Bryn Mawr Physics 302- Advanced Quantum Mechanics - 2010
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 .
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 .† Donít 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; weíll 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 .† Donít forget that †is very small.
4B.† i) In class, I introduced corrected kets for spin-Ĺ 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