Some extra practice problems, 4410, Fa '99 If you want more practice with addition of angular momentum, here's a couple problems. These are *not* extra credit, nor are they to be turned in - just look at them if you want some extra problems. (I'll post solutions soon). None of the individual parts is meant to be very computationally difficult.

i) Two electrons are each in a "p orbit" (that means *each* electron has L=1). The total 2-electron system has total angular momentum 1, with total z component -1. If you measure the z component of spin of electron number 1, what values may be found, with what probabilities? (Neglect spin, and the Pauli principle, entirely in this problem)

ii) If you have 3 electrons, all in p states, what are the possible values of total angular momentum? (Again, neglect spin entirely)

iii) How many independent basis states are there for a system of 2 particles, one of spin 3 and the other spin 1?

iv) A particle of spin 1 and a particle of spin 2 are at rest. What possible values can the total spin of the system have?

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

b) Totally forget part iv-a. 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 also the total .

Would it matter what order you choose to measure these?

What values can you get for each, with what probabilities?

v) The Hamiltonian of a spin system is given by

Find the energy eigenvalue of the system of two particles:

a) If one of the particles is spin 1/2, and the other is spin 1, and the total combined system has S(total)=1/2, m(total)=+1/2.

b)if both particles are spin 1/2, and distinguishable, and the combined system has S(total)=1, m(total)=0.

c)if both particles are spin 1/2, and identical. (Think about Pauli exclusion principle here.)