Title Elements of Group Theory for Physicists English 17.6 MB 348
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AWJoshi
,

A HALSTED PRESS BOOK

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BY TIlE SAME AUTIIOR

Matrices and Tensors ill Physics, 2nd ed. (1984)

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160 ELEMENTS OF GROUP THEORY FOR PHYSiCISTS

Consider next the case of an electron in a hydrogen atom. The
wave function of the electron is of the form Rnl (r) Yt"l (e, 4». Here the
position vector r is measured in a coordinate system whose origin is at
the nucleus. If we displace the system through some vector so that the
nucleus is no longer at the origin of coordinates, the displaced function
cannot be put in the standard form and hence does not denote a possible
state of the system. The linear momentum is therefore not a constant of
motion for an electron in a hydrogen atom which is of course a well-
known result.

5.3.2 Time displacements. In a~alogy with the space displace-
ment of a physical system considered above, we may displace a system
in time and try to' find out whether the displaced function represents
a possible state of the system.

Thus, let ~(t) be the wave functionS of a physical system and let
P,('t') denote the operator for translating the functions of time by an
amount't'. We then have

(5.30)

We may expand the function ~(t--r) in a Taylor series about the ·
point t and we then find that

PI ('t') Ht)=exp (--rO/atH (t). (5 . 31)
We therefore have

PI ('t')=exp( --r%t). (5.32)

Now, the quantum mechanical energy operator is given by
.9{=ifia/ot. If \$I. is itself independent of time, that is, if the energy is
a constant of motion, then we can replace a/at in the exponential in
(5.32) by .9{and obtain

PI ('t')=exp (i't'.9{/ 'fi), . (5.33)

which is a unitary operator because 't' is real and .9{ is hermitian.
This again shows that if a physical system is invariant under all

time displacements, then the energy of the system is a constant of.
motion. The transformed function in this case still obeys the Schroe-
dinger equation. All the time translation operators PI('t') commute
with the Hamiltonian, i.e.,

[P/(-r),&]=O, all T. (5.34)

The set of all time translation operators is again an abelian, conti-

8We are not interested in the other variabies on which y may depend;
these a re therefore suppressed here.

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GROUP THEORY IN QUANTUM MECHANICS. I 161

nuous, connected, one-parameter, noncompact group which is the
symmetry group of the physical system.

Once a?~in, we may consider the example of the hydrogen atom.
If we have an isolated hydrogen atom, with no perturbations, its
Hamiltonian is invariant under all time ·displacements. If the atom is
in a particular state at a given instant of time, it will continue to remain
in the same state for all time and the total energy of the system will be an
invariant. On the other hand, if we apply a time-dependent perturba-
tion, the Hamiltonian is no longer invariant under time translations,
the atom may make transitions froll' one state to another and the
energy of the atom does not remain a constant of motion.

5.4 Symmetry ~f the Hamiltonian

In the previous section we have seen by means of two examples
that when a system possesses a certain symmetry, there is a corres-
ponding physical observable which remains a constant of motion. We
shall develop this concept here in its complete generality. We shall
hereafter use the operator .ge of (5'.1) to mean the Hamiltonian (the
energy operator) of the system.

The Hamiltonian .ge is itself a function of the various parameters
of the system such as the position vector, time, momentum, angular
momentum, etc., and it reflects the symmetry of the system it des-
cribes. Its familiar form in the single-particle approximation is

1i2V 2
.ge - 2m +V, (S.3S)

where the.first term is the kinetic energy operator for the particle it des-
cribes and V contains all the other terms. The Laplacian V2 is invariant
under all orthogonal transformations of the coordinate system (that is,
under the rotation-inversion group 0(3». Hence, the symmetry of .geis
essentially governed by the symmetry of the function V. Thus, if(S. 3S)
refers to an electron in a hydrogen atom, the potential energy of the
electron is spherically symmetric, and .ge would be invariant under
the group 0(3); if it refers to an electron in a crystal, .ge would be
invariant under the symmetry transformations of. the crystal (that is,
the operations of the space group of the crystal, to be discussed in
Chapter 7).

Let us consider the operation of PR , which corresponds to some
coordinate transformation R, on the Schroedinger equation (5.1):

PR.ge Jr=PR EJr,

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INDEX

multivalued, 11 1l, I III
of a continuous group, I I I.
reducible, 61, 63
reduction of, 77
regular, 87
unfaithful, 60

Representation vector, 74
Rose, M.E. , 150, 228
Rosenthal, J. E., 25 4 /
Rotation, improper, 121

proper, 121
Rotation group(s), 11 7-1 27

classes of SO(3), 124
connectedness of 50(3), 125
generators of SO(2), 11 9
generators of SO(3), 1,23
0(4), 127
O(n>, 126
parameters of SO(3), 122
SO(2), 117
SO(3),I 20

Rotation-inversion group, 121, 193
Rotenberg, M., 215/. 228
Runge-Lenz vector, 178

ScaJar, 31
operator, 215
particle, 1 91

Schenkman, E., 29
Schiff, L.I., 144, 150, 152/. 154/. 187/.

