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**1. Homework Statement**

The spin of an electron is described by a vector: [tex]\psi = \left(\frac{\uparrow}{\downarrow}\right)[/tex] and the spin operator:

[tex]\hat{S} = \hat{S_{x}}i + \hat{S_{y}}j + \hat{S_{z}}k[/tex]

with components:

[tex]\hat{S_{x}} = \frac{\hbar}{2} \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right][/tex]

[tex]\hat{S_{y}} = \frac{\hbar}{2} \left[ \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right][/tex]

[tex]\hat{S_{z}} = \frac{\hbar}{2} \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right][/tex]

__I.__

i. State the normalisation condition for: [tex]\psi[/tex].

ii. Give the general expressions for the probabilities to find: [tex]\hat{S_{z}} = \pm \frac{\hbar}{2}[/tex] in a measurement of: [tex]\hat{S_{z}}[/tex]

iii. Give the general expression for the expectation value of: [tex]\left<\hat{S_{z}}\right>[/tex]

__II.__

Calculate the commutators:

[tex]\left[\hat{S_{y}},\hat{S_{z}}\right][/tex] and [tex]\left[\hat{S_{z}},\hat{S^{2}}}\right][/tex]

.. Are [itex]\hat{S_{y}} [/itex] and [itex]\hat{S_{z}} [/itex] simultaneous observables? Are [itex]\hat{S_{z}}[/itex] and [itex]\hat{S^{2}}[/itex] simultaneous observables?

__III.__

i. Normalise the state: [tex]\frac{1}{1}[/tex]

ii. Calculate the expectation values: [tex]\hat{S_{x}}[/tex], [tex]\hat{S_{y}}[/tex], and [tex]\hat{S_{z}}[/tex] for this normalised state.

**2. Homework Equations**

Within the question details and solution attempts.

**3. The Attempt at a Solution**

__I:__

i. Normalisation condition: [tex]\langle\psi|\psi\rangle[/tex] .. yes?

ii. I have calculated that:

[tex]\left[\hat{S_{x}}, \hat{S_{y}}\right] = i\hbar S_{z}[/tex]

and then know that eigenvalues of [itex]\hat{S_{z}}[/itex] are simply [itex]\frac{\hbar}{2}[/itex] times the eigenvalues of [itex]\sigma_{z}[/itex] (which is just the matrix part of [itex]\hat{S_{z}}[/itex] if that makes sense). Not sure what more to do now here though.

iii. I don't know how to go about doing this part at the moment.

__II.__

I have calculated the commutators as:

[tex]\left[\hat{S_{y}},\hat{S_{z}}\right] = \frac{\hbar^{2}}{2} \left[ \begin{array}{cc} 0 & i \\ i & 0 \end{array} \right][/tex]

and

[tex]\left[\hat{S_{z}},\hat{S^{2}}}\right] = 3 \hbar^{3} \left[ \begin{array}{cc} i & -1 \\ 1 & -i \end{array} \right][/tex]

.. but I don't know in either case if they are simultaneous observables? Or indeed, how I would be able to determine if so for each case?

__III.__

I don't understand what I need to do, as far as normalising the state as required. Once I know how to do this, can then obviously have a go at calculating the expectation values.