Two spin ½ particles

Problem:

The Heisenberg Hamiltonian representing the “exchange interaction” between two spins (S1 and S2) is given by H = -2f(R)S1S2, where f(R) is the so-called exchange coupling constant and R is the spatial separation between the two spins.  Find the eigenstates and eigenvalues of the Heisenberg Hamiltonian describing the exchange interaction between two electrons.
HINT: The total spin operator is S = S1 + S2.

Solution:

Problem:

Let Si, i = 1, 2 denote the spin vectors of two spin-½ particles. The interaction is given by
H = U0 (S1·S2 − 3 S1zS2z).
Find the energy eigenstates and eigenvalues.

Solution:

Problem:

Two spin ½ particles, 1 and 2, are in the singlet state shown below. Here |+> and |-> refer to spin up and spin down with respect to the z-axis.
|ψ(1,2)> = ½½(|+>1|->2 - |->1|+>2).
(a)  What is the probability of measuring the spin of particle 1 along an axis u(θ,φ), and finding the particle in the state |+>u?
Assume the spin of particle 1 is measured along the z-axis and found to be ħ/2.
(b)  A simultaneous measurement of the spin of particle 2 along the z-axis would yield which results with which probabilities?
(c)  A simultaneous measurement of the spin of particle 2 along the x-axis would yield which results with which probabilities?

Solution:

Problem:

Consider the state space E = E1imageE2 of two non-identical spin ½ particles spanned by the basis vectors {|++>, |+->, |-+>, |-->}.  Use what you know about the common eigenvectors of S2 and Sz, and find the common eigenvectors of S2 and Sx.  Express these eigenvectors in terms of the basis vectors {|++>, |+->, |-+>, |-->}.

Solution: