A discrete subgroup of a continuous symmetry group can always be de¯ned by choosing the group parameter at periodic intervals (see problem 7). However, here we are going to discuss some discrete symmetries that are not subgroups of continuous groups. Such symmetries are not associated to any conserved quantities, as they have no generators.
The following two discrete symmetries are of general importance in physics.
7.5.1 Space inversion
The space inversion operator, Is, has the following operation on the position vector r.
Isr=¡r: (7.95)
The quantum states representation ofIs will be called UI and for an arbitrary statejsi hrjUIjsi=jh¡rjsi=jÃs(¡r): (7.96) The extra factor j is needed due to the discreteness of the symmetry. In continuous sym-metry operations the value ofjis unity as in the limit of all symmetry parameters going to zero the wavefunction must stay unchanged. For a discrete symmetry such a limit cannot be de¯ned. For the spinless particles that we have discussed till now, two space inversions should produce the original wavefunction i.e.
UI2jsi=jsi; for anyjsi: (7.97) Hence, from equation 7.96 we get
j2= 1; j=§1: (7.98)
The value of j is called the INTRINSIC PARITY of the system.
Theorem 7.4 The energy eigenstates of an inversion symmetric system can be chosen such that they change at most by a sign under the inversion operation.
Proof: If jEi is an eigenstate of energy with eigenvalue E, then from the de¯nition of a quantum symmetryUIjEiis also an eigenstate with the same eigenvalue. Hence, the following are also eigenstates of energy with the same eigenvalue.
jE1i=jEi+UIjEi; jE2i=jEi ¡UIjEi: (7.99) From equations 7.97 and 7.99 it can be seen that
UIjE1i= +jE1i; UIjE2i=¡jE2i: (7.100) This proves the theorem.
CHAPTER 7. SYMMETRIES AND CONSERVED QUANTITIES 93 De¯nition 37 The energy eigenstates of the type jE1i in equation 7.100 are called SYM-METRIC and they are also said to have POSITIVE (TOTAL) PARITY. The eigenstates of the typejE2i in equation 7.100 are called ANTISYMMETRIC and they are also said to have NEGATIVE (TOTAL) PARITY.
It should be noted that the intrinsic parity is included in the total parity. However, the intrinsic parity of particles cannot be absolutely determined. The intrinsic parities of some particles have to be assumed and then those of others can be determined if the system is inversion symmetric (i.e. total parity is conserved).
From equation 7.97 we see that UIy = UI and hence, in the position representation (using equations 7.96 and 7.98)
UIyrUIÃs(r) =UIyrjÃs(¡r) =¡rÃs(r): (7.101) As this is true for any stateÃs(r), the following must be true for the position operator R.
UIyRUI =¡R: (7.102)
Similarly, for the momentum operatorP
UIyPUI =¡P: (7.103)
Hence, for the angular momentum operator,L=R£P, one obtains
UIyLUI =L: (7.104)
7.5.2 Time reversal
The time reversal operator is expected to be di®erent in nature from all other symmetry operators discussed upto now. This is due to the fact that the SchrÄodinger equation is ¯rst order in time and hence, a time reversal would change the sign of only the time derivative term. To be precise it can be seen that the time reversal operator, T, is antiunitary. We have seen this to be possible from theorem 7.3. However, we have also seen that continuous group transformations cannot be antiunitary. So, due to its discrete nature, it is possible for T to be antiunitary. To demonstrate the antiunitary nature ofT, let us consider the energy eigenstatejEi at timet= 0 that has the eigenvalueE. The result of a time translation of tfollowed by a time reversal must be the same as that of a time reversal followed by a time translation of ¡t. Hence, from equation 7.23
T Ut(t)jEi = Ut(¡t)TjEi; Texp(¡iHt=¹h)jEi = exp(iHt=¹h)TjEi; Texp(¡iEt=¹h)jEi = exp(iEt=¹h)TjEi;
Texp(¡iEt=¹h)jEi = [exp(¡iEt=¹h)]¤TjEi: (7.105)
CHAPTER 7. SYMMETRIES AND CONSERVED QUANTITIES 94 As the above equation is true for any t and E, an arbitrary state jsi that can be written as the linear combination
jsi=X
E
aEjEi; (7.106)
would be time reversed as
Tjsi=X
E
a¤ETjEi: (7.107)
Also the arbitrary state
jri=X
E
bEjEi; (7.108)
is time reversed as
Tjri=X
E
b¤ETjEi (7.109)
Hence,
hrjTyTjsi=X
E0E
bE0a¤EhE0jTyTjEi: (7.110) Due to time reversal symmetry the set of statesfTjEigfor allE would be the same as the set of states fjEig for allE. Hence,
hE0jTyTjEi=±E0E (7.111)
and then from equation 7.110 it follows that hrjTyTjsi=X
E
bEa¤E =hsjri: (7.112) This demonstrates, from de¯nition, thatT is an antiunitary operator.
