The successful application of Schrödinger's wave mechanics to the hydrogen atom was a key to the acceptance of wave mechanics as the proper model of atomic reality. It was recognized that that analysis equally applied to any hydrogenic atom; i.e., any atom or ion with but a single electron. This is a further generalization to any system consisting of two objects of opposite electrostatic charges. This would include hydrogenic atoms but also positronium (an electron/positron pair), muonium and tauonium hydrogenic atoms, and any atom involving an antiproton and a positively charged lepton.

Let the two subatomic objects and their properties be labeled 0 and 1. The force between them is proportional to the product of their charges q₀ and q₁ and inversely proportional to the square of their separation distance s; i.e.,

F = Jq₀q₁/s²

where J is a constant. Since q₀ and q₁ are of opposite sign the force is an attraction. The force can be expressed as

F = −Q/s²

In the subsequent analysis it is only Q=−Jq₀q₁ that is relevant. The potential energy V is given by

V(s) = −Q/s

The kinetic energy K of the system is given by

K = ½m₀v₀² + ½m₁v₁²
or, in terms of
the momenta p_j=m_jv_j
K = p₀²/2m₀ + p₁²/2m₁

The Hamiltonian (total energy) function for the system is then

H = p₀²/2m₀ + p₁²/2m₁ −Q/s

The time-independent Schrödinger equation for the system is then

−h²[(∂²ψ/∂r₀² (∂²ψ/∂r₁²] −Q/s

But this is not a convenient form for analysis. What is needed is a coordinate system suited to the geometry of the system.

Formulation of a Convenient Coordinate System

Let the origin of the coordinate system be located at the center of mass for the two objects. Then

m₀r₀ = m₁r₁
and
r₀ + r₁ = s

It is easily shown that

r₀ = (m₁/(m₀+m₁))s
r₁ = (m₀/(m₀+m₁))s

Thus there are not two "radial" variable but only one, s. Likewise there are not four angle variables but only two.

The Momenta and the Components of Kinetic Energy

The Radial Components

The two radial momenta are

p_s0 = m₀(dr₀/dt)
= m₀(m₁/(m₀+m₁))(ds/dt)
which can be put into the form
p_s0 = [1/(1/m₀+1/m₁)](ds/dt)

The reduced mass μ of the system is defined by

1/μ = 1/m₀+1/m₁

Thus

p_s0 = 1/(1/μ)(ds/dt) = μ(ds/dt)

Likewise p_s1 = μ(ds/dt).

Therefore the radial component of kinetic energy is

K_s = p_s0²/(2m₀) + p_s1²/(2m₁)
= μ²(ds/dt)²/(2m₀) + μ²(ds/dt)²/(2m₁)
= (μ²/2)(ds/dt)²[1/m₀ + 1/m₁]
= (μ²/2)(ds/dt)²(1/μ) = ½μ(ds/dt)²

Define p_s as μ(ds/dt) and thus

p_s² = μ_s²(ds/dt)²
and hence
K_s = p_s²/(2μ)

The Angular Momenta and Tangential Components of Kinetic Energy

First of all note that

m₀r₀ = m₁r₁ = (m₀m₁/(m₀+m₁))s = μs

The kinetic energy due to the tangential velocities is

K_φ = ½m₀(r₀ω)² + ½m₁(r₁ω)²
where ω=(dφ/dt).

This can be represented as

K_φ = ½((m₀r₀)²/m₀)ω² + ½((m₁r₁)²/m₁)ω)²
which reduces to
K_φ = ½(μ²s²ω²)(1/m₀ + 1/m₁) = μs²ω²

Now define p_φ as

p_φ = μs(sω) = μs²ω
and thus
p_φ² = μ²s⁴ω²

This means that

K_φ = p_φ²/(2μs²)

The Hamiltonian and Schrödinger Equation

From the previous material, the Hamiltonian is given by

H = p_s²/(2μ) + p_φ²/(2μs²) − Q/s

The process of solving the Schrödinger equation for the system is intricate and involved. It is easy in that process to lose sight of the objective. What is needed is a fast-track solution for a simplified case without every step being derived.

The Solution for a Special Case

Consider the case in which the mass one of the objects is relatively large compared to the mass of the other. In effect, the mass of one object is infinite compared to the other. The center of mass is then at the center of the larger object and the separation distance is the same as the orbit radius for the smaller object. In this case the variable s is replaced by r and the coordinate system is spherical.

The Hamiltonian function for the system is then

H = p²/2μ − Q/r

The time-independent Schrödinger equation for the system is then

−(h²/2μ)∇²ψ −Qψ/r = Eψ

The Laplacian operator ∇² for spherical coordinates (r, φ, θ) is

(1/r)(∂²(rψ)/∂r² + (1/r²)[(1/sin θ)(∂(sin θ (∂ψ/∂θ))/∂θ + (1/sin² θ)(∂²ψ/∂φ²)]

From this Laplacian it can be shown that −(h²/2μ)∇²ψ is equivalent to the operator for p_r²ψ+p_φ²ψ. Furthermore it is shown at that p_φ²ψ is quantized to h²l(l+1).

If ψ(r, θ, φ) is assumed to be of the form R(r)Y_l^m(θ, φ) the analysis is separated into two parts.

