Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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456 10 Group Theory
35. Commutation relations for vectors
a) Verify the commutation relation
L
i
,V
j
i, j,k
V
k
(10.568)
where
L is the orbital angular momentum and
V is an arbitrary vector operator. Thus,
show that
L ,
V
V ,
L for any vector operator
V.
b) Verify
L,
A ,
BL
2
,
A ,
B0 for arbitrary vector operators
A and
B. Do not assume
that
A and
B commute.
c) Use commutation relations to verify that
A
B transforms as a vector under rotations.
Do not assume that
A and
B commute.
d) Next, show that
V
L
L
V 2
V (10.569)
for any
V. For example, the identity
ˆr
1
2
ˆr
L
L ˆr
(10.570)
is often useful.
e) Show that
L
2
,
V
V
L
L
V
2
V
V
L
(10.571)
for any
V. Thus, L
2
,
L0.
36. Commutation relations involving
r,
p, and
L
Derive each of the commutation relations listed below. Here
p
+,
L
r
p,ˆr
r/r.
a)
p
2
, ˆr
2
r
2
ˆr ˆr
L
r
2
ˆr
L
L ˆr
b)
p
L,
1
r
r
2
ˆr
L
L
p,
1
r
r
2
ˆ
L
r
p
L
L
p,
1
r
r
2
ˆr
L
ˆ
L
r
c)
p
L ,
p
L p
2
L
2
p
L ,
L
p p
2
L
2
4
L
p ,
p
L p
2
L
2
L
p ,
L
p p
2
L
2
37. Runge–Lenz vector for hydrogenic atoms: An example of dynamical symmetry
In this problem we use group-theoretical methods to analyze the spectrum of hydrogenic
atoms without ever solving a differential equation. The method relies upon a generaliza-
tion of the Runge–Lenz vector which, for the Kepler problem, points in the direction of
the major axis of an elliptical orbit. The conservation of the classical Runge–Lenz vector
Problems for Chapter 10 457
demonstrates that there is a dynamical symmetry for the inverse-square law that leads to
closed elliptical orbits; small violations of this symmetry in general relativity are respon-
sible for the famous precession of the perihelion of Mercury. One can show that the n
2
degeneracy of energy level E
n
in the Bohr model of hydrogenic atoms arises from a similar
dynamical symmetry described by conservation of a quantum mechanical generalization
of the Runge–Lenz vector. Here we guide the student through an analysis that is relatively
straightforward, even if some of the algebra is tedious.
a) Show that the symmetrized Runge–Lenz vector
0Α
p
L
L
p
ˆr (10.572)
is hermitian and that it commutes with the hamiltonian
H
p
2
2Μ
Κ
r
(10.573)
for a suitable choice of Α (real). Thus, there is a dynamical symmetry and the full sym-
metry group is based upon the six generators
L and
0.
b) Show that
0
2
1
2H
ΜΚ
2
L
2
1
(10.574)
c) Show that the symmetry operators satisfy the following commutation relations:
L
i
,L
j
i, j,k
L
k
(10.575)
L
i
, 0
j
i, j,k
0
k
(10.576)
0
i
, 0
j
2H
ΜΚ
2
i, j,k
L
k
(10.577)
d) The presence of H in the preceding commutation relations shows that
0,
L do not
form a Lie group with respect to the full Hilbert space. However, if we define new
Runge–Lenz operators according to
0
2H
Μk
2
Λ (10.578)
where H ! E within any subspace with common energy E<0, the commutators for
Λ,
L do form a closed algebraic system. Demonstrate that the new system is closed.
Then evaluate the commutation relations for the linear combinations
J
1
2
L
Λ
(10.579)
K
1
2
L
Λ
(10.580)
and prove that their structure constants satisfy the conditions required for a Lie group.
Construct the Casimir operators, identify a complete set of quantum numbers, and spec-
ify their allowed values. Note that for this system one of the Casimir operators is actu-
ally redundant.
458 10 Group Theory
e) Use J
2
to compute the energy eigenvalues and determine the degeneracy of multiplets.
38. Landé formula and deuteron magnetic moment
In this problem you will derive a formula that is often useful in nuclear or atomic spec-
troscopy and then will apply that formula to analyze the magnetic moment of the deuteron.
a) Suppose that
V is a vector operator. Show that
j )V )j
j, m
J ,
V j, m
j j 1
(10.581)
for j>0.
b) The magnetic dipole operator for the deuterium nucleus takes the form
Μg
p
S
p
g
n
S
n
1
2
L (10.582)
where
S
p
and
S
n
are proton and neutron spins and where the factor of
1
2
for the orbital
angular momentum
L arises because only the proton carries charge and it carries half
the total orbital angular momentum. The ground-state wave function can be expressed
as
Ψ
d
a
3
S
1
b
3
D
1
(10.583)
where a
2
b
2
1 and where the standard spectroscopic notation specifies
2S1
L
J
.
