Download Quantum Physics. Approuch Modern Physics, JS Townsend, (417p), 2010 PDF

TitleQuantum Physics. Approuch Modern Physics, JS Townsend, (417p), 2010
File Size86.5 MB
Total Pages418
Table of Contents
                            Constants and Conversion Factors
Contents in Brief
Contents
Preface
Chapter 1: Light
Chapter 2: Wave Mechanics
Chapter 3: The Time-Independent Schrodinger Equation
Chapter 4: One-Dimensional Potentials
Chapter 5: Principles of Quantum Mechanics
Chapter 6: Quantum Mechanics in Three Dimensions
Chapter 7: Identical Particles
Chapter 8: Solid-State Physics
Chapter 9: Nuclear Physics
Chapter 10: Particle Physics
Appendix A: Special Relativity
Appendix B: Power-Series Solutions
Answers to Odd-Numbered Problems
Index
                        
Document Text Contents
Page 2

Constants and Conversion Factors

Planck 's constant " = 6.626 X 10- 34 J· s = 4.136 x 10- " eV·s

hbar Ii = " / 2IT = 1.055 X 10- 34 J·s = 6.582 x 10- 16 eV·s

Speed of light e = 2.998 x 108 m/ s

Elementary charge e = 1.602 x 10- 19 C

Fine-structure constant 0: = e2 / 4rrEolle = 7.297 x 10- 3 = 1/ 137.036

Boltzmann constant kg = 1.381 x 10- 23 J/ K = 8.617 x 10- 5 cV/ K

Avogadro constant N .. , = 6.022 x lOll particles/mole

Electron mass III , = 9.109 X 10- 31 kg =0.5 110MeV/ c2

Proton mass 111 1' = 1.673 x 10- 27 kg = 938.3 MeV/ c2

Neutron mass 111 " = 1.675 X 10- 27 kg = 939.6 MeV / c
2

Bohr radius ao = 4rr EOIi
2

/ III ,C
2 = 0.5292 x 10-

10
III

Rydberg energy heRoo = lII ,c
2

0:
2/2 = 13.6 1 eV

Bohr magneton /Lil = eli / 2111 , = 5.788 x 10-
5

eV / T

I keV = 103 eV I MeV = 106 eV I GeV = 109 eV I TeV = 10
12

eV

I IL I11 = 10- 6 m I 11 111 = 10- 9 III I pm = 10- 12 111 I fm = 10-
15

m

leV = 1.602 x 10-
19

.1 I A = 0.1 n111 = 10- 10 111

Page 209

196 Chapter 6: Quantum Mechanics in Three Dimensions

EXAMPLE 6.6 An electron wi th the azimuthal wave funct ion

I
,p(</» = ft cos </>

is placed in an external magnetic field. Take the Hamiltonian to be

e B
H = - L - = woL-

2111 ·"P . op

where Wo = eB / 2m. Determine ,p(</> , f).

SOLUTION In Exampl e 6.1 we saw that

I I (ei¢ + e-i¢)
,p(</» = ft cos</> = ft 2

I I

= Ji""+ J/'-'
namely the wave function is a superposition of eigenfunctions of L ' ''I' with eigenval-
ues ± Ii. Since the eigenfunctions of L", are also eigenfunctions ofthe Hamiltonian
(H and L", differ by only the mult iplicative constant wo), the second line of this
equation expresses the wave function as a superposition of two energy eigenfunc-
tions with the distinct eigenvalues ± hwo. Tacking on the time-dependent factor
e-iEt / fi for each energy state, we find

We see that the argument of the cosine is rotating about thez axis with angul ar speed
woo This is in accord with the behavior that we would expect in classical physics
for a magnetic moment in 3n external magnetic fie ld in the z direction, namely, the
magnetic moment would precess about the z axis with angular frequency Wo o

6_5 Intrinsic Spin

Atomic spectra are more complex than our discussion so far suggests. For hydrogen, for
example, what should be a si ngle wavelength (a sing le line in the spectrum) corresponding

to the II = 2 to 11 = I transition in the absence of an external magnetic fie ld turns out to
be a doublet: two closely spaced lines that can be resolved with a good spectrograph. 7

And when an external magnetic field is applied, the response is more complex than was

7 A similar doublet with a larger spacing between Ihe lines is formed by the famous sodium D·lines.

Page 210

Section 6.5 Intrinsic Spin 197

suggested in the prev ious sect ion, leadi ng to what was historically referred to as the
anomalous Zeeman effect.8 In 1925, in an effort to explain atom ic spectra, two Dutch
graduate students, Samuel Goudsmit (an experimentalist who had worked in Zeeman's
laboratory) and George Uhlenbeck (a theorist), suggested that the electron had its own

intri nsic spin angular momentum S with a corresponding magnetic moment

It =g(~~)S (6.89)
Thus ifan atom such as hydrogen is placed in an external magnetic fie ld B, the interaction

energy (6.85) of the electron with thi s magnetic fi eld becomes

( e e) -It · B = - L + g-S ·B 2m 2m (6.90)
where we have included the contribulions of the orbital and the spi n magnetic moments.
The factor of g in (6.89) is known, pcrhaps not too imaginatively, as the g fac to r. In
non relativistic quantum mechani cs it is a "fudge" factor (another techn ical term) that
must be inserted in order to give good agreement with experiment. Relat ivistic quantum
mechanics (via the Dirac equation) predicts g = 2 for the electron, exactly9

