How Does Stern-Gerlach Experiment Determine Proton Spin Orientation?

[ihatelolcats]

Homework Statement

A beam delivering protons is sent through a stern-gerlach splitter oriented to ask whether the spin is oriented parallel to the y axis. What fraction of protons have spin down with respect to the y axis?
$$\chi$$=$$\stackrel{1}{\sqrt{17}}$$ $$\stackrel{4}{i}$$
the 4, i thing is the spinor matrix.

Homework Equations

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The Attempt at a Solution

is it just the expectation value, <S_y>? that doesn't seem right at all, but the textbook is not very helpful
my other idea is that its just 1/2 and I've wasted a couple hours

ps: sorry my latex skill is poor

 

[vela]

What exactly is $\chi$? You just kind of throw it out there without saying what it has to do with the problem.

 

[ihatelolcats]

sorry, it is the state the protons are in

 

[vela]

Expand $\chi$ in terms of the eigenstates of S_y.

 

[ihatelolcats]

i'm not sure what you mean by expand in terms of S_y
i have $$\chi$$ = 1/sqrt(17) ($$\stackrel{4}{0}$$) + 1/sqrt(17) ($$\stackrel{0}{i}$$)

S_y = $$\hbar$$ /2 $$\stackrel{0}{i}$$ $$\stackrel{-i}{0}$$

where to go from here?
i'm still going to have to take an expectation of something to get a probability...

do you know of a reference or something online i can use to learn how to do this? i have griffiths intro to QM textbook, but like i said it isn't helpful on this. due friday heh.

 

[vela]

i'm not sure what you mean by expand in terms of S_y
i have $$\chi$$ = 1/sqrt(17) ($$\stackrel{4}{0}$$) + 1/sqrt(17) ($$\stackrel{0}{i}$$)

It's actually

$$\chi = \frac{4}{\sqrt{17}} \begin{pmatrix}1 \\ 0\end{pmatrix} + \frac{1}{\sqrt{17}} \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

where, presumably,

$$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$

is the S_z spin-up eigenstate and

$$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

is the S_z spin-down eigenstate. You can determine the probability of finding the particle in a spin state by squaring the modulus of the corresponding coefficient. For example, in this case, the probability of finding it in the spin-up state would be 16/17, and in the spin-down state, 1/17.

But this problem is asking you to find the probabilities using the y-axis instead of the z-axis, so what you want to do is find the eigenstates of S_y by diagonalizing the matrix

$$\hat{S}_y = \frac{\hbar}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$$

and express $\chi$ as a linear combination of them to determine the coefficients you need. In linear-algebra-speak, you want to change basis from the S_z basis to the S_y basis.

This is effectively what you always do to find probabilities of a measurement. You find the eigenstates of the observable, express the state in terms of those eigenstates, and calculate the probabilities by squaring the modulus of the coefficients.

i'm still going to have to take an expectation of something to get a probability...

do you know of a reference or something online i can use to learn how to do this? i have griffiths intro to QM textbook, but like i said it isn't helpful on this. due friday heh.

An expectation value just gives you an average. It doesn't give you a probability.

 

[ihatelolcats]

wow. thanks so much! :)