Quantum mechanics and SG device.(eigenvects)

[ttttrigg3r]

Homework Statement

This is a problem straight from my homework.
Imagine that you have a box that emits quantons that have a definite but unknown spin state. If we run quantons from this box through an Stern Gerlach (z-axis) device, we find that 20 percent of the electrons come out the plus channel and 80 percent from the minus channel. If we run quantons from the same box through an Stern Gerlach (x-axis) device, we find that 50 percent of the electrons come out of each channel. If we run quantons from the box through an Stern Gerlach (y-axis) we find that 90 percent come out the plus channel and 10 percent out of the minus channel. Find a quantum state vector for quantons emerging from the box that is consistent with these data

Homework Equations

The outcome probability rule states: the probability that the quanton's state will collapse after going through the SG device is the "absolute square of the inner-product" of the original and the result's eigenvector.

The Attempt at a Solution

This is what I have so far:
The original spin state of the quanton is unknow, we call it |ψ>=[ψ1; ψ2]
where psi1 and 2 are imaginary numbers: ψ1=a+ib and ψ2=c+id

this is where I get stuck. i know that there is z, y, and x SG devices and this is the table of spin eigenvectors.
|+z>=[1;0]
|-z>=[0;1]
|+y>=[sqrt(1/2);isqrt(1/2)]
|-y>=[isqrt(1/2);sqrt(1/2)]
|+x>=[sqrt(1/2);sqrt(1/2)]
|-x>=[sqrt(1/2);-sqrt(1/2)]

Can someone help me set up the equations to solve this? I asked my professor and she told me that there are 4 equations. one each for z,y,x and one for the normalized condition.

 

[ttttrigg3r]

Can I bump this to the top? Is this in the right section, introductory physics, or should I post this in the advanced physics section.

 

[Greg-ulate]

Your state vector looks like ψ = [a+ib;c+id]. It has to produce 3 known expectation values and it needs to be normalized. The expectation values are the observable features responsible for the percentages of positives and negatives given in the problem. So you end up with 4 equations:
<ψ|S_x|ψ> = X_average
<ψ|S_y|ψ> = Y_average
<ψ|S_z|ψ> = Z_average
<ψ|ψ> = 1

You should be able to figure out the average value based on the percentages they give you in the beginning, for example, a hypothetical SG experiment examining a state $$\varphi = \sqrt{.7} |\uparrow\rangle + \sqrt{.3} |\downarrow\rangle$$ would find 70% of the electrons in the positive channel and 30% in the negative channel, corresponding to an expectation value $$\langle\varphi | S_z | \varphi \rangle = \frac{\hbar}{2}\langle\varphi|\sigma_z|\varphi \rangle = \frac{\hbar}{2}(.7-.3) = \frac{2\hbar}{10}=Z_{Average}$$ Find the corresponding expectation values for the percentages given for X, Y, and Z in the problem and plug them into your system of equations. Use some matrix algebra to transform into a system of polynomial expressions and solve for the coefficients.

Once you have solved for a,b,c,d, notice that there is an additional degree of freedom in the overall phase of the state vector. You can use this to make your state vector look a little prettier by fixing the first element in the vector to be 1 (what i did was add another equation to my solver that set b = 0 so my ψ1 was real)

You can check that you have the right state vector by verifying the probability you expect for each axis' positive and negative eigenvector
$$|\langle +X |\Psi\rangle|^2 = .5 \\|\langle +Z |\Psi\rangle|^2 = .2$$