In quantum mechanics, electron wave functions often involve spherical harmonics, and the l and m values are related to the angular momentum of the state. The rate at which atoms absorb photons to go from a state with (l1, m1) to a state with (l2, m2) is often proportional to the following integral: ro=2n rcos 0=+1 (Y (0, 6) xY (0, ø)d cos edo үт ym2 0=0 Jcos 0=-1 where x is the x coordinate and * means complex conjugation, not multiplication. And, yes, I wrote x, not z. And, yes, I know that I wrote z on a problem set. This is a slightly different experiment, where the light beam is polarized along a different axis. For each set of (l1,m1, l2,m2) values in the table below, tell me whether the integral is zero or non-zero.
In quantum mechanics, electron wave functions often involve spherical harmonics, and the l and m values are related to the angular momentum of the state. The rate at which atoms absorb photons to go from a state with (l1, m1) to a state with (l2, m2) is often proportional to the following integral: ro=2n rcos 0=+1 (Y (0, 6) xY (0, ø)d cos edo үт ym2 0=0 Jcos 0=-1 where x is the x coordinate and * means complex conjugation, not multiplication. And, yes, I wrote x, not z. And, yes, I know that I wrote z on a problem set. This is a slightly different experiment, where the light beam is polarized along a different axis. For each set of (l1,m1, l2,m2) values in the table below, tell me whether the integral is zero or non-zero.
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Transcribed Image Text:In quantum mechanics, electron wave functions often involve spherical harmonics, and the l
and m values are related to the angular momentum of the state. The rate at which atoms absorb
photons to go from a state with (l1, m1) to a state with (l2, m2) is often proportional to the following
integral:
rcos 0=+1
(Y (0, 4)) ¤¥? (0, 4)d cos 0dø
ymi(A
l1
0=0
Jcos 0=-1
where x is the coordinate and * means complex conjugation, not multiplication. And, yes, I wrote
x, not z. And, yes, I know that I wrote z on a problem set. This is a slightly different experiment,
where the light beam is polarized along a different axis.
For each set of (l1, m1, l2, m2) values in the table below, tell me whether the integral is zero or
non-zero.
I'll let you work out what x is in spherical coordinates. (And don't worry about the value of
r; just give it some non-zero value and then worry about whether the angular factors work out to
make that integral zero or non-zero.)
l1
m1
l2
m2
Zero or non-zero?
1
-1
1
1
1
1
2
1
1
2
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