212
3 Quantum Mechanics – II
Writing sin
2
θ = 1−cos
2
θ and simplifying we get u
∗ u =
2
9
A
2
3 e
−2x x
4 which
is independent of both θ and ϕ. Therefore the 3d functions are spherically
symmetrical or isotropic.
3.71 ψ 100 =
πa
3
0
−
1
2 exp
−
r
a 0
The probability p of finding the electron within a sphere of radius R is
P =
R
0
|ψ 100 |
2
.4πr
2 dr =
4π
πa
3
0
R
0
r
2 exp
−
2r
a 0
dr
Set
2r
a0
= x; dr =
a 0
2
dx
P =
4
a
3
0
·
a
2
0
4
·
a 0
2
x
2 e
−x dx =
1
/ 2
x
2 e
−x dx
Integrating by parts
P =
1
/ 2 [−x
2 e
−x
+ 2
xe
−x dx]
=
1
/ 2
−x
2 e
−x
+ 2
−x e
−x
+
e
−x dx
=
1
/ 2
−x
2 e
−x
− 2xe
−x
− 2e
−x
2R/a 0
0
=
1
/ 2
−
2R
a 0
2
exp
−
2R
a 0
− 2
2R
a 0
exp
−
2R
a 0
−2 exp
−
2R
a 0
+ 2
P = 1 − e
−
2R
a 0
1 +
2R
a 0
+
2R
2
a
2
0
3.72 The hydrogen wave function for n = 2 orbit is
ψ 200 =
1
4
2πa
3
0
−
1
2
2 −
r
a 0
e
−r/2a0
The probability of finding the electron at a distance r from the nucleus
P = |ψ 200 |
2
· 4πr
2
=
1
8
r
2
a
3
0
2 −
r
a 0
2
exp
−
r
a 0
Maxima are obtained from the condition d p/dr = 0.
The maxima occur at r = (3 −
√
3)a 0 and r = (3 +
√
3)a 0 while minimum
occur at r = 0, 2a 0 and ∞ (Fig. 3.25) (the minima are found from the condition p = 0).
3.73 (a) hν = 13.6 Z
2
mμ
m e
1
2 2 −
1
3 2
Put m μ = 106 MeV (instead of 106.7 MeV for muon)
Z = 15 and m e = 0.511 MeV