Evaluate the magnetic moment, defined by the matrix element
Μ
J,M Μ
0
J,M
(10.584)
for the aligned substate with maximum magnetic quantum number. Given that g
p
5.586Μ
N
, g
n
3.826Μ
N
, and Μ
d
0.857Μ
N
where Μ
N
is the nuclear magneton, esti-
mate the D-state probability, b
2
. Note that this model omits the contributions due to
meson exchange between nucleons, but still gives a good approximation to the D-state
admixture in the ground state.
11 Bibliography
Although these sources are not available to the reader, I feel obliged to acknowledge that
my most important references were the notes and homework that I saved from similar
courses, AMa95 by Prof. Saffman and PH129 by Prof. Peck, at CalTech in the mid-1970s.
Below I have compiled a brief bibliography that might be more useful to the reader, with
some personal comments. No attempt has been made to quote original sources because
I did not use them and they probably would be less useful to most readers than these
secondary texts, anyway.
General
1. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th edition, (Else-
vier, Amsterdam, 2005)
An encyclopedic work that covers a broader range of topics but often at a somewhat
lower level than the current text.
2. E. Butkov, Mathematical Physics, (Addison-Wesley, Reading MA, 1968)
Particularly good treatment of generalized functions and the theory of distributions.
3. F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics, (Dover,
N.Y., 1969)
4. R. V. Churchill, J. W. Brown, and R. F. Verhey, Complex Variables and Applications,
3rd edition, (McGraw-Hill, NY, 1974)
Clear and concise development of the theory of analytic functions.
5. A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua,
(McGraw-Hill, NY, 1980)
The sections on the general string equation, Sturm–Liouville problems, and solitons are
the most relevant to the present text.
6. J. D. Jackson, Classical Electrodynamics, 2nd edition (Wiley, NY, 1975)
Extensive treatments of boundary-value problems and dispersion theory.
7. S. M. Lea, Mathematics for Physicists, (Thomson Brooks-Cole, Belmont CA, 2004)
8. J. Mathews and R. L. Walker, Mathematical Methods of Physics, (Benjamin, Menlo
Park, 1964)
Insightful general text at a slightly higher level.
Graduate Mathematical Physics. James J. Kelly
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40637-9
460 11 Bibliography
9. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, (McGraw-Hill, NY,
1953)
A massive two-volume text at a rather advanced level. This is an invaluable reference
but is not suitable as a textbook, at least for recent generations of students.
Group Theory
1. M. Hamermesh, Group Theory and Its Application to Physical Problems, (Addison-
Wesley, Reading MA, 1962)
2. W. Ludwig and C. Falter, Symmetries in Physics, (Springer-Verlag, Berlin, 1988)
3. L. I. Schiff, Quantum Mechanics, 3rd edition, (McGraw-Hill, NY, 1968)
4. M. Tinkham, Group Theory and Quantum Mechanics, (McGraw-Hill, NY, 1964)
5. E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic
Spectra, (Academic Press, NY, 1959)
Numerical Methods
1. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes,
(Cambridge University Press, Cambridge, 1986)
Reference Books
1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (National
Bureau of Standards, Washington DC, 1970)
2. I. S. Gradshteyn and I. M. Ryzkhik, Tables of Integrals, Series, and Products, 4th edi-
tion, (Academic Press, NY, 1965)
Index
a
abelian 371
addition theorem 282
aliasing 153
analytic 24
analytic continuation 84, 191
– Euler transformation 217
– gamma function 195
analytic function
– derivatives 37
– extrema 27
– isolation of zeros 45
angular momentum 418, 425
–CGcoefficient 442
– coupling 438
– generators, commutation relations 428
– orbital 415, 418
– reduced matrix element 445
– spin 417
Argand diagram 1
argument function 6
argument principle 63
associated Legendre functions 275
– summary 312
asymptotic series 95
– expansion of integrand 104
– gamma function 99
– Hankel functions 297
– partial integration 101
– Richardson extrapolation 103
– stationary phase 109
– steepest descent 96
b
Baker–Haussdorff identity 420
bandwidth 153
Bessel expansion 294
Bessel functions 289
– cylindrical 289
– modified 301
– spherical 303
Bessel’s equation 293
Bessel’s integral 294
