Multiple Electron Spin Measurements

[Uncertain Pen]

Homework Statement

The z-component of the spin of an electron is measure and the value h-bar/2 is found. Immediately afterwards, the spin along a direction making an angle θ with the z-axis is measured. What are the possible outcomes of this second measurement and with which probabilities do they arise?

Homework Equations

None given, but I am working with the spin operators. The equations I have been dealing with are:

Spin measured about axis n:
n(dot)S = h-bar/2 * [ cosθ, sinθ*exp(i*phi); sinθ*exp(-i*phi), -cosθ] (2x2 matrix)

S|ψ> = h-bar/2 |ψ>

Spin about x,y,z axes:
S_x |ψ> = h-bar/2 [0, 1; -1, 0]
S_y |ψ> = h-bar/2 [0 -i; i, 0]
S_z |ψ> = h-bar/2 [1, 0; 0, -1]

The Attempt at a Solution

So far I've applied the formula for spin about an axis, assuming phi = 0 (since only and angle of θ is mentioned in the description, I assume it is planar), getting

h-bar/2 * [cosθ, sinθ; sinθ, -cosθ]

However, from there, I'm not sure how to reduce this into distinct spin values, and the probabilities of each being chosen. If anyone could help to point me in the right direction here, I would be extremely grateful - thank you!

 

[TSny]

Hello, Uncertain Pen. Welcome to PF!

So far I've applied the formula for spin about an axis, assuming phi = 0 (since only and angle of θ is mentioned in the description, I assume it is planar), getting

h-bar/2 * [cosθ, sinθ; sinθ, -cosθ]

The two eigenvectors of this spin matrix will represent the states of spin up and spin down along the direction defined by θ. See if you can find them. Also think about how to write the state vector corresponding to the outcome of the first measurement (along the z-axis). Finally, think about how to use these three vectors to calculate the probability outcomes of the second measurement.