When writing the time independent Schr\"odinger equation in spherical coordinates, we need to plug the Laplacian in Spherical Coordinates into the time independent Schr\"odinger equation . The Laplacian was found to be
∇
s
p
h
2
=
1
r
2
∂
∂
r
(
r
2
∂
∂
r
)
+
1
r
2
s
i
n
θ
∂
∂
θ
(
s
i
n
θ
∂
∂
θ
)
+
1
r
2
s
i
n
2
θ
∂
2
∂
ϕ
2
{\displaystyle \nabla _{sph}^{2}={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial }{\partial r}}\right)+{\frac {1}{r^{2}sin\theta }}{\frac {\partial }{\partial \theta }}\left(sin\theta {\frac {\partial }{\partial \theta }}\right)+{\frac {1}{r^{2}sin^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}}
Using the three dimensional Schr\"odinger equation we then have
H
^
ψ
(
r
,
θ
,
ϕ
)
=
−
ℏ
2
2
m
[
1
r
2
∂
∂
r
(
r
2
∂
ψ
(
r
,
θ
,
ϕ
)
∂
r
)
+
1
r
2
s
i
n
θ
∂
∂
θ
(
s
i
n
θ
∂
ψ
(
r
,
θ
,
ϕ
)
∂
θ
)
+
1
r
2
s
i
n
2
θ
∂
2
ψ
(
r
,
θ
,
ϕ
)
∂
ϕ
2
]
+
V
(
r
,
θ
,
ϕ
)
ψ
(
r
,
θ
,
ϕ
)
=
E
ψ
(
r
,
θ
,
ϕ
)
{\displaystyle {\hat {H}}\psi (r,\theta ,\phi )=-{\frac {\hbar ^{2}}{2m}}\left[{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial \psi (r,\theta ,\phi )}{\partial r}}\right)+{\frac {1}{r^{2}sin\theta }}{\frac {\partial }{\partial \theta }}\left(sin\theta {\frac {\partial \psi (r,\theta ,\phi )}{\partial \theta }}\right)+{\frac {1}{r^{2}sin^{2}\theta }}{\frac {\partial ^{2}\psi (r,\theta ,\phi )}{\partial \phi ^{2}}}\right]+V(r,\theta ,\phi )\psi (r,\theta ,\phi )=E\psi (r,\theta ,\phi )}
We can gain insight into this somewhat ugly equation by rewriting it using the square of the angular momentum operator in spherical polar coordinates:
L
^
2
=
1
sin
θ
∂
∂
θ
(
sin
θ
∂
∂
θ
)
+
1
sin
2
θ
∂
2
∂
ϕ
2
{\displaystyle {\hat {L}}^{2}={1 \over \sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial \over \partial \theta }\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}}
This leads to
(
−
ℏ
2
2
m
(
1
r
2
∂
∂
r
(
r
2
∂
∂
r
)
)
+
1
2
m
L
^
2
r
2
+
V
(
r
,
θ
,
ϕ
)
)
ψ
(
r
,
θ
,
ϕ
)
=
E
ψ
(
r
,
θ
,
ϕ
)
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\left({1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial \over \partial r}\right)\right)+{\frac {1}{2m}}{\frac {{\hat {L}}^{2}}{r^{2}}}+V(r,\theta ,\phi )\right)\psi (r,\theta ,\phi )=E\psi (r,\theta ,\phi )}
This equation is only exactly solvable if
V
=
V
(
r
)
{\displaystyle V=V(r)}
function without angular dependence. We then write
ψ
(
r
,
θ
,
ϕ
)
=
R
(
r
)
Y
(
θ
,
ϕ
)
{\displaystyle \psi (r,\theta ,\phi )=R(r)Y(\theta ,\phi )}
(
−
ℏ
2
2
m
(
1
r
2
∂
∂
r
(
r
2
∂
∂