beta function 81, 319
Born approximation 359
boundary conditions, classification 228,
332
branch cut 6
– Legendre functions 287
– principal branch 15
– elementary functions 20
branch point 11
Bromwich integral 165
c
Cartan’s theorem 422
Casimir operator 423
Cauchy convergence principle 38
Cauchy inequality 44
Cauchy integral formula 34
Cauchy sequence 23
Cauchy–Goursat theorem 28
Cauchy–Riemann equations 24
causality 119, 196
Cayley–Klein parametrization 432
character 388
class 373
Clebsch–Gordan coefficient, SU(2) 442
Clebsch–Gordan expansion 401
– SU(2) 438
commutation relations 412
– angular momentum 419
– Pauli 416
– SO(3) 418
– spherical tensors 443
– SU(2) 417
commutator 377
completeness 236
– Legendre polynomials 274
– spherical Bessel functions 308
– spherical harmonics 280
complex numbers 1
Graduate Mathematical Physics. James J. Kelly
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40637-9
460 Index
Condon–Shortley 276, 430
conformal transformation 28
conjugate elements 373
conjugate variables 125
continuity 23
continuous group 406
contour integration 31
– custom contour 72
– deformation of contour 32
– great semicircle 69
– unit circle 67
contour wall 32
convergence 38
– absolute 38
– asymptotic 95
– conditional 38
– in the mean 235, 261
– radius 39
–tests 38
– uniform 41
convergence factor 66
convolution 119
– charge densities 181
– discrete Fourier transform 153
– electromagnetic response 199
– Fourier transform 132
– Laplace transform 171
correlation function 133
– autocorrelation 134
– cross correlation 133
– discrete Fourier transform 156
coset 374
cyclic group 373
cylindrical Bessel functions
– summary 314
cylindrical coordinates 338
d
de Moivre’s theorem 5, 56
degenerate perturbation theory 255
differentiability 23
diffusion equation 140, 176
digamma function 299
dimensionality theorem 393
Dirac delta function 112
– derivatives 116
– multidimensional 120
– nascent delta function 112
direct product 376, 400
Dirichlet boundary conditions 228
discrete Fourier transform 148
– convolution 153
– FFT 151
– periodicity 151
– power spectrum 160
– sampling 149
– windowing 187
– working array 152
dispersion 197
– anomalous 201
– wave packet 209
dispersion integrals 205
dispersion relations 196
– Kramers–Kronig 203
– oscillator model 200
– subtracted 218
– sum rules 206
distributions 113
divisor theorem 374
domain 22
– simply connected 22
e
eigenfunction expansion 236, 250
eigenfunctions
– completeness 235
– reality 238
eigenvalues 225
– degenerate 235
– discreteness 235
– reality 233
– shifting theorem 264
eigenvectors 225
entire function 24
equivalence class 373
error function
– asymptotic series 102
Euler angles 430
Euler transformation 217
Euler’s formula 4
exponentially bounded 166
f
factor group 375
far field 352
FFT 151
Index 461
– convolution 186
flavor symmetry 446
form factor 309, 352
Fourier series 126
– cosine 128
– sine 128
Fourier transform 128
– convolution theorem 132
– cosine 147
– derivatives 138
– discrete 148
– fast (FFT) 151
– Green functions 139
– orthogonality relations 128
– Parseval’s theorem 131
– sine 147
– summary table 139
Fourier-Bessel transform 306
Fredholm alternative theorem 264
Frobenius method 284
fundamental theorem of algebra 44
g
gamma function 79
– analytic continuation 195
– asymptotic series 99
– integral representation 83
generating function 267
– associated Legendre functions 319
– Bessel 289
– Legendre polynomials 268
– recursion relations 269
generator 410
– infinitesimal 411
– SO(3) 418
– SU(2) 416
– translation 410
Gibb’s phenomenon 235
Gram–Schmidt orthogonalization 230
great semicircle 69
Green functions 118
– damped oscillator 143, 175, 200
–diffusion in one dimension 139
–diffusion in three dimensions 141
–diffusion, constant boundary value 176
– eigenfunction expansion 246
– electrostatic 333
– grounded box 329
– Helmholtz equation 338
– interface matching 242
– Laplace transform 173
– Poisson equation 268, 283, 337
– RC circuit 174
– reciprocity 244
– retarded 357
– vibrating string 245, 252
Green’s electrostatic theorem 333
Green’s identities 332
Green’s theorem 29
group 370
group multiplication table 371
group velocity 209
h
Hankel functions 295
harmonic function 26
Heaviside step function 115
Helmholtz equation 327
Hermitian operator 227
