View Single Post
kakarukeys
kakarukeys is offline
#1
Sep12-03, 11:50 PM
P: 190
An easy (not easy) 1st year undergraduate project:

Energy eigenvalues of a single spin-1/2 system which Hamiltonian is given by

H = - k S^z

are -/+ (1/2 k).

I got that using the spin operator S^z = 1/2 Sigma^z (let h_bar = 1), Hamiltonian is in diagonalized form:

( -1/2 k 0
0 1/2k )

So, Eigenvectors are given by (1, 0), (0, 1), spin up and spin down respectively.

How do I compute the energy eigenvalues, eigenvectors of an double identical spin-1/2 system which Hamiltonian is given by

H = - J vec(S1) dot vec(S2) - k S1^z - k S2^z

J is a const. > 0

After going some references, Griffiths, Sakurai, Merzbacher, etc,
I have no idea how to begin,

I have the following problems in mind:
(1) if |u u> represents both spins up, (1/sqrt 2)(|u d> +/- |d u>) represent one spin up, |d d> represent both spins down. |u u> and |d d> should be eigenstates of the system (because they are ground states), for |u d>, |d u> states I am not sure.

(2) I can't use same spin matrices for S1^z, S2^z. But there is only one S^z matrix namely, 1/2 Sigma^z. Could S1^z be a tensor product of 1/2 Sigma^z with an identity matrix, and S2^2 the other way round? If it is so, I have no idea how to do the maths!

(3) Do vec(S1) and vec(S2) commute?

Anyone could give me some hints?
Phys.Org News Partner Science news on Phys.org
Lemurs match scent of a friend to sound of her voice
Repeated self-healing now possible in composite materials
'Heartbleed' fix may slow Web performance