Homework Help: Possible values and their Probability of Measuring S^2 - Spin

1. Aug 27, 2010

cp51

1. The problem statement, all variables and given/known data
I have a two spin 1/2 particles. The Hamiltonian for the system is given as H = w1 S1z + w2 S2z. I need to find the possible values and their probabilities when I measure S^2 at some later time T. Also the Initial state \Psi (0) = a | $$\uparrow$$ $$\downarrow$$ > + b | $$\downarrow$$ $$\uparrow$$>

2. Relevant equations

3. The attempt at a solution

Now I know for a 2 spin 1/2 particle system, s = 1 and 0.

The eigenvalue equation for S2 is S2|sm> = hbar2 ( s ( s+1) )|sm>

So the possible values are 2 \hbar^2 and 0.

I know at some later time, the state will look like \Psi(t) = a e{-iE_1 t/ \hbar} + b e{-iE_2 t/ \hbar}

and I can find E_1 and E_2

However, how do i find the probabilities?

If I was just looking for S_z probabilities, I know it would be a2 for spin up and b2 for spin down. I also know that if I was looking for S_x I would need to evolve the coefficients in time. However, how do I measure the probabilities of S2?

2. Aug 27, 2010

vela

Staff Emeritus
You need to express $|\psi(t)\rangle$ in terms of the eigenstates of S2. Do you know how to do that?

3. Aug 29, 2010

cp51

Hmm, I'm not positive... is that writing it in |10> and |00> states? Im not exactly sure how to do this. Can you help get me moving in the right direction?

thanks for the help.

4. Aug 29, 2010

vela

Staff Emeritus
Yes, that's what I mean. Do you know how to express those states as linear combinations of $|\uparrow\,\downarrow\,\rangle$ and $|\downarrow\,\uparrow\,\rangle$? If not, you should figure out how to do that. It's probably covered in your textbook as it's a pretty common example of the addition of angular momentum.

5. Aug 31, 2010

cp51

Ok, I think I got it,

so using:

e1 = [TEX]\frac{1}{\sqrt{2}}[/TEX](|+,-> + |-,+>) with eigenvalue 2hbar^2

and

e2 = [TEX]\frac{1}{\sqrt{2}}[/TEX](|+,-> - |-,+>) with eigenvalue 0

I rewrite: |+,-> as (e1 + e2)*(sqrt(2)/2) and |-,+> as (e1 - e2)*(sqrt(2)/2)

Combine e1 and e2 terms. And then the coefficients squared give me the probabilities of measuring each eigenvalue as a function of time.

Sound good?