Quantum Mechanics pp 163-166 | Cite as

# Another Example: Successive Polarization Filters for Beams of Spin *s*= 1/2 Particles

## Abstract

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?

## Keywords

Pure State World Scientific Unitary Transformation Azimuth Angle Beam Line## Preview

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