Total angular momentum OP

[rubertoda]

HI,i am aiming to show that 1/(2)^1/2(|spin up>|spin down> + |spin down>|spin up>) is an eigenvalue to the total angular momentum operator in a two-electron system.


I know that i should end up with getting the eigenvalues of the separate spins; L1|spin up> and
L2|spin down> and so on..also i have been suggested to use L+ and L- operators...anyone who can help how to start with the solution? thanks a lot!

Do they mean the total angular momentum to be J=L+s OR L1 + L2??

 

[Oxvillian]

I think the operator in question would be
$$S^2 = ({\bf S_1} + {\bf S_2})^2.$$
Try operating on your state with that. You'll need to express the crossterm in terms of ladder operators.

 

[rubertoda]

Thanks. But wouldn't the total angular momentum operator be like, L or J? I mean, not the total spin operator?


kind regards

 

[Oxvillian]

I think when they say "total" here, they're talking about the total spin angular momentum S₁ + S₂. Not the spin angular momentum plus the orbital angular momentum.

 

[paranormal]

As a scientist, it is important to carefully consider the information provided and to clarify any potential misunderstandings or ambiguity. In this case, it is not clear whether the total angular momentum is denoted as J=L+s or J=L1+L2. This distinction is important because the calculation and interpretation of the eigenvalues will differ depending on the definition of the total angular momentum.

Assuming that J=L+s, the provided expression 1/(2)^1/2(|spin up>|spin down> + |spin down>|spin up>) can be rewritten as 1/(2)^1/2(L1+L2)|spin up>|spin down> + 1/(2)^1/2(L1+L2)|spin down>|spin up>. This can be simplified to 1/2(L1|spin up>|spin down> + L2|spin down>|spin up>) + 1/2(L1|spin down>|spin up> + L2|spin up>|spin down>). We can now apply the L+ and L- operators to each term to obtain the eigenvalues of the total angular momentum operator. The L+ operator will raise the spin state by one unit, while the L- operator will lower it by one unit. This will result in the eigenvalues J+1 and J-1 for the total angular momentum.

On the other hand, if J=L1+L2, the expression will be 1/(2)^1/2(L1+L2)|spin up>|spin down> + 1/(2)^1/2(L1+L2)|spin down>|spin up>, which can be simplified to 1/2(L1|spin up>|spin down> + L2|spin down>|spin up>) + 1/2(L1|spin down>|spin up> + L2|spin up>|spin down>). In this case, the eigenvalues will be L1+L2+1 and L1+L2-1 for the total angular momentum.

In either case, the use of L+ and L- operators is appropriate and will help in obtaining the eigenvalues of the total angular momentum operator. It is important to carefully define the total angular momentum and to use the appropriate operators in the calculations.