Bryn Mawr Physics 302- Advanced Quantum Mechanics - 2010

Walter Smith

Assignment #4 (Corrected)

Due: Friday, Feb. 19 at 10:00 am.

Reading: Chapter 4 of Townsend

Note: As Townsend points out on p. 80, operators such as can refer to angular momentum due to spin or orbital angular momentum. When specifically referring to spin, they are called, for example, . So, for this assignment you can think of as being equivalent to .

Group problems:

3.14 (You may wish to wait until after class Tuesday to do this. Use raising and lowering operators.)

3.15 Hint: For example, you can find in the z-basis using . Don’t forget to normalize your results, e.g. set .

3.17 Do problem 3.15 before doing part c of this problem.

4A. In class, we used an arm-waving argument (literally) to show the reasonableness of . In this problem, you will quantify our argument.

i) Explain briefly why the mathematical version of what we showed in class would be , where a is to be determined and dj is an infinitesimal rotation. (In class, we used , which is hardly infinitesimal, but similar results would apply even better to smaller rotations.) Note: in writing this, we have ignored a small difference in the vertical displacement between the left and right hands; we’ll deal with this in part iii below.

ii) Using diagrams and equations, show that .

iii) Using diagrams and equations, show that the vertical displacement ignored in part i is of order or higher.

iv) Now show that the relation in part i leads quantitatively to . Don’t forget that is very small.

4B. i) In class, I introduced corrected kets for spin-½ particles, ones that are consistent with cyclic permutations. Using these kets, confirm that , where is an (unimportant) overall phase factor, as would be expected from considering how the rotation affects ordinary vectors.

ii) Use the corrected kets to show that , as would be expected.

Individual-problems: 2.11, 3.10