States of a spin 1/2 particle in a magnetic field

  • Thread starter ulcrid
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Hi, for my quantum mechanics course I'm confronted with the following question:

Homework Statement


A spin 1/2 particle has the following eigenstates of Sz: |+> = (1 0) and |-> = (0 1). A magnetic field is pointing in the z direction, B = (0,0,B). The Hamiltonian is H = -B * n, with n = -(e/mc)S and S = h/4pi * s (with B, n, S, and s vectors and H of course an operator).

The questions are a) to find the normalized energy eigenstates and eigenvalues and b) to find the normalized eigenstates and eigenvalues of Sx in terms of the eigenstates of Sz.

Homework Equations


The Pauli spin matrices are most important here, for a) only the z component is relevant as the magnetic field only has a z component and the Hamiltonian is defined as -B * n. For b) I think the x component is also relevant.
sz=
(1 0)
(0 -1)
sx=
(1 0)
(0 1)

The Attempt at a Solution


I think I got the most far on question a). I calculated H and found it to be H = (ehB/4pi*mc)*
(1 0)
(0 -1)
Then, using the determinant of (H - lambda * I) I calculated the eigenvalues of H: lambda = +/- (ehB/4pi*mc). After that, I used the eigenvalue equation Av = lambda*v to find the eigenvectors (1 0) and (0 1). I don't think that should be surprising though because somewhere in the book (Introductory Quantum Mechanics by Liboff) it says that Sz and H have the same eigenfunctions.
But are these normalized? And are these values/vectors correct?

On question b) I don't really know what to do - if I try finding out the eigenstates of Sx I get stuck on the vectors. I get equations like h/4pi (b a) = (a b) which doesn't have a solution except for a = b = 0 which obviously is incorrect. How do I find the correct eigenstates? And how do I then find out how to write them as combinations of eigenstates of Sz? Hope someone can help!
 

Answers and Replies

  • #2
vela
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The Attempt at a Solution


I think I got the most far on question a). I calculated H and found it to be H = (ehB/4pi*mc)*
(1 0)
(0 -1)
Then, using the determinant of (H - lambda * I) I calculated the eigenvalues of H: lambda = +/- (ehB/4pi*mc). After that, I used the eigenvalue equation Av = lambda*v to find the eigenvectors (1 0) and (0 1). I don't think that should be surprising though because somewhere in the book (Introductory Quantum Mechanics by Liboff) it says that Sz and H have the same eigenfunctions.
But are these normalized? And are these values/vectors correct?
If the vectors are normalized, they'd have a magnitude of 1. What are the magnitude of your vectors?
On question b) I don't really know what to do - if I try finding out the eigenstates of Sx I get stuck on the vectors. I get equations like h/4pi (b a) = (a b) which doesn't have a solution except for a = b = 0 which obviously is incorrect. How do I find the correct eigenstates? And how do I then find out how to write them as combinations of eigenstates of Sz? Hope someone can help!
You need to show more details of your work. You might just be dropping a constant somewhere.

The matrix representation of Sx you're using is with respect to basis of the Sz eigenstates, so the eigenvectors you find will already be representations in terms of the eigenstates of Sz.
 
  • #3
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Well, the magnitude of the vectors seems to be 1. After all, |(1 0)|2 is 1. So that would be okay?

For the second question I used the Pauli-spinmatrix of Sx:
Code:
[U]h[/U]   (0 1)
4pi (1 0)

Then, I can use det(Sx-lambda*I) = 0 to get lambda2-1 = 0 so lambda is either 1 or -1.
Using that in the eigenvalue equation I get Sx*v = +/- v
Which gets me to
Code:
[U]h[/U]   (0 1)(a) = +/- (a)
4pi (1 0)(b)       (b)
Which is equal to
Code:
[U]h[/U]   (b) = +/- (a)
4pi (a)       (b)
Which in my view is only solvable for a = b = 0. ?
 
  • #4
vela
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Your eigenvalues should be [itex]\pm\hbar/2[/itex]. Can you see the mistake you made in calculating them? If you fix those, the eigenvectors should work out for you.
 
  • #5
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Ah, yes, I see where I went wrong. And because I used the Pauli-spinmatrix for Sx, which apparently already is in the Sz/S2 representation, these vectors and eigenvectors are already in terms of the eigenstates of Sz?
 

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