Imagine you have a beam of spin 1/2 particles moving in the y-direction. We can set up an inhomogeneous magnetic field to interact with the particles, separating them according to their spin component in the direction of the magnetic field, B·Sˆ. This is the Stern-Gerlach experiment (a) You set up a magnetic field in the z-direction. As the beam of particles passes through it, it splits in two equal beams: one goes up, corresponding to the spin-up particles (those whose Sˆ z eigenvalue was +ℏ 2 ), and the other goes down, corresponding to the spin-down particles. Now, you take the beam that went up and pass it through another magnetic field in the z-direction. Does the beam split? If so, what fraction of the particles go to each side?

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Imagine you have a beam of spin 1/2 particles moving in the y-direction. We can set up an inhomogeneous magnetic field to interact with the particles, separating them according to their spin component in the direction of the magnetic field, B·Sˆ. This is the Stern-Gerlach experiment

(a) You set up a magnetic field in the z-direction. As the beam of particles passes through it, it splits in two equal beams: one goes up, corresponding to the spin-up particles (those whose Sˆ z eigenvalue was +ℏ 2 ), and the other goes down, corresponding to the spin-down particles. Now, you take the beam that went up and pass it through another magnetic field in the z-direction. Does the beam split? If so, what fraction of the particles go to each side?

(b) Instead, you pass the beam through a z-field, take the beam that went up, and pass it through a magnetic field in the x-direction. Does the beam split? If so, what fraction of the particles go to each side?

(c) You select one of the beams from part b above, and pass it through another magnetic field in the z-direction. Does the beam split? If so, what fraction of the particles go to each side? Compare with part a and explain.

Suppose we start with N particles. We first pass them through a magnetic field in the z-direction, and block the beam that goes down. After this process, you find that only N 2 particles remain. They then go through a magnetic field in the x-z plane, an angle θ from the z-axis, and the beam that goes against the direction of the field is blocked. Then you have a magnetic field in the z-direction again, and block the beam that goes up this time. How many particles come out? Compare with the case without the middle magnetic field.