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The Quantum Mechanical Analysis of a Two Body System

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 pj=mjvj
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 = Eψ = −|E|

where ψ is the wave function and E is the energy. For stable systems E is negative. Thus to emphasize the negativity of E, it is expressed as −|E|.

But the above equation 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 revolutions of the two bodies about their center of mass takes place in a plane. Therefore a polar coordinate system is appropriate rather than a spherical one.

The Momenta and the Components of Kinetic Energy

The Radial Components

The two radial momenta are

ps0 = m0(dr0/dt)
= m0(m1/(m0+m1))(ds/dt)
which can be put into the form
ps0 = [1/(1/m0+1/m1)](ds/dt)

The reduced mass μ of the system is defined by

1/μ = 1/m0+1/m1

Thus

ps0 = 1/(1/μ)(ds/dt) = μ(ds/dt)

Likewise ps1 = μ(ds/dt).

Therefore the radial component of kinetic energy is

Ks = ps0²/(2m0) + ps1²/(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 ps as μ(ds/dt) and thus

ps² = μs²(ds/dt)²
and hence
Ks = ps²/(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

Kθ = ½m0(r0ω)² + ½m1(r1ω)²
where ω=(dθ/dt).

This can be represented as

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

Now define pθ as

pθ = μs(sω) = μs²ω
and thus
pθ² = μ2s4ω2

This means that

Kθ = pφ²/(2μs²)

The Hamiltonian and Schrödinger Equation

From the previous material, the Hamiltonian is given by

H = ps²/(2μ) + pθ²/(2μs²) − Q/s

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

(∂²/∂s²) + (1/s)(∂/∂s) + (1/s²)((∂²/∂θ²)

Thus the equation to be satisfied by ψ is:

−(h²/2μ)[(∂²ψ/∂s²) + (1/s)(∂ψ/∂s) + (1/s²)((∂²ψ/∂θ²)] + (|E| − Q/s)ψ = 0

At this point it will be assumed that ψ(r, θ) is equal to R(r)Θ(θ). This is the separation of variables assumption. This is a mathematical convenience that is fraught with danger of precluding the physically relevant solutions. In this case it is alright because only circular orbits will be dealt with later.

When R(s)Θ(θ) is substituted into the equation it can be reduced to

−(h²/2μ)[R"(s)/R + (1/s)R'(r) + (1/s²)(Θ"(θ)/Θ] + (|E| − Q/s) = 0

This equation may be put into the form

s²R"(s)/R + rR'(r)/R + (2μ/(h²)(−|E|s² + sQ) = − (Θ"(θ)/Θ)

The LHS of the above is a function only of s and the RHS a function only of θ. Therefore their common value must be a constant. Let this constant be denoted as n².

Thus

(Θ"(θ)/Θ) = −n²
or, equivalently
Θ"(θ) + n²Θ(θ)

This equation has solutions of the form

Θ(θ) = A·cos(n·θ + θ0)

where A and θ0 are constants. Through a proper orientation of the polar coordinate system θ0 can made equal to zero. So Θ(θ) = A·cos(n·&theta). In order for Θ(θ+2π) to be equal to Θ(θ) n must be an integer. The probability density is the squared magnitude of the wave function. Therefore the probability density is proportional to cos²(nθ).

Below is the shape of this function for n=6.

The Radial Component of the Wave Function

The radial equation is then

s²R"(s)/R + rR'(r)/R + (2μ/(h²)(−|E|s² + sQ) = n²
or, equivalently

R"(s) + R'(s)/s + [(2μ/(h²)(−|E| + Q/s) −n²/s²]R = 0

For large values of s the above equation approaches the equation

S"(s) − [(2μ/(h²)|E|]S = 0

Let

λ = [(2μ/(h²)|E|]½

The solutions to the above equation for S are exp(+λs) and exp(−λs). Only the negative exponential is relevant for physical situations.

On the other hand at s goes to zero the radial equation approaches the solution to this equation

U"(s) − (n²/s²)U = 0

This equation has the solution

U = sl

where l(l−1)=n².

This has the problem that l would not be an integer. It is not necessary that all parameters in QM solutions be integral but it is a desirable simplification of the analysis. Therefore a slightly different approach will be used to establish the radial component of the wave function for the electron.

From the Laplacian it can be shown that −(h²/2μ)∇²ψ is equivalent to the operator for ps²ψ+pθ²ψ. Furthermore it is shown at that pθ²ψ is quantized to h²l(l+1).

The function R(s) must satisfiy the equation

−(h²/2μ)((1/s)∂²(sR)/∂s²) + [(h²/2μ)l(l+1)/(2μs²) − Q/s − E]ψ = 0

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

−(d²u/ds²) + [l(l+1)/s² − (2μ/h²)(Q/s) − (2μE/h²)]u = 0

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

κ² = 2μ|E|/h²
σ = 2κs
λ = Q/(h²κ)

The equation for u(s) then simplifies to

(d²u/ds²) − (l(l+1)/σ²)u + (λ/σ − 1/4)u = 0

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

(d²U/ds²) − (1/4)U = 0

The solution to this equation is of the form

U(σ) = A*exp(−σ/2) + B*exp(σ/2)

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

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

(d²V/ds²) − [l(l+1)/σ²]V = 0

If V(σ) is of the form σβ then

β(β-1)σβ-2 − [l(l+1)/σ²]σβ
which reduces to
β(β-1) = l(l+1)
which has the solution
β = (l+1)

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

u(σ) = exp(−σ/2)σl+1F(σ)

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

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

σ(d²F/dσ²) + (2l + 2 − σ)(dF/dσ) − (l + 1 − λ)F(σ) = 0

If

F(σ) = Σ0 Cjσj

then the coefficients Cj must satisfy the condition

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

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

qLq(σ)
0(1/0!)1
1(1/1!)(1-σ)
2(1/2!)(2-3σ+σ²)
3(1/3!)(6-18σ+9σ²-σ³)

The radial function is then

Rnl(σ) = exp(-σ/2)σl+1Ln-l-1(σ)

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.

Below is a depiction of the probability bumps in the plane of the electron's motion for a principal quantum number of 6.

The electron quantum mechanically moves relatively slowly in a probability bump, otherwise known as a state, and then relatively rapidly to the next state (bump).

The expected value of the radial dimension is the same for any angle θ. The probability density as a function of θ is as shown below. The red horizontal is the classical probability density ; i.e., .the proportion of time spent at the different locations. As can seen the QM PDF averaged over a range of angles would be equal to classical PDF.

Conclusion

What came out of the quantum mechanical analysis of the two body problem is a probability distribution for the states of the system; i.e., the separation distance of the bodies and the angle in the plane of their revolutions about their center of mass. This probability distribution corresponds to the proportion of the time the system spends in its various states. This sort of probability distribution for the bodies may be derived from the probability distribution of the system, but there is no probability distribution corresponding to an intrinsic indeterminancy of the bodies.

(To be continued.)


Reference:

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