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A Generalization of the Quantum Mechanical Analysis of a Hydrogen Atom 

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 q_{0} and q_{1} and inversely proportional to the square of their separation distance s; i.e.,
where J is a constant. Since q_{0} and q_{1} are of opposite sign the force is an attraction. The force can be expressed as
In the subsequent analysis it is only Q=−Jq_{0}q_{1} that is relevant. The potential energy V is given by
The kinetic energy K of the system is given by
The Hamiltonian (total energy) function for the system is then
The timeindependent Schrödinger equation for the system is then
But this is not a convenient form for analysis. What is needed is a coordinate system suited to the geometry of the system.
Let the origin of the coordinate system be located at the center of mass for the two objects. Then
It is easily shown that
Thus there are not two "radial" variable but only one, s. Likewise there are not four angle variables but only two.
The two radial momenta are
The reduced mass μ of the system is defined by
Thus
Likewise p_{s1} = μ(ds/dt).
Therefore the radial component of kinetic energy is
Define p_{s} as μ(ds/dt) and thus
First of all note that
The kinetic energy due to the tangential velocities is
This can be represented as
Now define p_{φ} as
This means that
From the previous material, the Hamiltonian is given by
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 fasttrack solution for a simplified case without every step being derived.
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
The timeindependent Schrödinger equation for the system is then
The Laplacian operator ∇² for spherical coordinates (r, φ, θ) is
From this Laplacian it can be shown that −(h²/2μ)∇²ψ is equivalent to
the operator for p_{r}²ψ+p_{φ}²ψ. Furthermore it is shown at
that p_{φ}²ψ 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
The first step toward a solution is to let rR(r) be denoted as u(r). The resulting equation for u is
The analysis can be further simplified by introducing some nondimensional variables; i.e.,
The equation for u(r) then simplifies to
As ρ increases without bound the equation for u asymptotically approaches the equation
The solution to this equation is of the form
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
If V(ρ) is of the form ρ^{β} then
This suggests a solution for u(ρ) of the form
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.
If
then the coefficients C_{j} must satisfy the condition
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.,
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.,
The quantization condition for energy is then
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:
q  L_{q}(ρ) 
0  (1/0!)1 
1  (1/1!)(1ρ) 
2  (1/2!)(23ρ+ρ²) 
3  (1/3!)(618ρ+9ρ²ρ³) 
The radial function is then
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.
The angular equation is dealt with in Angular.
The timeindependent Schrödinger equation for the system is
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
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
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
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
In order for Ψ to be a singlevalued function of φ, λ must be an integer. Now let λ be designated as n.
The equation for the radial function is
(To be continued.)
Reference:
Richard L. Liboff, Introductory Quantum Mechanics, HoldenDay Inc., San Francisco, 1980.