The angular part of the integral can be done. All the terms of the wavefunction contain aY 00
andrdoes not depend on angles, so the angular integral just gives 1.
〈ψ|r|ψ〉=1
2
∫∞
0(R 10 −R 20 e−i(E^2 −E^1 )t/ ̄h)∗r(R 10 −R 20 e−i(E^2 −E^1 )t/ ̄h)r^2 drThe cross terms are not zero because of ther.
〈ψ|r|ψ〉=1
2
∫∞
0(
R^210 +R^220 −R 10 R 20
(
ei(E^2 −E^1 )t/ ̄h+e−i(E^2 −E^1 )t/h ̄))
r^3 dr〈ψ|r|ψ〉=1
2
∫∞
0(
R^210 +R 202 − 2 R 10 R 20 cos(
E 2 −E 1
̄ht))
r^3 drNow we will need to put in the actual radial wavefunctions.
R 10 = 2
(
1
a 0) (^32)
e−r/a^0
R 20 =
1
√
2
(
1
a 0)^32 (
1 −
r
2 A 0)
e−r/^2 a^0〈ψ|r|ψ〉 =1
2 a^30∫∞
0(
4 e−^2 r/a^0 +1
2
(
1 −
r
a 0+
r^2
4 a^20)
e−r/a^0− 2
√
2
(
1 −
r
2 a 0)
e−^3 r/^2 a^0 cos(
E 2 −E 1
̄ht))
r^3 dr=
1
2 a^30∫∞
0(
4 r^3 e−a 2 r(^0) +
1
2
r^3 e−ar(^0) −
1
2 a 0r^4 e−ar(^0) +
1
8 a^20r^5 e−ar
0+
(
− 2
√
2 r^3 e− 2 a 3 r(^0) +
√
2
a 0r^4 e− 2 a 3 r
0)
cos(
E 2 −E 1
̄ht))
dr=
1
2 a^30[
24
(a
0
2) 4
+ 3a^40 −1
2 a 024 a^50 +1
8 a^20120 a^50+
(
− 2
√
26
(
2 a 0
3) 4
+
√
2
a 024
(
2 a 0
3) 5 )
cos(
E 2 −E 1
̄ht)]
=
a 0
2[
3
2
+ 3−12 + 15 +
(
− 12
√
2
16
81
+
√
2
a 024
32
243
)
cos(
E 2 −E 1
̄ht)]
=
a 0
2[
3
2
+ 3−12 + 15 +
(
−
√
2
64
27
+
256
√
2
81
)
cos(
E 2 −E 1
̄h
t)]
= a 0[
15
4
+
32
√
2
81
cos(
E 2 −E 1
̄ht