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

  1. Oct 11, 2014 #1
    1. The problem statement, all variables and given/known data
    In a spin-[itex]\frac{1}{2}[/itex] system all particles are in the state [itex]|\psi\rangle[/itex]. 3 experiments performed and are separate, the results are as follows:

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

    Determine [itex]|\psi\rangle[/itex] in the [itex]S_z[/itex] basis.

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

    where [itex]S_z = \frac{\hbar}{2}\sigma_z[/itex]

    [itex]\sigma = \sigma_x \hat{i} + \sigma_y \hat{j} + \sigma_z \hat{k}[/itex]

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

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

    Should I try to find the eigenstate such that in 3-D Euclidean space
    [tex]\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][/tex]

    I don't want the answer only some direction as to how to proceed, as I am lost.
     
  2. jcsd
  3. Oct 11, 2014 #2

    TSny

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    Homework Helper
    Gold Member

    Welcome to PF!:D
    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.
     
    Last edited: Oct 11, 2014
  4. Oct 13, 2014 #3
    Thanks, feels good to join!

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

    I didn't write the complete solution online because going through the motions really helped understanding the problem!
     
    Last edited: Oct 13, 2014
  5. Oct 13, 2014 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Looks good!
     
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