r
)
)
+
1
2
m
L
^
2
r
2
+
V
(
r
,
θ
,
ϕ
)
)
ψ
(
r
,
θ
,
ϕ
)
R
(
r
)
Y
(
θ
,
ϕ
)
=
E
R
(
r
)
Y
(
θ
,
ϕ
)
−
ℏ
2
2
m
(
Y
(
θ
,
ϕ
)
(
1
r
2
∂
∂
r
(
r
2
∂
∂
r
)
)
R
(
r
)
)
+
R
(
r
)
2
m
L
^
2
Y
(
θ
,
ϕ
)
r
2
+
V
(
r
)
R
(
r
)
Y
(
θ
,
ϕ
)
=
E
R
(
r
)
(
Y
(
θ
,
ϕ
)
{\displaystyle {\begin{matrix}\left(-{\frac {\hbar ^{2}}{2m}}\left({1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\frac {\partial }{\partial r}}\right)\right)+{\frac {1}{2m}}{\frac {{\hat {L}}^{2}}{r^{2}}}+V(r,\theta ,\phi )\right)\psi (r,\theta ,\phi )R(r)Y(\theta ,\phi )&=ER(r)Y(\theta ,\phi )\\-{\frac {\hbar ^{2}}{2m}}\left(Y(\theta ,\phi )\left({\frac {1}{r^{2}}}{\partial \over \partial r}\left(r^{2}{\frac {\partial }{\partial r}}\right)\right)R(r)\right)+{\frac {R(r)}{2m}}{\frac {{\hat {L}}^{2}\,Y(\theta ,\phi )}{r^{2}}}+V(r)R(r)Y(\theta ,\phi )&=ER(r)(Y(\theta ,\phi )\end{matrix}}}
To solve this equation we need to remove the angular dependence. This is simply done by substituting the eigenfunctions of
L
^
2
{\displaystyle {\hat {L}}^{2}}
Y
l
m
(
θ
,
ϕ
)
{\displaystyle Y_{l}^{m}(\theta ,\phi )}
ℏ
2
l
(
l
+
1
)
{\displaystyle \hbar ^{2}l(l+1)}
L
^
2
Y
l
m
(
θ
,
ϕ
)
=
ℏ
2
l
(
l
+
1
)
Y
l
m
(
θ
,
ϕ
)
{\displaystyle {\hat {L}}^{2}\,Y_{l}^{m}(\theta ,\phi )=\hbar ^{2}l(l+1)Y_{l}^{m}(\theta ,\phi )}
We now substitute this result into the Schr\"odinger equation and divide through by a common factor of
Y
l
m
(
θ
,
ϕ
)
{\displaystyle Y_{l}^{m}(\theta ,\phi )}
(
−
ℏ
2
2
m
(
1
r
2
∂
∂
r
(
r
2
∂
∂
r
)
+
ℏ
2
l
(
l
+
1
)
r
2
)
+
V
(
r
)
)
R
(
r
)
=
E
R
(
r
)
{\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\left({\frac {1}{r^{2}}}{\partial \over \partial r}\left(r^{2}{\frac {\partial }{\partial r}}\right)+{\frac {\hbar ^{2}l(l+1)}{r^{2}}}\right)+V(r)\right)R(r)=ER(r)}
This is the radial equation .\\
{\mathbf References}
[1] Griffiths, D. "Introduction to Quantum Mechanics" Prentice Hall, New Jersey, 1995.
![{\displaystyle {\hat {H}}\psi (r,\theta ,\phi )=-{\frac {\hbar ^{2}}{2m}}\left[{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial \psi (r,\theta ,\phi )}{\partial r}}\right)+{\frac {1}{r^{2}sin\theta }}{\frac {\partial }{\partial \theta }}\left(sin\theta {\frac {\partial \psi (r,\theta ,\phi )}{\partial \theta }}\right)+{\frac {1}{r^{2}sin^{2}\theta }}{\frac {\partial ^{2}\psi (r,\theta ,\phi )}{\partial \phi ^{2}}}\right]+V(r,\theta ,\phi )\psi (r,\theta ,\phi )=E\psi (r,\theta ,\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33a960585e3b935651e3b8ed57f845dc287718a0)
