In three dimensions we often encounter spherically symmetric potential energy
functions. For example the potential energy of the electron in the
hydrogen atom is the electrostatic potential energy of the proton-electron
system. It depends only on the distance r of the electron from the proton,
not on the direction of the position vector **r**. For the
proton-electron system we have U(r) = -q^{2}/(4πε_{0}r)
where q is the magnitude of the charge of the proton and the electron,
q = 1.6 *10^{-19 }C,
and ε_{0 }= 8.85*10^{-12}C^{2}/(Nm^{2})
is a constant called the permittivity of free space. We put the origin of our
coordinate system at the position of the proton.

Let us assume that U(x,y,z) = U(r). This means that the potential
energy of the particle depends only on its
distance from the origin. For such a potential, rectangular coordinates (x,y,z) are an inconvenient
choice. Spherical coordinates are a much better fit. Spherical
coordinates are defined in the figure below. We specify a position **r**,
by giving its distance r from the origin, the angle θ
the position vector makes with the z-axis, and the angle φ
the projection of the position vector onto the x-y plane makes with the x-axis.

The relationships between spherical coordinates ad rectangular coordinates are

z = r cosθ, y = r sinθ sinφ, x = r sinθ cosφ, (0 < r < ∞, 0 < θ < π, 0 < φ < 2π),

or

r = ( x^{2} + y^{2} + z^{2})^{1/2}, tanθ
= ( x^{2} + y^{2})^{1/2}/z, tanφ
= y/x.

**Problems:**

Find the spherical coordinate defining the point x = 1, y = 1, z = 1.

- Solution:

r = √3 = 1.732, θ = tan^{-1}(√2) = 0.955 rad = 54.73^{o}, φ = tan^{-1}(1) = π/4 = 45^{o}.

Find the rectangular coordinates of a point defined by r = 2,
θ = 30^{o}, φ =
180^{o}.

- Solution:

z = 2cos(30^{o}) = 1.73, y = 0, x = 2sin(30^{o})cos(180^{o}) = -1.

The time-independent Schroedinger equation for a particle
moving in a spherically symmetric potential with potential energy U(r)
= U((x^{2}+y^{2}+z^{2})^{1/2}) is

(-ħ^{2}/(2m))[∂^{2}ψ(x,y,z)/∂x^{2}
+ ∂^{2}ψ(x,y,z)/∂y^{2}
+ ∂^{2}ψ(x,y,z)/∂z^{2}]
+ U((x^{2}+y^{2}+z^{2})^{1/2})ψ(x,y,z) =
Eψ(x,y,z)

In spherical coordinates it is written as

(-ħ^{2}/(2m))[(1/r)∂^{2}(rψ(r,θ,φ))/∂r^{2}
+ (1/(r^{2}sin^{2}θ))∂(sinθ ∂ψ(r,θ,φ)/∂θ)/∂θ
+ (1/(r^{2}sin^{2}θ))∂^{2}ψ(r,θ,φ)/∂φ^{2}]

+ U(r)ψ(r,θ,φ) = Eψ(r,θ,φ)),

since

∂^{2}ψ/∂x^{2}
+ ∂^{2}ψ/∂y^{2}
+ ∂^{2}ψ/∂z^{2}
= (1/r)∂^{2}(rψ)/∂r^{2}
+ (1/(r^{2}sin^{2}θ))∂(sinθ ∂ψ/∂θ)/∂θ
+ (1/(r^{2}sin^{2}θ))∂^{2}ψ/∂φ^{2}.

In spherical coordinates separation of variables is possible. We write ψ(x,y,z) = R(r) Θ(θ) Φ(φ) and find

(-ħ^{2}/(2m))[(1/(Rr)∂^{2}(rR))/∂r^{2}
+ (1/(Θr^{2}sin^{2}θ))∂(sinθ∂Θ/∂θ)/∂θ
+ (1/(Φr^{2}sin^{2}θ))∂^{2}Φ/∂φ^{2}]
= E - U(r),

or

(r^{2}/(Rr)∂^{2}(rR))/∂r^{2}
+ (2mr^{2}/ħ^{2})(E - U(r))
+ (1/(Θsin^{2}θ))∂(sinθ∂Θ/∂θ)/∂θ
+ (1/(Φsin^{2}θ))∂^{2}Φ/∂φ^{2}
= 0.

