Spin hamiltionian with respect to certain basis

[Heimisson]

Homework Statement

A particle with spin 1/2 and magnetic moment is in a magnetic field B=B_0(1,1,0). At time t=0 the particle has the spin $$1/2 \hbar$$ in the z direction.

i) Write the hamiltonian with respect to the basis that is defined by the eigenvectors of $$\widehat{S}_z$$

Homework Equations

$$\widehat{S}_z = \hbar /2<br /> \left(<br /> \begin{array}{ c c }<br /> 1 & 0 \\<br /> 0 & -1<br /> \end{array} \right)$$

The Attempt at a Solution

So finding the hamiltonian is trivial:
$$H=-\gamma \textbf{B} \cdot \textbf{S} =<br /> = B_0 \hbar /2<br /> \left(<br /> \begin{array}{ c c }<br /> 0 & 1-i \\<br /> 1+i & 0<br /> \end{array} \right)$$
if my calculations are right. But what I don't really don't understand is what I wrote in the bold font. I thought I had to find the matrix A so:

if:
$$B=<br /> \left(<br /> \begin{array}{ c c }<br /> 0 & 1-i \\<br /> 1+i & 0<br /> \end{array} \right)$$

$$A \widehat{S}_z = B \Rightarrow A = B \widehat{S}_z^{-1}$$

but this is really just a shot in the dark I'm not really sure why I should do this.
It would be great if someone could shed some light on this I'm sort of rusty in linear algebra of finite vectors.

 

[grey_earl]

With respect to the basis defined by the eigenvectors of S_z is just the usual basis of spin up = (1,0) and spin down = (0,1), for which the spin matrices are the Pauli matrices, i.e.
S_z = (1,0)(0,-1). In general you define S_z (and analogue S_x and S_y) by S_z |spin up> = |spin up> and S_z |spin down> = - |spin down>, so your explicit matrix expression for S_z changes if you use a basis of, say, 1/sqrt(2)(up+down) = (1,0) and 1/sqrt(2)(up-down) = (0,1).
(as an exercise, you may calculate S_z in that basis)