Another Example: Successive Polarization Filters for Beams of Spin s= 1/2 Particles
So far, our first example of a unitary transformation from one basis to another involved a finite-dimensional unitary submatrix. Let us consider one more example of this type, an even simpler example involving spin s = 1/2 particles, hence, a 2 × 2-dimensional transformation. Suppose we have a beam of spin s = 1/2 particles. They can be prepared, so all are in a state of definite spin orientation, say, with m s = +1/2, or with m s =− 1/2, along some specific z-direction in 3-D space by passing the beam through a polarization filter. The historically first such filter is that employed by Stern and Gerlach involving a set of three magnets, with nonuniform magnetic fields, placed in succession along the beam line, so a set of baffles can eliminate the particles with one of the two spin orientations. Other types of sophisticated polarization filters exist. (For a reference to modem polarization filters, see, e.g., Polarized Beams and Polarized Gas Targets, Hans Paetz gen. Schieck and Lutz Sydow, eds. World Scientific, 1996). We will assume the filter is perfect and prepares particles in a pure state of very definite m s along a specific z-direction. Suppose the first such filter is followed with a second filter, identical to the first, but now with its new z’ axis oriented along some new direction, given by polar and azimuth angles, θ and φ, relative to the original x, y, z axes, and set for some definite m’s along the new direction. What fraction of the s = 1/2-particles will pass through the second filter?
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