The first term in the sum depends only on r and the second only on the angles θ and φ. Each term must be equal to a constant, and the two constants must sum to zero. We write

(r^{2}/(Rr)∂^{2}(rR))/∂r^{2}
+ (2mr^{2}/ħ^{2})(E - U(r)) = k,

(1/(Θsin^{2}θ))∂(sinθ∂Θ/∂θ)/∂θ
+ (1/(Φsin^{2}θ))∂^{2}Φ/∂φ^{2}
= -k.

The second equation can then be rewritten as

(1/Θ)∂(sinθ∂Θ/∂θ)/∂θ
+ ksin^{2}θ + (1/Φ)∂^{2}Φ/∂φ^{2} =
0.

The first term in the sum depends only on θ and the second only on φ. Each term must be equal to a constant, and the two constants must sum to zero. We write

(1/Θ)∂(sinθ∂Θ/∂θ)/∂θ
+ ksin^{2}θ = m^{2},

and

(1/Φ)∂^{2}Φ/∂φ^{2}
= -m^{2}, ∂^{2}Φ/∂φ^{2}
= -m^{2}Φ, Φ_{m}(φ)
= Aexp(imφ).

Since Φ_{m}(φ)
must be periodic with period 2π, we need m to be an
integer, m = 0, ±1, ±2, ... .

The functions A exp(imφ) are eigenfunctions of
the operator (ħ/i)∂/∂φ,
which is the operator for the z-component of the orbital
angular momentum, L_{z}.
Its eigenvalues are mħ, **the z-component of the
angular momentum is quantized**.

The equation (1/(sin^{2}θ))∂(sinθ∂Θ/∂θ)/∂θ
+ (k - m^{2}/sin^{2}θ)Θ
= 0, with m equal to an integer has acceptable solutions in the region 0
< θ < π if
k = l(l
+ 1) and |m| ≤ l.
We denote these functions by Θ_{l}(θ)
and the product of the functions Θ_{l}(θ)Φ_{m}(φ)
by
Y_{lm}(θ,φ).

The
Y_{lm}(θ,φ) are eigenfunctions of the
operator L^{2} = (ħ^{2}/(sin^{2}θ))∂(sinθ∂/∂θ)/∂θ + (ħ^{2}/(sin^{2}θ))∂^{2}/∂φ^{2},
which is the operator for the square of the magnitude of the orbital angular
momentum. Its eigenvalues are l(l
+ 1)ħ^{2}. **The square of the
magnitude of the orbital angular momentum is quantized.**

The radial equation now becomes (-ħ^{2}/(2m))∂^{2}(rR))/∂r^{2}
+ (l(l + 1)ħ^{2}/(2mr^{2}))(rR)
= (E - U(r))(rR). The solutions
R_{nl}(r) to this equation depend on the exact form of the
potential energy function U(r).

In spherical coordinated each solution of the time independent Schroedinger equation with U(x,y,z) = U(r) can be written as as a product of a function that depends only on r, and another function that depends on the angles θ and φ. We write

ψ_{nlm}(r,θ,φ) = R_{nl}(r)Y_{lm}(θ,φ).

Different quantum numbers n,
l, and m denote different stationary
states. (In three dimensions we need three quantum numbers.) The eigenstates of the energy operator are also eigenstates of the
square of the angular momentum operator, L^{2}, and of the z-component
of the angular momentum operator, L_{z}. The quantum number
l
characterizes the eigenvalues of L^{2}, and the quantum number m
characterizes the eigenvalues of L_{z}. **E, L ^{2} and L_{z}
are compatible observables for a particle with potential energy U(r), we can
know all three eigenvalues exactly at the same time.** But the energy of a stationary state never depends on m in a
spherically symmetric potential. All states with the same n and
l but
different m are degenerate.

The functions R_{nl}(r) depend on the exact form of the spherically
symmetric potential U(r), but the functions
Y_{lm}(θ,φ)
are the same for all spherically symmetric potentials. The functions
Y_{lm}(θ,φ)
are called the spherical harmonics. They are the eigenfunctions of L^{2}
and L_{z}, operators that do not operate on the radial coordinate r. |Y_{lm}(θ,φ) |^{2} is equal to the probability density of finding the particle
at angles (θ,φ).
Some of the normalized spherical harmonics are given below.

,