Let în be a unit vector in a direction specified by the polar angles (0, ø). Show that the component of the angular momentum in the direction ân is 6.12 L, = sin0 cos pL, + sin0 sin øL,y + cos 0L: = } sin (e-* L, + e* L_) + cos 0 L̟. If the system is in simultaneous eigenstates of L² and L; belonging to the eigen- values I(I + 1)h² and mh, (a) what are the possible results of a measurement of L„? (b) what are the expectation values of L, and L?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let în be a unit vector in a direction specified by the polar angles (0, ¢).
Show that the component of the angular momentum in the direction ân is
6.12
= sin 0 cos øL, + sin0 sin øL, + cos 0L:
= } sin@(e-i® L4 + e* L_) + cos 0 L;.
If the system is in simultaneous eigenstates of L² and L; belonging to the eigen-
values I(1 + 1)ħ² and mħ,
(a) what are the possible results of a measurement of L„?
(b) what are the expectation values of L, and L?
Transcribed Image Text:Let în be a unit vector in a direction specified by the polar angles (0, ¢). Show that the component of the angular momentum in the direction ân is 6.12 = sin 0 cos øL, + sin0 sin øL, + cos 0L: = } sin@(e-i® L4 + e* L_) + cos 0 L;. If the system is in simultaneous eigenstates of L² and L; belonging to the eigen- values I(1 + 1)ħ² and mħ, (a) what are the possible results of a measurement of L„? (b) what are the expectation values of L, and L?
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