Two Spin-1/2 Particles: Probability of J1z Measured Results

[conana]

Homework Statement

Two spin-1/2 particles are governed by $$\dfrac{J^2}{2I}+\omega_0 J_z$$ where $$J=J_1+J_2$$. At time t = 0 the total angular momentum is measured and found to be 0.

(a) At time t, $$J_{1z}$$ is measured. What are the possible results and what is the probability of each?

Homework Equations

The Attempt at a Solution

So it seems that what I want to do is find $$|\psi (t)\rangle$$ in the eigenbasis of the hamiltonian, which would involve the total angular momentum of the two particles, j, and the z-component of the angular momentum, $$m_j$$, and then change to a basis involving the individual angular momenta so that I can find the possible results and probabilities for $$J_{z1}$$.

Since the total angular momentum was measured at t = 0, the system is in an eigenstate of the hamiltonian and, thus, the time evolution should not affect the probabilities at time t, so I can just look at $$|\psi (0)\rangle$$.

For 0 total angular momentum at t=0, it seems $$|\psi(0)\rangle=|0, 0\rangle$$ in the basis $$|j, m_j\rangle$$.

Now I run into a couple of problems.

(1) How can total angular momentum J = 0? If J = J1 + J2, then J1 + J2 = 0. It is my understanding that J1,J2 >=0 in general because J1 = L1 + S1 or abs(L1 - S1) (where L1 is the orbital angular momentum and S1 is the spin angular momentum, both of which are magnitudes i.e. > 0) and similarly for J2, so it must be that J1 = J2 = 0. For this problem, however, even if L1 = L2 = 0, it would be the case that J1 = J2 = 1/2 and thus J = 1. Is my understanding here totally flawed?

(2) Change of basis. I'm guessing I want to use a basis $$|j_1 m_1, j_2 m_2\rangle$$ where j1, j2 are the total angular momenta and m1, m2 are the z-component of angular momentum for particles 1 and 2, respectively. My professor gave us a table of Clebsch-Gordan coefficients for just such an occasion, but in this case I am not exactly sure how to apply it. I guess my problem is that I don't know how to find j1, j2 to be able to use the correct entry of my table. This stems, at least partly, from my first problem with the total J = 0.

This is part of a sample final exam question for my final exam tomorrow and, unfortunately, my professor is unavailable for help due to medical issues. Any help is GREATLY appreciated.

 

[diazona]

I'll have to come back and expand on this later, but let me address your problem (1) for now: for starters, J₁ and J₂ etc. are operators. When the problem tells you that total angular momentum was measured to be zero, it doesn't mean that the operator J = 0; it means that the particles were in a state that satisfies
$$J\lvert\psi\rangle = 0$$
It's the eigenvalue (lowercase j) that is zero, not the operator. You can't really talk about an operator being positive or negative, at least not in the same sense that you would talk about positive or negative numbers. And operators are certainly not magnitudes.

Besides, what you're doing here is not just adding operators. You're actually multiplying state spaces, which is more complicated than adding (or multiplying) numbers. Each individual particle has a quantum state that "lives" in a two-dimensional space. When you multiply them together, you get a four-dimensional space - not just a single result, but a whole set of results. Some of them have total angular momentum 1 (1/2 + 1/2) and some have total angular momentum 0 (1/2 - 1/2).

Anyway, I think you basically have the right idea of how to start, i.e. you need to find $\lvert\psi(t)\rangle$ in the Hamiltonian's eigenbasis and then convert it. But I'm not sure offhand whether you can just use $\lvert\psi(0)\rangle$.

 

[conana]

Thank you for the reply. Between your advice and me talking to another professor I think I have straightened everything out. I was confusing summing eigenvalues and operators and also some of the labeling conventions. Feeling pretty good, but my exam is in an hour so it's back to the books for me! Thanks again.