Help Stern-Gerlach Experiment and Spherical Harmonics

[lavenderblue]

Hey, this is going to be abit long winded...

In a Stern-Gerlach experiment a beam of hydrogenic atoms in the l=1 state traveling along the y-axis is first passed through an inhomogeneous magnetic field in the z-direction to yield three beams corresponding to three eigenstates Y(z,1), Y(z,0) and Y(z,-1). One of the beams is selected and then passed through an inhomogeneous magnetic field in the x-direction. Discuss the outcome of the experiment in each of the folowing cases:

(a) The Y(z,-1) beam is selected
(b) The Y(z,0) beam is selected
(c) The Y(z,1) beam is selected

In each case compute the probability of the beam going into the states of a,b, and c.

Could you please show your working, am so confused! Thanks :)

 

[BigJDubs]

(a) The Y(z,-1) beam is selected: In this case, the beam will be split into three components by the inhomogeneous magnetic field in the x-direction. These components will correspond to the following eigenstates: Y(x,-1/2), Y(x,+1/2), and Y(x, -3/2). The probability of the beam going into each of these states is equal to 1/9. (b) The Y(z,0) beam is selected: In this case, the beam will be split into three components by the inhomogeneous magnetic field in the x-direction. These components will correspond to the following eigenstates: Y(x, -1/2), Y(x, 0), and Y(x, +1/2). The probability of the beam going into each of these states is equal to 1/3. (c) The Y(z,1) beam is selected: In this case, the beam will be split into three components by the inhomogeneous magnetic field in the x-direction. These components will correspond to the following eigenstates: Y(x, +1/2), Y(x, +3/2), and Y(x, +5/2). The probability of the beam going into each of these states is equal to 1/9.