(a) For a function f(ф) that can be expanded in a Taylor series, show that(where φ is an a...

(a) For a function f(ф) that can be expanded in a Taylor series, show that

(where φ is an arbitrary angle). For this reason,  is called the generator of rotations about the z-axis Hint: Use Equation 4.129, and refer to Problem 3.39.

More generally,  is the generator of rotations about the direction , in the sense that exp() effects a rotation through angle φ (in the right-hand sense) about the axis . In the case of spin, the generator of rotations is . In particular, for spin 1/2

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tells us how spinors rotate.

(b) Construct the (2 × 2) matrix representing rotation by 180° about the x-axis, and show that it converts “spin up” (χ+) into "spin down" (χ‒), as you would expect.

(c) Construct the matrix representing rotation by 90° about the y-axis, and check what it does to χ+

(d) Construct the matrix representing rotation by 360° about the z-axis. If the answer is not quite what you expected, discuss its implications.

(e) Show that

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