. This book contains a thorough discussion of the whole fieldat a moderately advanced level.

• Prigogine, I., From Being to Becoming, Freeman, San Francisco, 1980. This bookdevelops Prigogine’s ideas referred to in Chapter 11.

Bibliography 17

• Rae, A. I. M., Quantum Physics, Illusion or Reality, University Press, Cambridge,1986. This book discusses the conceptual problems of quantum mechanics at greaterlength, although at a somewhat less advanced level than is done in Chapter 11.

Additions

• Auletta, G., Foundations and Interpretations of Quantum Mechanics, World Scien-tific, Singapore, 2001. A wideranging text that discusses many topics in considerabledepth, but (despite its 981 pages) only touches on others.

• Baggott, J., Beyond Measure, Oxford University Press, Oxford, 2004. A treatment ofquantum foundations that covers much of the material in Chapter 13, but with moredetail and rigour.

• Griffiths, R. B. Consistent Quantum Theory, Cambridge University Press, Cam-bridge, 2002. A text by one of the inventors of the consistent histories interpretation,which aims to teach the whole subject from this point of view. Contains illuminatinginsights which are sometimes masked by mathematical detail.

Applications

The applications of quantum mechanics discussed in this book are mainly, if not entirely,drawn from the fields of atomic, nuclear, particle, and solid-state physics. More detaileddiscussion of these subjects can be found in a number of textbooks of which the followingare typical examples.

• Corney, A., Atomic and Laser Spectroscopy, Oxford University Press, London, 1977.

• Kuhn, H., Atomic Spectra (2nd ed.), Longman, Harlow. 1970.

• Woodgate, G. K., Elementary Atomic Structure (2nd ed.), Oxford University Press,London, 1980.

• Burcham, W. E., Nuclear Physics: An Introduction (2nd ed.), Longman, Harlow,1973.

• Cohen, B. L., Concepts of Nuclear Physics, McGraw-Hill, New York, 1971.

• Perkins, D. H., Introduction to High-Energy Physics, Addison-Wesley, Reading,Massachusetts, 1972.

• Hook, J. R., and Hall, H. E., Solid State Physics, Wiley, 1991.

• Kittel, C., Introduction to Solid State Physics (5th ed.), Wiley, New York, 1976.

December 2, 2011

Quantum Mechanics 5th Edition Alastair I M Rae

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