Electron and Nuclear spin interaction

[MMS]

Homework Statement

Hello,
I'm asked to show the equivalence of the given Hamiltonian below which describes the interaction between an electron and a nucleus

and the following Hamiltonian

The Attempt at a Solution

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I've simply written down each Hamiltonian as a sum of four tensor product and calculated it.
The first (given) Hamiltonian gives

And as for the second one, I've also written it the same way (sum of four tensor products) and received

I've went over my calculations a few times now and I can't seem to find a mistake so it got me thinking if it's more than simply a calculation mistake.

For the given Hamiltonian, I'm pretty sure of what I did. I know how to take the tensor product of two matrices and I've checked my answers online for tensor product of Pauli matrices and it seems to be right.

As for the second one, I've written it as (the summation only with the 1/2 taken out the brackets)
(1 ⊗ σ0 + σ0 ⊗ 1)^2+(1 ⊗ σ1 + σ1 ⊗ 1)^2+(1 ⊗ σ2 + σ2 ⊗ 1)^2+(1 ⊗ σ3 + σ3 ⊗ 1)^2
and for each (1 ⊗ σi + σi ⊗ 1)^2 I've calculated the tensor product, added them up and only then taken the square of it (multiplication of the matrix by itself) and the rest with multiplying by the factor and substituting 3I from it is trivial.

Any ideas on where I could've went wrong? Is what I described above right?
Also can I write (1 ⊗ σi + σi ⊗ 1)^2 somehow in the form of a^2+b^2+2ab?

Thanks in advance.

 

[TSny]

As for the second one, I've written it as (the summation only with the 1/2 taken out the brackets)
(1 ⊗ σ0 + σ0 ⊗ 1)^2+(1 ⊗ σ1 + σ1 ⊗ 1)^2+(1 ⊗ σ2 + σ2 ⊗ 1)^2+(1 ⊗ σ3 + σ3 ⊗ 1)^2

The summations go from $i = 1$ to $i = 3$. So, there is no contribution from $\sigma_0$ (in either the first or the second expression for $H_I$).

 

[MMS]

The summations go from $i = 1$ to $i = 3$. So, there is no contribution from $\sigma_0$ (in either the first or the second expression for $H_I$).

Dam, I should've noticed that. I will try again now.

Thank you TSny!

 

[MMS]

I still don't get the same matrix even after this correction... Any help?

 

[TSny]

I still don't get the same matrix even after this correction... Any help?

It seems to work out for me. As a check, what 4x4 matrix do you get for $\sigma_2 \otimes \sigma_2$?

You can also work this problem without explicitly constructing the 4x4 matrices.

 

[MMS]

It seems to work out for me. As a check, what 4x4 matrix do you get for $\sigma_2 \otimes \sigma_2$?

You can also work this problem without explicitly constructing the 4x4 matrices.

This is what I get

Is it correct?

 

[TSny]

Yes. So, I don't know why it isn't working out for you. What did you get for $\left( I \otimes \frac{1}{2}\sigma_1 + \frac{1}{2}\sigma_1 \otimes I \right)^2$?

 

[MMS]

Yes. So, I don't know why it isn't working out for you. What did you get for $\left( I \otimes \frac{1}{2}\sigma_1 + \frac{1}{2}\sigma_1 \otimes I \right)^2$?

I got

How about this?

 

[TSny]

OK. I think we found it.

 

[MMS]

Oh my goodness, it's supposed to be I!

Now I'm curious though, How can I do this without using the matrix form? Do I need to expand the squared component?

And once again, thank you TSny!

 

[TSny]

Now I'm curious though, How can I do this without using the matrix form? Do I need to expand the squared component?

Yes. Use the property $\left( A \otimes B \right) \left( C \otimes D \right) = AC \otimes BD$.

 

[MMS]

Thank you, I will try this as well.

 

[TSny]

OK, I think you'll like it.

 

[MMS]

I sure did. 4 lines. That's all it took!

 

[TSny]

Nice work!

 

[MMS]

If I may, another question regarding this.

They ask me to find the eigenvalues and eigenvectors of the Hamiltonian. I can think of at least two ways to do this. However, they've written a hint that says "angular momentum"... I'm not sure how to use that hint or what they meant by it. Any idea as to what it means and how to use it?

Thanks in advance.

 

[TSny]

The hint might be suggesting that the eigenstates are the states of definite total angular momentum. What are the states of definite total angular momentum that come from coupling two spin 1/2 particles?

Edit: I should have been more careful in the way I worded that. For "states of definite total angular momentum", I should have said "states with definite values of $S^2$, where $\vec{S}$ is the total spin angular momentum of the system. Does the $S^2$ operator commute with the Hamiltonian?

 

[MMS]

The hint might be suggesting that the eigenstates are the states of definite total angular momentum. What are the states of definite total angular momentum that come from coupling two spin 1/2 particles?

Edit: I should have been more careful in the way I worded that. For "states of definite total angular momentum", I should have said "states with definite values of $S^2$, where $\vec{S}$ is the total spin angular momentum of the system. Does the $S^2$ operator commute with the Hamiltonian?

The singlet and triplet states, of course. That was one of the ways I was thinking of (the other is solving the characteristic polynomial which is very simple in this case). Simply taking them and working them on the Hamiltonian but I wasn't sure of that.

Edit: The Hamiltonian does commute with S^2 operator.

 

[TSny]

Yes. The second way of writing $H_I$ in the statement of the problem makes it very easy to see that the eigenfunctions of $H_I$ are the eigenfunctions of $S^2$. It is also easy to see what the eigenvalues are of $H_I$.