The function Y_l^m(θ, φ) is called a spherical harmonic and it will be dealt with later. The function R(r) must satisfiy the equation

−(h²/2μ)((1/r)∂²(rR)/∂r²) + [h²/2μl(l+1)/(2μr²) − Q/r − E]ψ = 0

The first step toward a solution is to let rR(r) be denoted as u(r). The resulting equation for u is

−(d²u/dr²) + [l(l+1)/r² − (2μ/h²)(Q/r) − (2μE/h²)]u = 0

The analysis can be further simplified by introducing some nondimensional variables; i.e.,

κ² = 2μ|E|/h²
ρ = 2κr
λ = Q/(h²κ)

The equation for u(r) then simplifies to

(d²u/dr²) − (l(l+1)/ρ²)u + (λ/ρ − 1/4)u = 0

As ρ increases without bound the equation for u asymptotically approaches the equation

(d²U/dr²) − (1/4)U = 0

The solution to this equation is of the form

U(ρ) = A*exp(−ρ/2) + B*exp(ρ/2)

where A and B are constants. The only solutions of this form that are bounded as ρ→∞ are those for which B=0.

As ρ goes to zero the equation for u asymptotically approaches the equation

(d²V/dr²) − [l(l+1)/ρ²]V = 0

If V(ρ) is of the form ρ^β then

β(β-1)ρ^β-2 − [l(l+1)/ρ²]ρ^β
which reduces to
β(β-1) = l(l+1)
which has the solution
β = (l+1)

This suggests a solution for u(ρ) of the form

u(ρ) = exp(−ρ/2)ρ^l+1F(ρ)

where F(ρ) is a polynomial in ρ that is finite everywhere.

When the proposed solution is substituted into the equation for u the result is that F(ρ) must satisfy the following equation.

ρ(d²F/dr²) + (2l + 2 − ρ)(dF/dρ) − (l + 1 − λ)F(ρ) = 0

If

F(ρ) = Σ₀ C_jρ^j

then the coefficients C_j must satisfy the condition

C_j+1 = C_j[(j+ l +1 − λ)/(j+1)(j+2l+2)]

The polynomial F(ρ) will be of finite order if and only if there is an integer q such that the numerator of the fraction in the above condition is equal to zero; i.e.,

q + l + 1 = λ

This means that λ must be an integer and that q=λ−(l+1). Usually λ is called the principal quantum number and it is denoted as n. The definitions of of κ and λ imply a quantifization of energy; i.e.,

κ² = 2μ|E|/h²
n = λ = Qμ/(κh²)
hence
|E| = h²κ²
and
κ² = (Qμ)²/(n²h⁴)
and therefore
|E| = Q²μ²/(n²h²)

The quantization condition for energy is then

E = −Q²μ²/(n²h²)

The polynomials represented by F(ρ) are known as Laguerre polynomials. They depend upon an integral parameter q, which is equal to (n−(l+1)). The first few are:

The radial function is then

R_nl(ρ) = exp(-ρ/2)ρ^l+1L_n-l-1(ρ)

The probability density function is proportional to the square of this radial function. The shapes of R and R² for several values of n and l are shown in the display below. For historical reasons the values of l are coded as letters: s=0, p=1, d=2.

from Introductory Quantum Mechanics by Richard L. Liboff, p. 191.

The Angular Equation

The angular equation is dealt with in Angular.

The Full Analysis

The time-independent Schrödinger equation for the system is

−h²[(∂²ψ/∂s²)/(2μ) + (∂²ψ/∂φ²)/(2μs²)] −(Q/s)ψ = Eψ

where ψ is the wave function for the system and E is its total energy.

It is now assumed that the wave function is of the form

ψ(s, φ, θ) = R(s)Ψ(φ, θ)

where φ is the latitudinal angle and θ the longitudinal angle of the line connecting the centers of the two objects.

When derivatives are denoted with subscripts the Schrödinger equation becomes

−h²[R_ssΨ/(2μ) + RΨ_φφ/(2μs²)] −(Q/s)RΨ = ERΨ
which, upon division
by RΨ, becomes
−h²[(R_ss/R)/(2μ) + (Ψ_φφ/Ψ)/(2μs²)] −(Q/s) = E
and after multiplication by s²
−h²[(s²R_ss/R)/(2μ) + (Ψ_φφ/Ψ)/(2μ)] −(Qs) = Es²

The term (Ψ_φφ/Ψ)/(2μ) is not a function of s whereas all of the other terms are functions of s. The above equation may be rearranged to the form

−(Ψ_φφ/Ψ) = s²R_ss/R + (2μ/h²)[Qs + Es²]

Since one side of this equation is a function of s and the other is not, both sides must be equal to a constant, say λ². Thus

Ψ_φφ/Ψ = −λ²
which has as solution
Ψ = A(θ)cos(λφ) + B(θ)sin(λφ)

In order for Ψ to be a single-valued function of φ, λ must be an integer. Now let λ be designated as n.

The equation for the radial function is

s²R_ss/R + (2μ/h²)[Qs + Es²] = n²
or, equivalently
R_ss + [(2μ/h²)(Q/s + E) −n²/s²]R = 0

(To be continued.)

Reference:

Richard L. Liboff, Introductory Quantum Mechanics, Holden-Day Inc., San Francisco, 1980.