Spin Prob 1

Problem 1:(Added by Hang Chen's group)

P is a beam of atoms with spin quantum number 1/2 and zero orbital angular momentum, all with angular momentum ${\frac {\hbar }{2}}$ along the x axis. Q is a beam of similar but unpolarised atoms.

(a) What's the spin state function of P in terms of $\alpha$ and $\beta$, the eigenfunctions of $S_{z}$?

(b) If the two beams are passed separately through a Stern-Gerlach apparatus with its magnetic field along the z axis, is there any difference between the emerging beams in the two cases?

(c) How could the difference between P and Q be detected experimentally?

Solution:

(a) The spin state function $\psi _{p}$ of P is the eigenfunction of $S_{x}$ with eigenvalue $+{\frac {\hbar }{2}}$. So $\psi _{p}={\frac {\alpha +\beta }{\sqrt {2}}}$

(b) No. In each case the two beams emerging from the Stern-Gerlach apparatus would contain equal numbers of atoms.

(c) The difference between P and Q could be determined by passing each beam through a Stern-Gerlach magnet with its magnetic field along the x axis. For beam P all the atoms would emerge in the same channel, corresponding to x-component of spin angular momentum equal to $+{\frac {\hbar }{2}}$. For beam Q two beams would emerge with equal numbers of atoms. For this beam the same result would occur whatever the direction of the magnetic field of the Stern-Gerlach magnet.