dimensions are described by the group SO(3), while rotations of half-integral spin states are described by a different group SU(2) which is the “covering group” for SO(3). Don’t be put off by the jargon—If you have taken linear algebra you know that rotations in three 2
4. (10 pts) Fun with commutators. We will be working a lot with com- mutators, so you need to get familiar with what you can and cannot do. Show the following ( ˆ A, ˆ B . . . are operators; a, b . . . are numbers): (a) [ ˆ A, ˆ B ] = - [ ˆ B, ˆ A ] (b) [ ˆ A, ˆ B + ˆ C ] = [ ˆ A, ˆ B ] + [ ˆ A, ˆ C ] (c) [ ˆ A + ˆ B, ˆ C ] = [ ˆ A, ˆ C ] + [ ˆ B, ˆ C ] (d) [ ˆ A, b ˆ C ] = b [ ˆ A, ˆ C ] (e) [ ˆ A ˆ B, ˆ C ] = ˆ A [ ˆ B, ˆ C ] + [ ˆ A, ˆ C ] ˆ B (f) [ ˆ A, ˆ B ˆ C ] = ˆ B [ ˆ A, ˆ C ] + [ ˆ A, ˆ B ] ˆ C Can you see the pattern in parts 4e and 4f? (g) [ ˆ I + a ˆ B, ˆ I + c ˆ D ] = ac [ ˆ B, ˆ D ] where ˆ I is the identity. We used this on rotation operators to get the commutation rela- tions for the generators ˆ J k . One way to see this is to use parts parts 4b and 4c to expand the LHS into four commutators, three of which vanish because ˆ I commutes with everything. 5. (10 pts) In some basis three operators are given by the matrices ˆ A ↔ " 1 0 0 - 1 # , ˆ B ↔ " 3 0 0 3 # , ˆ C ↔ " 0 1 1 0 # . (a) You can see that these operators are Hermitian, and so can rep- resent observable quantities A, B, C . Find the eigenspinors and eigenvalues of all three matrices. What is different about the eigenvalues of ˆ B compared to the eigenvalues of the other two operators? If you are careful, you will find that the eigenspinors for ˆ B are not uniquely defined, which goes along with this fact.
You've reached the end of your free preview.
Want to read all 4 pages?