Clebsch-Gordan coefficients for two spin-1/2 Hilbert spaces.

Basically I understand the first part to be 4: |-->, |++>, |-+> and |+->. The second part to be something like; J.J = (S1+S2).(S1+S2) = S1^2 + S2^2 + 2*S1.S2 = (3/2)*(h^2) + 2*S1z*S2z + S1+*S2- + S1-*S2+. After this it gets a little hazy and then for the Clebsch-Gordan coefficients I have read and read and am still a little confused. See attached file for full problem description.

According to the Hawking process, black holes evaporate because negative energy virtual particles fall into it while the positive energy virtual particles escape. Why don't positive energy virtual particles fall into the black hole?

Stephen Hawking has stated that black holes disappear over time. My understanding is that because of the uncertainty principle there are vacuum fluctuations causing positive and negative energy particles to be produced at the event horizon. If the negatively energy particles fall into the black hole they reduce the mass of the object and over time the black hole evaporates. My question is don't these particles have a 50% chance of being of positive or negative energy? And if I'm right the same percentage of positive or negative energy particles will fall into the black hole, yielding no change to the object.

Exchange Interaction

Exchange Interaction. See attached file for full problem description.

Eigenfunction expansion, time evolution, Hamiltonian

Eigenfunction expansion, time evolution, Hamiltonian. See attached file for full problem description.

Ferromagnet

Ferromagnet. See attached file for full problem description.