|Sep12-03, 11:50 PM||#1|
Help me (Spin Operators)
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?
|Sep13-03, 03:41 AM||#2|
Each operator acts on the part of the state vector that concerns it. If an operator belongs to the first particle it will act on the part of the state vector that belongs to the first particle. So S1 and S2 must commute.
σz = σ1z * I2 + I1 * σ2z
/ 1 0 0 0 \ / 1 0 0 0 \ | 0 1 0 0 | | 0 -1 0 0 | | 0 0 -1 0 | + | 0 0 1 0 | \ 0 0 0 -1 / \ 0 0 0 -1/
There are quite a few holes in my QM as I've just finished reading Sakurai, but did it on my own and I haven't got a lot of exercise to nail down the theory. Maybe that's why that letter J and the S1S2 would take me towards Perturbation Theory if the product is small and the variational method if it's not. That's where you'll get the energy shifts from the base E0=+/-k
|Sep13-03, 01:10 PM||#3|
edit: fixed bold font bracket
|Sep13-03, 01:13 PM||#4|
Help me (Spin Operators)
|Sep14-03, 03:13 AM||#5|
|Sep14-03, 09:57 AM||#6|
Thanks for everyone's help, I really appreciate.
I figured out a way, using a "trick":
Since vec(S1), vec(S2) commute,
2 vec(S1) dot vec(S2) = [vec(S1) + vec(S2) ]^2
- vec(S1)^2 - vec(S2)^2
vec(S1)^2 = s1 (s1 + 1)
vec(S2)^2 = s2 (s2 + 1)
vec(S1) + vec(S2) = 1 for triplet, 0 for singlet
using the above, it is very easy to compute the the matrix elements of Hamiltonian, provided if
1/sqrt 2 (|u d> +/- |d u>)
really form a basis.
I think they should, since |u>, |d> span the single particle hilbert space, their direct products should span the double particle Hilbert space which is the direct products of two single particle Hilbert spaces.
So the Hamiltonian looks like
x 0 0 0
0 x x 0
0 x x 0
0 0 0 x)
x means non-zero entry.
Solving the eigenvalue equations will give the eigenvalues and correct eigenvectors.
I will figure out another way using matrices, I think that's a better way, since you can no longer use 'trick' when things get complicated.
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