191, 206/. 221/
Schmeidler, W., 57
Schur, 320
Schur's lemmas, 66,6 7, 69
Screw rotation, 240
Secular equation, 297
Segall, B., 303
Selection rules, 175 , 176 , 200, 201, 226
Sequence, 3 3

Cauchy, 34
convergent, 33
limit of, 3-1
of n-tuplets, 34

Set, complete, 36
incomplete, 36
of inte~ers, 1
of unitary matrices, 2
overcomplete, 36

Shapiro, Z. Ya., 150

~ 1 " I ()v. G. E., 57
H IlimOIl ~, G. F., 52/. 57, 115/
S11l11)ly connected groups, 114
Hillin determinant, 1 92 ,200
Slnhll . .I . '., 228, 264, 304
Hl llol " l'IlOw ki, R., 246/. 248f
SOIlIIll.·dllld, 203
.Wl(,'), 11 7
SO( 1), 120
SO(,,). 127
SpuccN. closed, 38, 61

complete, 35
direct product of, 41
dirccl slImof. 41,45
dual, 4-1
function, 42
Hilbert, See Hilbert space
inner product, 32
linear, 31/
linear vector, 31/
of n-tuplets, 32,
orthogonal, 52
reducible, 61
vector" 30, 31
spin, 74

Space displacement, 158
Space group, 238 , 24 0

representations of, 273 , 279
symmorphic, 24 1

Space inversion symmetry, 190
Spectral theory, of operators, 5 1
Spherical components, 1 50 , 217 , 220
Spberical harmonics, 196 , 220
Spinor, 7-1, 147
Spin space, 74,1 47 ,1 54
Standard components, 150. 217, 220
Stereographic projections, 23 1, 2j-1
Strain tensor,'305',p06
Stress tensor, 305. 306
SUt2), 130

applications of. 1-16
classes 0 f, 13 3
generators or. J 41
homomorphism on 50(3), I \II
rank of, J-14
reprosontations of, I 31
structure constants or, 14-1

SU(3), 14 5
applications of. 1 -1 (;, I -1 S

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334 ELEMENTS OF GROUP THEORY FOR PHYSICISTS

generators of, 145
rank of, 146
structure constants of, 145

SU(n), 140
generators of, 140

Subgroup, 13
coset of, 14
index of, 16
invariant, 16
normal, 16
proper, 13

Subspace, 61
invariant, 61
proper, 61

Sudarshan, E. C. G., 150
Sylvester, 1
Symmetrized !lasis functions, 89

Talman, I. D ., 109/
Tensor operator(s), 215, 219

. direct product of, 221
irreducible, 220, 221
matrix elements of, 224
scalar product of. 227

Tetrahedral groups, 236
Three-dimensional rotation group, See

SO(3)
Time displacement, 158, 160
Time-reversal operator, 185, 187
Time-reversal symmetry. 183, 248,318

. in band theory, 320
Tinkham. -M., 29, 92/, 108,150,191 ,

254/. 264. 304
Transformatioc, group of, 4

of a function, 1 55
of coordinate system, 6, 236
orthogonal, 38/
similarity, 11
symmetry. 4
unitary, 38

Translation group, 238, 239, 267
Transposition, 21
Triangular inequality, 33
Trigg, G. L., 57

U(n), 140
generators of, 140

Universal covering group, 139

Van der Waerden, B. L., 57
Vector(s), 32

axial, 218
basis, 37
Fourier expansion of, 37
linear independence of, 36
norm3lized, 37
norm of, 33
operator, 217
polar, 218
spherical components of, 150
standard components of, 150
unit; 37

Vector coupling coefficients, See
Clebsch-Gordan coefficients

Vector space, 30 ,31
dimension of, 36
direct product of, 41
direct sum of. 41, 45

Venkatarayudu, T., 191, 304

Walter, I., 191, 228
Wave vector, 269

group of, 276, 278
star of, 276,277

Wheeler, R., 264 , 304
While, H. E., 197/.
Wigner,E.P .. l, 29, 108, 109,150,

191,215,224,245,248/,319
Wigner coefficients, See

Clebsch-Gordan coefficients
Wigner-Eckart theorem, 224,225
Wigner-Seitz cell, 271
Wooten, I. K., ~28
Wybourne, B. G., 130/, 150

Zak, I ., 264
Zeeman effect, 202
Ziman, I. M., 123/