Hilbert space 226
Hilbert transform 207
homomorphism 376
– SU(2) and SO(3) 431
hypergeometric equation 285
i
image charge 343
impulse 111
integral representation 82
– z 84, 195
– cylindrical Bessel functions 294, 296
– Hankel functions 295
– Legendre functions 288, 319
– Legendre polynomials 273
integral transform 125
interleaving of zeros 238, 321
intermediate boundary conditions 261
intrinsic spin 417
invariant function 420
invariant subgroup 374
irreducibility test 390
irreducible representation 379
– number 389
– SU(2) 433
irrep 379
isomorphism 371
462 Index
isospin 445
j
Jacobi identity 413
Jordan’s lemma 71
k
kernel 125
Killing form 422
Kramers–Kronig 205
l
ladder operator 428
ladder relations 271
Lagrange’s theorem 374
Laguerre polynomials 260
Laplace transform 165
– convolution theorem 171
– derivatives 170
– partial fractions 169
– summary table 173
Laurent series 46
Legendre expansion 273
– powers, xn 318
Legendre functions 287
– second kind 285
– summary 311
Legendre polynomials 231, 268
Legendre’s equation 271
Leibnitz’s formula 277, 317
Levi–Civita symbol 416
Lie algebra 422
Lie group 406, 413
– density functions 426
– structure constants 412
Lie series 412
limit 22
limit cycle 163
Liouville’s theorem 44
Lippmann–Schwinger equation 358
logistic map 159
Lorentz group 451
m
Maclaurin series 42
mapping 13
maximum modulus principle 59
mean-square error 261
Mellin inversion formula 165
meromorphic function 51
– pole expansion 51
– product expansion 54
metric 4
metric function 228
Mittag–Leffler theorem 53
mixed boundary conditions 232
modified Bessel functions 301
Morera’s theorem 37, 195
multipole expansion 280, 283
n
neighborhood 21
Neumann boundary conditions 228
Neumann functions 297
nonresonant background 251
normal derivative 333
normal modes 225
Nyquist frequency 153
o
orthogonality 227
orthogonality theorem 388
– characters 388, 390
– continuous groups 427
– finite representations 388
– rotation matrices 438
orthonormal 225
outgoing boundary conditions 350
p
parity 269, 279
Parseval’s theorem 131, 237
partial-wave expansion 352
Pauli matrices 416
Pauli parametrization 431
periodic boundary conditions 228
permanence of algebraic form 195
perturbation 253
perturbation theory 254
Picard’s theorem 49
Picone’s modification 241
plane wave 307
Poisson integral formula 60
pole 49
pole expansion 51
power spectrum 160
Index 463
principal branch 7, 20
principal value 75, 204
product expansion 54
q
quaternion 448
r
Rayleigh quotient 257
Rayleigh’s expansion 307
Rayleigh–Ritz 259
rearrangement theorem 373
reciprocity 244
recursion relations 269
reduced rotation matrix 433, 436
reflection principle 194
representation 376
– direct product 400
– faithful 378
– reducibility 379
– regular 391, 393
– unitarity 383
residue 50
residue theorem 67
resonance 251
retarded Green function 356
Riemann sheet 15
Rodrigues’ formula 273
– associated Legendre polynomials 276
rotation
– Euler parametrization 430
– infinitesimal 409, 415
– operator 409, 419
rotation matrix 433
– angular momentum 429
– irreducible representation 431
– orthogonality 438
– reduced 431
– spin 417
– vectors 417
Rouchés theorem 63
Runge–Lenz vector 456
s
saddle point 97
sampling 149
scalar product 280
scattering 350
Schläfli integral 109, 273
Schur’s lemma 385
Schwartz inequality 229
self-adjoint 227
separable coordinate systems 335
separation of variables 328
singularity 49
– essential 49
– pole 49
–removable 49
soliton 213
– KdV equation 221
– Sine–Gordon equation 222
special functions
– Bessel 289
– beta function 80
– complex sine integral 82
– error functions 80, 82
– exponential integrals 80
–Fresnel 80
– gamma function 79
– Hankel functions 295
– hypergeometric 286
– Laguerre polynomials 260
– Legendre polynomials 231
– modified Bessel functions 301
– Neumann functions 297
– spherical Bessel functions 303
– spherical harmonics 279
– zeta function 82
spherical Bessel functions 303
– Rayleigh’s formulas 305
– summary 315
spherical coordinates 336
spherical harmonics 279
– addition theorem 282
– summary 313
spherical tensor 280, 442
spontaneous symmetry breaking 213
stationary phase 109
steepest descent 96, 297
– Hankel functions 323
string equation 224
structure constants 412
Sturm’s comparison theorem 240
Sturm–Liouville operator 226, 232
subgroup 374