##### Document Text Contents

Page 1

AWJoshi

,

A HALSTED PRESS BOOK

Page 2

BY TIlE SAME AUTIIOR

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

Page 174

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.

Page 175

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,

Page 347

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

Page 348

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/

AWJoshi

,

A HALSTED PRESS BOOK

Page 2

BY TIlE SAME AUTIIOR

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

Page 174

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.

Page 175

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,

Page 347

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

Page 348

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/