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A Quantum Mechanical Analysis of a
Particle Moving in a Central Field:

This is an effort to extend the results of previous work establishing that the probability density functions (PDF's) that arise in quantum mechanical (QM) analysis do not represent the intrinsic indeterminacy of the particles but instead the average time spent at the possible locations. They are analogous to the fuzzy, blurred appearance of a rapidly rotating fan.

Bohr's Correspondence Principle

Neils Bohr nearly a century ago observed that classical analysis for many areas of physics had been empirically verified. Therefore, for any quantum mechanical analysis its appropriate extension to the realm of classical analysis should agree with the classical analysis. In atomic physics the extension is in terms of scale and/or the level of energy. In radiation physics the extension is the limit as h, Planck's constant, goes to zero. In statistical mechanics the limit as the number of molecules increases without bound should agree with thermodynamics.

The Copenhagen Interpretation

The Copenhagen Interpretation of the solutions, called the wave functions, to Schrödinger's equation is that the squared magnitude is equal to the probability density function (PDF) for the system. An alternate interpretation is that for a system undergoing a cycle the squared magnitude of the wave function at a state is proportional to the proportion of the time spent in that state.

The Classsical Analogue of a Probability Density Function

The proportion of time spent in a state can be construed as a probability density function in the sense that it is proportional to the probability of finding the system in that state at a randomly chosen time. The relationship is

P(x) = (1/|v(x)|)/T

P(x) is the probability density at state x, v(x) is velocity at state x and T is the time required to go through a cycle.

T = ∫dx/|v(x) = ∫dt

One Dimensional Systems

For a one dimensional system such as a harmonic oscillator it is clear that the alternative interpretation is valid and this applies to any one dimensional system involving a particle moving in a potential field.

Here is the QM PDF for a harmonic oscillator with principal quantum number equal to 30. The classical PDF is shown as a heavy line in the graph.

The classical PDF closely matches a spatial average of the QM PDF everywhere except at the end points.

A Particle Moving in a Central Field
with a Potential Energy Function V(r)

The Hamiltonian function for the system is then

H = p²/(2m) + V(r)

where p is the total momentum of the electron, m is its mass and V(r) is the potential energy of the particle as a function of its distance r from the center of the central field. The total momentum is made up of the radial momentum pr and the tangential momentum pθ.

At a macroscopic level a particle in a central field revolves about the center of the field in an elliptical orbit. That orbit is entirely in a plane. In order for the QM analysis to satisfly the Correspondence Principle it must also be limited to a plane.

The time-independent Schrödinger equation for the system is then

−(h²/2m)∇²ψ + V(r)ψ = Eψ

where h is Planck's constant divided by 2π and ψ is the wave function for the electron.

For stable systems the potential energy and the total energy re negative. To emphasize their negativity they will be written as −|V(r)| and −|E|. Thus the above equation is

−(h²/2m)∇²ψ −|V(r)|ψ = −|E|ψ

In order to satisfy the Correspondence Principle ψ must be a function of only r and θ, the longitudinal angle variable. The latitudinal angle φ must be a constant equal to the value for the equator.

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

(∂²/∂r²) + (1/r)(∂/∂r) + (1/r²)((∂²/∂θ²)

Thus the equation to be satisfied by ψ is:

−(h²/2m)[(∂²ψ/∂r²) + (1/r)(∂ψ/∂r) + (1/r²)((∂²ψ/∂θ²)] + (|E| − |V(r)|)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(r)Θ(θ) is substituted into the equation it can be reduced to

−(h²/2m)[R"(r)/R + (1/r)R'(r) + (1/r²)(Θ"(θ)/Θ] + (|E| − |V(/r)|) = 0

This equation may be put into the form

r²R"(r)/R + rR'(r)/R + (2m/h²)(−r²|E| + r²|V(r)|)) = − (Θ"(θ)/Θ)

The LHS of the above is a function only of r 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·θ). 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.

Note that cos²(nθ) has 2n peaks.

The Radial Component of the Wave Function

The radial equation is then

r²R"(r)/R + rR'(r)/R + (2m/(h²)(−(|E|−|V(r)|)r² ) = n²
or, equivalently

R"(r) + R'(r)/r + [(2m/(h²)(−(|E|−|V(r)| )r² −n²/r²]R = 0

The potential energy function for relevant physical systems involves functions of (1/r). Therefore r²E is a higher order function of r than r²V.

For large values of r the above equation approaches the equation

S"(r) − [(2m/(h²)|E|]S = 0

Let

λ = [(2m/(h²)|E|]½

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

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

U"(r) − (n²/r²)U = 0

This equation has the solution

U = rl

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.

From the Laplacian it can be shown that −(h²/2m)∇²ψ is equivalent to the operator for pr²ψ+pθ²ψ. Furthermore it is shown at that pθ²ψ is quantized to h²l(l+1)ψ.

The function R(r) then must satisfiy an equation of the form

−(h²/2m)((1/r)∂²(rR)/∂r²) + [h²/2ml(l+1)/(2mr²) −(|E| −|V(r)|)]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 + (2m/h²)|V(r)| − (2mE/h²)]u = 0

The analysis cannot proceed further with a general potential energy function. What is found when V(r) is −Q/r the wave function takes the form of a product of a negative exponential function of r and a finite order polynomial function of r. Eventually as r increases the negative exponential function overwhelms the polynomial function and the probability density decreases to near zero. The dependence of the probability density on cos²(nθ) means there 2n bumps of probability density that correspond to an orbit for the particle.

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

The particle 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.


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