**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 d*j* 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**