Issued Wed, Feb 4 Due Wed, Feb 11

Required reading for this week: Finish Gas. Ch. 15.

1a) Prove that .

Note: Gas. used this result, implicitly, in his Eq. 15-16.

b) Write each of the 4 "microscopic" states in terms of a linear combination of the 4 "total" states .

Note: We did it the other way around, in class. This should be a quick one!

c) Prove that the formula for the spin singlet state looks the same if the 's refer to eigenvectors of , rather than , i.e. prove that . (Gas. Eq. 15-25...)

2a) You would expect (from Eq. 11-48) that , even if the (given) spin 1 states are really "composite" (i.e. built up by combining 2 spin 1/2 states)

Explicitly verify that this is true, in a microscopic way. I.e, prove it by writing , and and .

You may freely assume any usual results for single spin 1/2 particles and operators, like e.g.

b) Repeat part a, but show .

(You should be able to take your results from part a, change one sign, and be done with part b!)

c) Repeat part a, except now explicitly prove that .

Here,

You may use Gas' method (Eq 15-16) which makes use of the result of problem 1a to write the dot product in terms of + and - components, or you may use my method (see notes, around p.65) where I use only x, y, and z components of S, whichever you prefer...3) Gas 15-2.

By "analogue of 15-47", Gas means you should explicitly write out (in terms of ) all 5 states with S=2 (m=2, 1, 0,-1, or -2 in order).

Extra Credit (If you do this, please turn it in to me separately! ) Go on and find all 3 states with S=1, and the state with S=0.

(In "group theory speak", you have just shown that )

4a) Gas 15-4

4b) A particle of spin 1 and a particle of spin 2 are at rest.

i) In general, what possible values can the total spin of the system have?

ii) Suppose you measure total spin, and find it to be 1, and measure the total , and get +1 . Now measure the z component of spin of the spin 2 particle. What values could you get, with what probabilities?

iii) Totally forget part ii. Instead, imagine you directly measure the z component of spin of the individual particles and get -1 for each. Now measure the total spin, and total . (Would it matter what order you choose to measure these?) What values can you get for each, with what probabilities?

5) An electron in hydrogen is in the combined spin/position state

a) If you measure (orbital angular momentum squared), what values might you get, with what probabilities?

b) Same question, for .

c) Same question, for (spin angular momentum squared)

d) Same question, for

e) Same question, for (total angular momentum squared, J=L+S)

f) Same question, for

(Note: You can answer f, and "what values might you get" in e, but you can't answer "with what probabilities" in part e, without using Clebsh-Gordan...)

g) If you measure the spatial position of the electron, what is the probability/unit volume to find it at the point ?

h) If you measure the z component of spin, and the distance from the origin (just r, ignoring the angles) (which you can do, because these measurements are independent) what is the probability/unit volume to find it at radius r from the origin, with spin up?