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## Homework Statement

The evolution of a particular spin-half particle is given by the Hamiltonian [itex]\hat{H} = \omega\hat{S}_{z},[/itex] where [itex]\hat{S}_{z}[/itex] is the spin projection operator.

a) Show that [itex]\upsilon = \frac{1}{\sqrt{2}}\begin{pmatrix}

e^{-i\frac{\omega}{2}t}\\

e^{i\frac{\omega}{2}t}

\end{pmatrix} [/itex] is a solution to the Schrodinger equation.

b) Calculate [itex]<\hat{S}_{x}>[/itex] as a function of time with respect to this state.

We are told

[itex]\hat{S}_{z} = \frac{\bar{h}}{2}\begin{pmatrix}

1&0\\

0&-1

\end{pmatrix}, [/itex][itex]\hat{S}_{x} = \frac{\bar{h}}{2}\begin{pmatrix}

0&1\\

1&0

\end{pmatrix}[/itex]

## Homework Equations

det([itex]\hat{S}_{z} - λI)=0[/itex]

## The Attempt at a Solution

This was a previous exam example - we went over it in class but I got a little bit lost. I know it has to do with eigenvalues and eigenvectors, and I can get up to a certain point but then I get stuck, and I've no clue about where to start for b, we didn't get time to do that in class.

I get that det([itex]\hat{S}_{z} - λI)=0[/itex], and I know that ([itex]\hat{S}_{z} - λI) = \begin{pmatrix}

\frac{\bar{h}}{2}-λ&0\\

0&-\frac{\bar{h}}{2}-λ

\end{pmatrix}.[/itex] Substituting this into det([itex]\hat{S}_{z} - λI)=0[/itex] gives [itex]\frac{-\bar{h}^{2}}{4}+λ^{2}=0.[/itex] Solving for λ gives [itex]λ=\frac{\bar{h}}{2}[/itex] or [itex]λ=-\frac{\bar{h}}{2}[/itex]. Then I substitute this back in, so that if [itex]λ=\frac{\bar{h}}{2}[/itex] then [itex] \begin{pmatrix}

0&0\\

0&-\bar{h}

\end{pmatrix} Z=0[/itex] where Z is some vector. Also if [itex]λ=-\frac{\bar{h}}{2}[/itex] then [itex] \begin{pmatrix}

\bar{h}&0\\

0&0

\end{pmatrix} Z=0[/itex] where Z is some vector. This is where I get stuck - I can't seem to solve for vector Z, which takes the form of \begin{pmatrix}a\\b\end{pmatrix}, without just ending up with zeros. Not sure how to find a meaningful value of Z. And even if I had one, not sure how to bring that around to prove [itex]\upsilon = \frac{1}{\sqrt{2}}\begin{pmatrix}

e^{-i\frac{\omega}{2}t}\\

e^{i\frac{\omega}{2}t}

\end{pmatrix} [/itex] is a solution to the Schrodinger equation.