Divergence in spherical coordinates

ivergence in Spherical Coordinates
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
=
∇
2
2
2
2
2
2
2
2
sin
1
sin
sin
1
1
φ
θ
θ
θ
θ
θ
r
r
r
r
r
r
*Free Particle on a Sphere (
(constant)): Spherical Harmonics
r = r e
I 2I
ˆ
sin
1
sin
sin
1
2
2
)
,
(
ˆ
2
2
2
2
2
2
2
L
H =
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
= −
∇
= −
φ
θ
θ
θ
θ
θ
µ
φ
θ
h
h
where
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
= −
2
2
2
2
2
sin
1
sin
sin
1
ˆ
φ
θ
θ
θ
θ
θ
h
L ,
∂φ
∂
= i
z
h
Lˆ
)
,
(
2
1)
(
)
,
(
2
ˆ
)
,
(
ˆ
2
2
φ
θ
φ
θ
φ
θ
lm
lm
lm
Y
I
l
l
Y
I
Y
+
=
=
h
L
H
Hydrogen(-like) atom
r
Ze
r
V
1
4
)
(
0
2
πε
−
= (Coulombic potential)
Hamiltonian operator
)
(
2
)
,
,
(
ˆ 2
2
r
r +V
∇
= −
µ
φ
θ
h
H
r
Ze
r
r
r
r
r 0
2
2
2
2
2
2
2
2
2
4
sin
1
sin
sin
1
2
2 πε
φ
θ
θ
θ
θ
θ
µ
µ
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
= −
h
h
r
Ze
r
r
r
r
r 0
2
2
2
2
2
2
4
2
ˆ
2 πε
µ
µ
−
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
= −
L
h
where
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
= −
2
2
2
2
2
sin
1
sin
sin
1
)
,
(
ˆ
φ
θ
θ
θ
θ
θ
φ
θ
h
L ,
φ
φ
∂
∂
= i
z
h
)
(
ˆL
Schrödinger equation:
)
,
,
(
4
)
,
(
ˆ
2
1
2
)
,
,
(
)
,
,
(
ˆ
0
2
2
2
2
2
2
φ
θ
ψ
ψ
πε
ψ
φ
θ
µ
ψ
µ
φ
θ
ψ
φ
θ
r
E
r
Ze
r
r
r
r
r
r
r =
−
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
= L
H
h
0
4
)
,
(
ˆ
2
1
2 0
2
2
2
2
2
2
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
−
ψ
πε
ψ
φ
θ
µ
ψ
µ
E
r
Ze
r
r
r
r
r
L
h
0
)
,
(
ˆ
1
4
2 2
2
0
2
2
2
2
=
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
ψ
φ
θ
ψ
πε
µ
ψ
L
h
h
E
r
Ze
r
r
r
r

Separation of variables (
)
,
(
)
(
)
,
,
( φ
θ
φ
θ
ψ
lm
Y
r
R
r = lm
lm
lm
Y
I
l
l
Y
I
Y
2
1)
(
2
ˆ
ˆ
2
2
+
=
=
h
L
H )
0
)
,
(
ˆ
1
)
(
)
,
(
)
(
1
4
2
)
(
)
,
(
2
2
0
2
2
2
2
=
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
φ
θ
φ
θ
πε
µ
φ
θ
lm
lm
lm
Y
r
R
Y
r
R
E
r
Ze
r
r
r
R
r
r
L
h
h
Y
0
)
,
(
1)
(
)
(
)
,
(
)
(
1
4
2
)
(
)
,
(
2
2
0
2
2
2
2
=
+
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
φ
θ
φ
θ
πε
µ
φ
θ
lm
lm
lm
Y
l
l
r
R
Y
r
R
E
r
Ze
r
r
r
R
r
r
h
h
h
Y
0
1)
(
1
4
2
)
(
)
(
1
0
2
2
2
2
=
+
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
l
l
E
r
Ze
r
r
r
R
r
r
r
R
πε
µ
h
0
1
4
2
1)
(
2
0
2
2
2
2
2
2
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
+
−
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
R
E
r
Ze
r
l
l
r
r
R
r
r
πε
µ
µ h
h
"Radial equation"
0
1
4
2
1)
(
2
2
0
2
2
2
2
2
2
2
2
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
+
−
∂
∂
+
∂
∂
R
E
r
Ze
r
l
l
r
r
R
r
r
R
r
πε
µ
µ h
h
0
2
1
4
2
1)
(
2
2
2
0
2
2
2
2
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
+
−
∂
∂
+
∂
∂
R
E
r
e
Z
r
l
l
r
R
r
r
R
h
h
µ
πε
µ
0
8
1
2
1)
(
2
2
0
0
0
2
2
2
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
+
−
∂
∂
+
∂
∂
R
e
a
E
r
a
Z
r
l
l
r
R
r
r
R
πε
where
2
2
0
0
4
e
a
µ
πε h
=
Asymptotic solution as an auxiliary function
0
8
2
0
0
2
2
=
+
∂
∂
asymp
asymp
R
e
a
E
r
R
πε
→ hen (bound state)
cr
asymp
e
r
R
−
∝
)
( w 0
E <
where
cr
e
r
K
r
R
−
= )
(
)
(
2
1
2
0
0
8
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
= e
a
E
c
πε
0
2
1)
(
2
2
2
2
0
2
2
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
+
−
−
′
+
+
′
−
′
′ −
−
−
−
−
−
cr
cr
cr
cr
cr
cr
Ke
c
r
a
Z
r
l
l
Ke
r
c
K e
r
Ke
c
cK e
e
K
0
)
(
1)
(
2
)
(
1
2
)
(
2
0
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ +
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+
′
⎟
⎠
⎞
⎜
⎝
⎛
−
+
′′ r
K
r
l
l
r
c
a
Z
r
K
c
r
r
K

