2. Spinors and rotations (a) For a function f(ø) that can be expanded in a Taylor series, show that f(ø + 4) = eiL=4/h f(6) where y is an arbitrary angle. For this reason, Lz/ħ is called the generator of rotations about the z-axis. More generally, L.ân/ħ is the generator of rotations about the direction în, in the sense that exp(iL.îng/ħ) effects a rotation through angle y (in the right hand sense) about the axis în. In the case of spin, the generator of rotations is S.în/ħ. In particular, for spin 1/2., tells us how spinors rotate. (b) Construct the (2x2) matrix representing rotation by 180° about the x-axis and show that it converts spin up (x+) into spin down (x-) as you would expect. (c) Construct the matrix representing the rotation by 90° about the y-axis, and check what it does to X+·
2. Spinors and rotations (a) For a function f(ø) that can be expanded in a Taylor series, show that f(ø + 4) = eiL=4/h f(6) where y is an arbitrary angle. For this reason, Lz/ħ is called the generator of rotations about the z-axis. More generally, L.ân/ħ is the generator of rotations about the direction în, in the sense that exp(iL.îng/ħ) effects a rotation through angle y (in the right hand sense) about the axis în. In the case of spin, the generator of rotations is S.în/ħ. In particular, for spin 1/2., tells us how spinors rotate. (b) Construct the (2x2) matrix representing rotation by 180° about the x-axis and show that it converts spin up (x+) into spin down (x-) as you would expect. (c) Construct the matrix representing the rotation by 90° about the y-axis, and check what it does to X+·
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Transcribed Image Text:2. Spinors and rotations
(a) For a function f(ø) that can be expanded in a Taylor series, show that
f(o +4) = e²L±p/hf(6)
where p is an arbitrary angle. For this reason, Lz/ħ is called the generator of
rotations about the z-axis.
More generally, L.ân/ħ is the generator of rotations about the direction în, in the
sense that exp(iL.ng/h) effects a rotation through angle y (in the right hand
sense) about the axis în. In the case of spin, the generator of rotations is S.în/ħ.
In particular, for spin 1/2.,
X = ei(oî)g/2x
tells us how spinors rotate.
(b) Construct the (2x2) matrix representing rotation by 180° about the x-axis and
show
it converts spin up (x+)
spin down (x-) as you would expect.
(c) Construct the matrix representing the rotation by 90° about the y-axis, and check
what it does to X+·
(d) Construct the matrix representing rotation by 360° about the z-axis. Discuss the
meaning of the answer.
(e) Show that
ei(o-î)e/2 = cos(p/2) + i(în · o) sin(4/2)
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