Spin state function of a beam of particles in terms of eigenfunctions.

[Mithra]

Hi there, I apologise that I should probably know this/its a stupid question but I seem to have forgotten all physics over the holiday and so any help would be great!

I have been told that there is a beam of atoms with spin quantum number 1/2 and zero orbital angular momentum, with spin +1/2 along the x axis. I am then asked what the spin state function of this beam is in terms of eigenfunctions of $\hat{S}^2$ and $\hat{S}_z$, being the kets |$1/2 , m_{s_z} = 1/2>$ and |$1/2 , m_{s_z} = -1/2>$

Would I be right in assuming that the answer is simply
$\phi_s = \frac{1}{\sqrt{2}}|1/2 , m_{s_z} = 1/2> + \frac{1}{\sqrt{2}}|1/2 , m_{s_z} = 1/2>$
because knowing the spin in the x direction doesn't tell you about the z direction (as they are incompatible observables) or is it more complex than that? Any help/advice greatly appreciated, thanks.

 

[vela]

You want to find the eigenstates of S_x in terms of the eigenstates of S_z. Your answer is a good guess, but there could be a different relative phase between the two terms.

 

[Mithra]

Thanks for the help :).

I got the eigenstates of $S_x$ in terms of $S_z$ as

$\frac{1}{\sqrt{2}} (|\frac{1}{2},\frac{1}{2}> + |\frac{1}{2},\frac{-1}{2}>$
and
$\frac{1}{\sqrt{2}} (|\frac{1}{2},\frac{1}{2}> - |\frac{1}{2},\frac{-1}{2}>$

With the first one corresponding to the same eigenvalue as the positive spin in the x direction given by $S_x$. Is that anything like what the correct answer would be? (I'm guessing that it should be similar to if the beam is not polarised as the following questions suggest that they would not be differentiated by a Stern-Gerlach experiment with the magnetic field along z.).

I don't know how much you're actually allowed to say "yes that's the correct answer" but thanks for the help anyway ;).

 

[vela]

Thanks for the help :).

I got the eigenstates of $S_x$ in terms of $S_z$ as

$\frac{1}{\sqrt{2}} (|\frac{1}{2},\frac{1}{2}> + |\frac{1}{2},\frac{-1}{2}>$
and
$\frac{1}{\sqrt{2}} (|\frac{1}{2},\frac{1}{2}> - |\frac{1}{2},\frac{-1}{2}>$

With the first one corresponding to the same eigenvalue as the positive spin in the x direction given by $S_x$. Is that anything like what the correct answer would be?

Yup, that's it.

(I'm guessing that it should be similar to if the beam is not polarised as the following questions suggest that they would not be differentiated by a Stern-Gerlach experiment with the magnetic field along z.).

I don't know how much you're actually allowed to say "yes that's the correct answer" but thanks for the help anyway ;).

 

[paranormal]

Hello! Your understanding is correct. The spin state function of this beam of particles can be written as a linear combination of the eigenfunctions of \hat{S}^2 and \hat{S}_z, with equal coefficients for both spin states. This is because the particles have a spin quantum number of 1/2, meaning they can either have spin +1/2 or -1/2 along the z axis. However, since the particles have zero orbital angular momentum, the eigenfunctions for \hat{S}^2 and \hat{S}_z are the same, so the coefficients for both spin states are equal.

In general, the spin state function can be written as a linear combination of eigenfunctions with different coefficients, depending on the specific spin state of the particles. But in this case, with only one spin state in the x direction, the coefficients are equal. I hope this helps clarify things for you. Keep up the good work in your studies!