Determine the state |n> given results and probabilities [QM]

[roguetechx86]

Homework Statement

In a spin-\frac{1}{2} system all particles are in the state |\psi\rangle. 3 experiments performed and are separate, the results are as follows:

Particle in state |\psi\rangle, measured S_z = \frac{\hbar}{2}, with P=1/4
Particle in state |\psi\rangle, measured S_x = \frac{\hbar}{2}, with P=7/8
Particle in state |\psi\rangle, measured S_y = \frac{\hbar}{2}, with P=\frac{4+\sqrt{3}}{8}

Determine |\psi\rangle in the S_z basis.

Homework Equations

S_z = \frac{\hbar}{2}\left[\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right], S_x = \frac{\hbar}{2}\left[\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right], S_y = \frac{\hbar}{2}\left[\begin{array}{cc} 0 & -i \\ i & 0\end{array}\right]
S = S_x \hat{i} + S_y \hat{j} + S_z \hat{k}

where S_z = \frac{\hbar}{2}\sigma_z

\sigma = \sigma_x \hat{i} + \sigma_y \hat{j} + \sigma_z \hat{k}

The Attempt at a Solution

So, if we are given P_z = 1/4, I would think this implies that
|\psi\rangle = \frac{1}{2}|+z\rangle + \frac{\sqrt{3}}{2}|-z\rangle\qquad \blacksquare
but I don't think this is correct or the whole thing, as I think that |\psi\rangle must satisfy all basis. Also I am not sure that S_x, S_y results are in their |+x\rangle, |+y\rangle respectively, such that the S_x measurement is
|\psi\rangle = \sqrt{\frac{7}{8}}|+x\rangle + \sqrt{\frac{1}{8}}|-x\rangle
or are written as |\pm x\rangle, |\pm y\rangle in the S_z basis

Do I need determine the |\pm x\rangle, |\pm y\rangle states in the S_z basis of the one mentioned earlier?

Should I try to find the eigenstate such that in 3-D Euclidean space
\frac{\hbar}{2}\left[\sigma_x \cos\phi + \sigma_y \sin\phi\right]\left[\begin{array}{c} \langle +z|\mu\rangle \\ \langle -z|\mu\rangle\end{array}\right] = \mu\frac{\hbar}{2}\left[\begin{array}{c} \langle +z|\mu\rangle \\ \langle -z|\mu\rangle\end{array}\right]

I don't want the answer only some direction as to how to proceed, as I am lost.

 

[TSny]

Welcome to PF!:D

So, if we are given P_z = 1/4, I would think this implies that
|\psi\rangle = \frac{1}{2}|+z\rangle + \frac{\sqrt{3}}{2}|-z\rangle\qquad \blacksquare.

You need to allow for possible phase differences in the different components. You can adjust the overall phase of the wavefunction so that the coefficient of $|+z\rangle$ is real and equal to 1/2, but you can't assume that the phase of the $|-z\rangle$ coefficient is simultaneously real.

 

[roguetechx86]

Thanks, feels good to join!

That clue, was extremely helpful, such that
|\psi \rangle = \exp(i \phi_+)\left(\frac{1}{2}|+z \rangle + \frac{\sqrt{3}}{2}\exp(i\phi) |+z \rangle \right)
where \phi = \phi_- - \phi_+. If ignoring the overall phase and rewriting the probabilities of
|\langle +x | \psi \rangle |^2 \to S_z basis
|\langle +y | \psi \rangle |^2 \to S_z basis
you can solve \phi = \frac{\pi}{6}.

I didn't write the complete solution online because going through the motions really helped understanding the problem!

 

[TSny]

Looks good!