What are the phase factors for the atoms in this quantum state?

[phil ess]

Homework Statement

Assume there is a source of some pre-selected atoms. When measuring atoms of that source in a Stern-Gerlach, you find the following probabilities for a spin-up result:

x-direction 5/6
y-direction 5/6
z-direction 1/3

Which state would you ascribe to the source?

Homework Equations

?

The Attempt at a Solution

Let the atoms be:

\[ \left( \begin{array}{ccc}<br /> \alpha \\<br /> \beta \end{array} \right)\]

Decomposing in the z-basis we get that the probability amplitudes for up and down are |\alpha|^2 and |\beta|^2, respectively.

From this we find that \alpha = 1/\sqrt{3} and \beta = \sqrt{2/3}

Ok this is where I am confused, my textbook says we need to do a measurement in all 3 directions, but I got these values with just the z result?

If someone could explain this to me itd be a big help, I am stuck :(

 

[Redbelly98]

Homework Statement

Assume there is a source of some pre-selected atoms. When measuring atoms of that source in a Stern-Gerlach, you find the following probabilities for a spin-up result:

x-direction 5/6
y-direction 5/6
z-direction 1/3

Which state would you ascribe to the source?

Homework Equations

?

The Attempt at a Solution

Let the atoms be:

\[ \left( \begin{array}{ccc}<br /> \alpha \\<br /> \beta \end{array} \right)\]

Decomposing in the z-basis we get that the probability amplitudes for up and down are |\alpha|^2 and |\beta|^2, respectively.

From this we find that \alpha = 1/\sqrt{3} and \beta = \sqrt{2/3}

Ok this is where I am confused, my textbook says we need to do a measurement in all 3 directions, but I got these values with just the z result?

If someone could explain this to me itd be a big help, I am stuck :(

Be careful, you have not actually found α and β. You have found |α| and |β|. There is a complex phase factor still to be accounted for.

 

[phil ess]

Hmmm, ok well what I have here is, if |a|² + |b|² = 1, then |a|² - |b|² is a number between 1 and -1, so we write:

|a|² - |b|² = cos 2x, where x is between 0 and pi/2

then

|a|² = 1/2 (|a|² + |b|²) + 1/2 (|a|² - |b|²)
|a|² = 1/2 + 1/2 (cos 2x) = cos²x

and similarly

|b|² = sin²x

so then

\alpha = e^i^\varphi cos x
\beta = e^i^\phi sinx

and then these are the phase factors you are talking about? I know they have magnitude 1, but I am not sure I know how to proceed from here :S