Under a time reversal, one expects the position operator to stay unchanged and the momentum operator to change in sign. Thus
TR=RT TP=¡PT: (7.113)
Hence, the angular momentum L=R£Phas the property
TL=¡LT: (7.114)
Problems
1. Prove corollary 7.1.
2. Show that the momentum eigenstates of a particle stay physically unchanged by a space translation.
CHAPTER 7. SYMMETRIES AND CONSERVED QUANTITIES 95 3. Derive the equations 7.65 and 7.66.
4. For a system of two angular momenta with given magnitudes of individual angular momenta (l1 and l2 ¯xed) show that the number of angular momentum eigenstates is (2l1+ 1)(2l2+ 1).
5. Show the minimum value oflfor the total angular momentum states isjl1¡l2j. [Hint:
For ¯xed values ofl1 andl2, the number of eigenstates of both the individual angular momenta and total angular momentum are the same.]
6. Find the Clebsch-Gordan coe±cients for l1= 1 and l2= 1.
7. A periodic potential V(r) has a three dimensional periodicity given by the vector a = (n1a1; n2a2; n3a3) where a1, a2 and a3 are ¯xed lengths andn1, n2 and n3 can take any integer values such that
V(r+a) =V(r):
(a) Show that the discrete translation symmetry operators Ud(a) = exp(¡iP¢a=¹h) commute with the hamiltonian.
(b) Show that the setfUd(a)gfor all possible integers (n1; n2; n3) inaform a group.
(c) The Bloch states are de¯ned by their position representation uB=u(r) exp(ip¢r=¹h);
whereu(r+a) =u(r) and pgives three labels for such states. Show that these states are physically unchanged byUd(a).
Chapter 8
Three Dimensional Systems
The generalization of problem solving methods to three dimensions is conceptually simple.
However, the mathematical details can be quite nontrivial. In this chapter we shall discuss some analytical methods in three dimensions. Quite obviously, such methods can have only limited applicability. However, the analytical solution of the hydrogen atom problem provides a better understanding of more complex atoms and molecules. Numerical methods in three dimensions can either be based on the analytical hydrogen atom solution or be independent of it, according to the nature of the system.
8.1 General characteristics of bound states
The characteristics of bound states in one dimension can be generalized to three dimensions.
IfE < V at large distances in all directions, then the energy eigenvalues must be discrete.
However, in three dimensions, there must be two other quantities, besides energy, that must also have discrete values. This is because in each dimension the condition of ¯niteness of the wavefunction will lead to some parameter being allowed only discrete values (using similar arguments as for energy in the one dimensional case discussed in chapter 5). For spherically symmetric potentials, it will be seen that the two extra discrete parameters are the eigenvalues of one angular momentum component (say Lz) and the magnitude of the total angular momentum (L2). A study of numerical methods (chapter 9) will clarify the nature of these discrete parameters for more general cases.
96
CHAPTER 8. THREE DIMENSIONAL SYSTEMS 97