Hi, for my quantum mechanics course I'm confronted with the following question:(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

A spin 1/2 particle has the following eigenstates of S_{z}: |+> = (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 S_{x}in terms of the eigenstates of S_{z}.

2. Relevant 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.

s_{z}=

(1 0)

(0 -1)

s_{x}=

(1 0)

(0 1)

3. 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 S_{z}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 S_{x}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 S_{z}? Hope someone can help!

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# Homework Help: States of a spin 1/2 particle in a magnetic field

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