time independent Schrödinger equation in spherical coordinates

When writing the time independent Schrödinger equation in spherical coordinates, we need to plug the Laplacian in Spherical Coordinates into the time independent Schrödinger equation. The Laplacian was found to be

1 ∂ ( ∂ ) 1 ∂ ( ∂ ) 1 ∂2 ∇2sph = ----- r2--- + ---------- sin𝜃--- + ----------- r2∂r ∂r r2sin 𝜃∂𝜃 ∂ 𝜃 r2sin2 𝜃∂ ϕ2

Using the three dimensional Schrödinger equation we then have

2 [ ( ) ( ) 2 ] ˆH ψ(r,𝜃,ϕ ) = − ℏ--- 1--∂- r2∂ψ-(r,𝜃,ϕ) + --1----∂-- sin𝜃 ∂ψ-(r,𝜃,-ϕ) + ---1---∂-ψ-(r,𝜃,ϕ) +V (r,𝜃,ϕ)ψ (r,𝜃,ϕ) = E ψ(r,𝜃, 2m r2∂r ∂r r2sin𝜃 ∂𝜃 ∂𝜃 r2sin2 𝜃 ∂ϕ2

We can gain insight into this somewhat ugly equation by rewriting it using the square of the angular momentum operator in spherical polar coordinates:

1 ∂ ( ∂ ) 1 ∂2 ˆL2 = -------- sin 𝜃--- + ---2-----2 sin𝜃 ∂𝜃 ∂𝜃 sin 𝜃 ∂ϕ

This leads to

( ( ( ) ) ) -ℏ2- -1 ∂-- 2-∂- -1--ˆL2- − 2m r2 ∂r r ∂r + 2m r2 + V (r,𝜃,ϕ) ψ (r,𝜃,ϕ) = E ψ(r,𝜃,ϕ )

0.1 Spherically symmetric separable solution

This equation is only exactly solvable if V = V (r), a function without angular dependence. We then write ψ(r,𝜃,ϕ) = R(r)Y (𝜃,ϕ) leading to the following equation:

ψ(r,𝜃,ϕ)R(r)Y (𝜃,ϕ)	= ER(r)Y (𝜃,ϕ)
− + + V (r)R(r)Y (𝜃,ϕ)	= ER(r)(Y (𝜃,ϕ)

To solve this equation we need to remove the angular dependence. This is simply done by substituting the eigenfunctions of L² into the equation. These are known to be the spherical harmonics, Y _l^m(𝜃,ϕ). We also know that these have eigenvalues ℏ²l(l + 1), i.e.

ˆ2 m 2 m L Y l (𝜃,ϕ) = ℏ l(l + 1)Yl (𝜃,ϕ )

We now substitute this result into the Schrödinger equation and divide through by a common factor of Y _l^m(𝜃,ϕ)

( 2 ( ( ) 2 ) ) − -ℏ-- 1--∂- r2-∂- + ℏ-l(l +-1) + V (r) R (r ) = ER (r) 2m r2∂r ∂r r2

This is the radial equation.
References

[1] Griffiths, D. “Introduction to Quantum Mechanics” Prentice Hall, New Jersey, 1995.

"time independent Schrödinger equation in spherical coordinates" is owned by bloftin. [ full author list (2) ]
(view preamble)