# Quantum State Tomography

1. Dec 12, 2009

### phil ess

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

?

3. The attempt at a solution

Let the atoms be:

$$$\left( \begin{array}{ccc} \alpha \\ \beta \end{array} \right)$$$

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 Im 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, im stuck :(

2. Dec 12, 2009

### Redbelly98

Staff Emeritus
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.

3. Dec 13, 2009

### 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 Im not sure I know how to proceed from here :S