2
2
⎛ ⎛
⎞
( ) ( ) ⎜
Z ⎞
r − cr K ′ r + 2⎜
− c⎟r
− l(
l + 1) ⎟K(
) = 0
K ′′ ( r)
+ 2
⎜ ⎜ ⎟ ⎟
r
⎝ ⎝ a0
⎠ ⎠
K( r)
= r L(
r)
l
⎛ 2Z
⎞
r L′ ( r)
+ (2l
+ 2 − cr)
L′
( r)
+
⎜ − 2c
− 2cl
⎟L(
r)
= 0: Laguerre differential equation
⎝ a0
⎠
Solutions: L l (r)
n
associated Laguerre functions
Solutions (r)exist when n = 1, 2, 3, L and 0 ≤ l ≤ n −1
.
R nl
E n
2 2
2 4
Z ⎛ e ⎞ Z µ e
= − ⎜ ⎟ = −
( a
2
8
2 2 2
n
πε 0a
⎝ 0 ⎠ n 8ε
0 h
0
=
2
0
2
4πε h
µe
= 0.529 Å: Bohr radius): n 2 -fold deg.
1
2
3
l+
Zr
na
⎡
⎤ ⎛ ⎞
−
( n − l −1)!
2Z
2 l 2 + 1⎛
2 ⎞
( ) = −⎢
⎥
⎜
⎟
0 l Zr
Rnl r
r e L +
⎜
⎟
3
n l (associated Laguerre functions)
⎢⎣
2n[(
n + l)!]
⎥⎦
⎝ na0
⎠
⎝ na0
⎠
n = 1
⎛
⎜ E
⎜
⎝
1
= −Z
2
⎛ 2
⎜
e
⎝ 8πε
0a
0
⎞⎞
⎟⎟
⎟
⎠⎠
← No (1-fold) degeneracy
0 :
1 ⎛ Z ⎞
l = L 1 ( x)
= −1
→ R ( ) 2
0
10 r =
⎜ e
a
a
⎟ (m = 0)
⎝ 0 ⎠
3
2
Zr
−
n = 2
⎛
⎜ E
⎜
⎝
2
2 ⎛ 2
Z
= − ⎜
e
4 ⎝ 8πε
0a
0
⎞⎞
⎟⎟
⎟
⎠⎠
← 4-fold degeneracy
3
2
Zr
−
2a0
1 1 ⎛ Z ⎞ ⎛ Zr ⎞
l = 0 : L 2 ( x)
= −2!(2
− x)
→ R20
( r)
=
1 e
2
⎜
a
⎟
⎜ −
0 2a
⎟ (m = 0)
⎝ ⎠ ⎝ 0 ⎠
3
2
Zr
−
2a0
3 1 ⎛ Z ⎞ ⎛ Zr ⎞
l = 1: L 3 ( x)
= −3!
→ R21(
r)
=
e
2 6
⎜
a
⎟
⎜
0 a
⎟ (m = −1, 0, 1)
⎝ ⎠ ⎝ 0 ⎠
n = 3
⎛
⎜ E
⎜
⎝
3
2 ⎛ 2
Z
= − ⎜
e
9 ⎝ 8πε
0a
0
⎞⎞
⎟⎟
⎟
⎠⎠
← 9-fold degeneracy

3
2
Zr
−
3a0
2 2
1 2
2 ⎛ Z ⎞ ⎛ 2Zr
2Z
r ⎞
l = 0 : L 3 ( x)
= −3!(3
− 3x
+ x ) → R30
( r)
=
⎜1
⎟ e
2
3 3
⎜ − +
a
⎟
(m = 0)
⎝
⎜
0 ⎠ 3a0
27a
⎟
⎝
0 ⎠
3
2
Zr
−
3a0
2 2
3 8 ⎛ Z ⎞ ⎛ Zr Z r ⎞
l = 1: L 4 ( x)
= −4!(4
− x)
→ R31(
r)
=
⎜ ⎟ e
2
27 6
⎜ −
a
⎟
(m = −1, 0, 1)
⎝
⎜
0 ⎠ a0
6a
⎟
⎝ 0 ⎠
3
2
2
Zr
−
3a0
5 4 ⎛ Z ⎞ ⎛ Zr ⎞
l = 2 : L 5 ( x)
= −5!
→ R32
( r)
=
e
81 30
⎜
a
⎟
⎜
0 a
⎟ (m = −2, −1, 0, 1, 2)
⎝ ⎠ ⎝ 0 ⎠