What are we to make of this intri nsic spin? You are probably thin king that the electron

is a ball of charge spinning about its ax is very much as the earth spi ns about its ax is
as it revolves around the sun. That is basica lly what Goudsmit and Uhlenbeck thought,

at least initially.lo But this model o f the electron's spin canllot be correct. After all ,
we can calculate spin angular momentum of a rotating ball by integrat ing the orbital

angular momentum of each sl11all piece of mass dill as it moves about its rotation ax is
(iw = Jr x dill v). Thus this sort of spi n angular momentum is j ust orbital angular
momentum, too; we simply g ive it a different name. But we have seen that the L=
eigenvalues of orbital angular momentum are restricted to be III th with lil t an integer (so

that the wave function ",(r, e, </J) is si ngle valued). Si nce the obscrved values of S, fo r
an electron are ± h/ 2 (or lII , h with III ., = ± 1/ 2, clearly not an in teger), we cannot think
of the electron 's spi n as arising in some way from the motion of the partic le (S # r x p).
We must give up on the notion of a wave function that te lls us, for example, something

about the orientation of the electron. In fac t, as far as we can tell , the electron itselfi s a
point particl e. This sp in angula r momentum is an intrinsic attribute of the particle, like

its charge. Any attempt to search for a deeper physical model that generates thi s spin is
probably not appropriate. As an anti dote to such attempts, you shoul d note that particles

Sit turns oUllhat there were classical arguments thaI seemed to "explain" the normal Zeeman effec t.
but tbe origin orthe anomalous Zeeman effect was a mystery before the advent of quantum mechanics
and the discovery of til e role of intrins ic spin.

9Thc observed value is 2.00232. This apparent discrepancy between the pred iction of the Dirac
equation and the observed value is beautifu lly reconci led through quantum clcclrodyn<ll1l ics. as we will
disclIss in Section 10.1.

IO When Goudsmit and Uhlenbcck proposed the idea of electron spin to P. Ehrcnfest, he encouraged
them to write up thei r results and to talk with H. Lorentz. After SOme analysis, Lorentz poinled out
Ihal a class ica l model of a sp inning electron required that the speed at the surface be approximately
ten times the speed of light in order to obtain the observed magnetic momen t. When Goudsm it and
Uhlenbeck went to tet! Ehrenfest of their foo lishness. he informed them that he had already submitted
their paper for publication. He assured them they shouldn't worry since they were ··both young enough
to be able to afford a stupidity." Physics Today. June 1976, p. 40.

Page 417



Uhlenbeek, G., 197

Ulnm, S., 308
Ult raviolet catastrophe. 254

Uncertainty, 78
Uncertainty relations, 163- 164

energY- lime, 167- 169,330- 332

Heisenberg, 163

orbital angular momentum. 186
Up quark, 327, 333

Uranium hexafluoride, 307

van cler Mecr, S .. 342
vnn der Waals force, 340

Vector potential, 358
Velocity lra nsfonnation

nonrclativisitc. 367

relativistic, 380- 38 1
Virtual panic le. 320
Volume term, 283

von Laue. M .. 59

von Neumann. J , 308

II' boson, 340-342

Wavc cquat ion
light

in a mcdium. 3
in vacuum, 3

Schrodinger, 62
Wi.I VC function, 62

physical s ignificancc, 64-65

Wave number, 2

Wave packet. 68

and sca ttering, \38- 139

Wavelength , 2

Waves, 1- 3

Weak interaction, 296, 305, 333,
340- 342

l1 onconscrvation of parity, 354-355
nonconsc rvation of st rangeness, 333

powering the sun, 346

Weinberg angle, 341
Whee ler, J.. 27, 306
White dwarf star, 225

Si rius B. 225
Wieman, C., 245

Wi en 's law. 239
Wigner, E., 359
Wilson. R., 240
Work function. \1

wu. C-S .. 354

Yang. C. N. , 354
Young, T. , 32, 43
Yukawa, 1-1 ., 330

Z boson, 340- 342

Zeeman effect, 194-195
anomalous, 197

Zecman, P., 195
Zero-point energy

harmonic osc ill ator, 128- 129

panicle in a box, 94

Index 411

Page 418

Constants and Conversion Factors
r

Planck 's constant II = 6.626 X 10- 34 J ·S = 4.136 x 10- 15 eV·s

hbar fl = 11 127T = 1.055 X 10- 34 J.s = 6.582 x 10- 16 eV ·s

Speed of light c = 2.998 X 108 m( s

Elementary charge e = 1.602 x 10- 19 C

Fine-struclureconstant ,,=e'/47TEohc = 7.297 x 10-3 = 1( 137.036

Boltzmann constant kB = 1. 381 x 10- 13 J ( K = 8.6 17 x 10- 5 eV ( K

Avogadro constant NA = 6.022 x 1023 panicles/mole

El ectron mass III , = 9. 109 X 10-31 kg = 0.51 10 MeV/ c'

Proton mass fJ/ {J = 1. 673 x 10 - 27 kg = 938.3 MeV / c2

Neutron mass 111 1/ = 1.675 X 10- 27 kg = 939 .6 MeV / c2

Bohr radius (10 = 4rrEoli2 j fJ/ e C2 = 0.52 92 x 10- 10 J11

Rydbergencrgy II cRoo = lII <c',,'(2 = 13.61 eV

Bohr magneton JLB = eh 12111 e = 5.788 X 10- 5 e V IT

I keY = 103 eV I MeV = 10(, eV I GeV = 109 eV I TeV = 10 " eV

I JLm = 10- 6 m I nm = 10- 9 m I pm = 10- " m I fm = 10- 15 III

I eV = 1.602 x 10- ' 9 J I A = 0. 1 nm = 10- 10 III

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