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The Probability Density Function
from Schroedinger's Equation for
a Particle Moving in a Central Field

Background

This material is for the further development of the proposition that the probability density function from Schroedinger's equation is related to the time-spent probability density function of classical physics .

Niels Bohr noted that the validity of classical physics has been well established at the macroscopic level . Therefore he asserted that for quantum physics to be valid its results must match or approach in the limit those of classical physics as the energy level increases without bound. This is Bohr's Correspondence Principle. It has been modified to include the limit of other parameters besides energy. It also needs to be modified to apply to the spatial average of the quantum results rather than the quantum results themselves . The observed results of classical physics are necessarily time or space averages. Phyical quantities cannot be measured at instants or points.

Consider what the correspondence principle implies . At each energy level the quantum physics determines a probability density distribution (PDD). In the limit it also determines a PDD. What classical physics result can this correspond to? Classical physics is deterministic; the only PDD that the limit can be is the time-spent PDD.

This has been verified for one dimensional physical systems because there are analytical solutions for them . The material here is for extending the results to two dimensional systems. Surprisingly enough there are no valid analytical solutions for particles in two and three dimensional systems. Physicists have resorted to what is called the Separation of Variables assumption to get solutions, but the Separation of Variables assumption is not compatible with particality (particleness). Therefore such solutions are mathematically valid but irrelevant physically .

Schroedinger's Time-Dependent Equation

The general form of this equation is

ih(∂ψ/∂t) = H^ψ

where i is the imaginary unit, h is Planck's constant divided by 2π and t is time. H^ is the Hamiltonian operator derived from Hamiltonian function for the physical system. The wave function ψ is such that |ψ|² is the probability density function for the system.

Schroedinger's Time-Dependent Equation provides a numerical method for approximating solutions. The only problem is what should be the initial conditions for starting the iterative process of numerical solution. Note that Schroedinger's Time-Dependent Equation works backwards in time as well as forward.

Here the initial condition will be the analytical solution that would prevail if Planck's constant were zero. If Planck's constant is zero the solution is the classical one.

Perturbation Analysis

Perturbation analysis also may be used to obtain the physical solution for the increase in Planck's constant from zero to its real world level. For this approach the relevant equation is Schroedinger's time independent equation

H^ψ = Eψ

where E is the energy of the system.

A Particle Moving in a Central Field

The system being considered is that of a particle of mass m moving in a potential field −K/r. The energy of this system is then

H = ½mv² − K/r = p²/(2m) − K/r

The Hamiltonian operator corresponding to this Hamiltonian function is

H^ = −(h²/(2m))∇² −(K/r)

Initial Conditions

Consider a circular orbit about the origin for the particle. The orbit radius is given by

K/R² = mv²/R
with
½mv² = E

Thus

K/R² = 2E/R
and hence
R = K/(2E)

The linear probability density for the system 1/(2πR) if r equals R and zero other wise. If the polar plane is divided into rectangles of dimensions Δr×(rΔθ) then the areal probability density is 1/(rΔθΔr) for R−Δr<r<R+Δr and zero otherwise. This means that the wave function ψ is 1/(rΔθΔr)½ in the band about r=R and zero otherwise.

Thus at t=0

The Laplacian for a Polar Coordinate System

∇²ψ = (∂²ψ/∂r²) + (1/r)(∂ψ/∂r) + (∂²ψ/∂θ²)

When the wave function is always circularly symmetric the derivative with respect to θ is zero and therefore Schroedinger's time dependent equation for the system is

ih(∂ψ/∂t) = −(h²/(2m))[(∂²ψ/∂r²) + (1/r)(∂ψ/∂r)] −( K/r)ψ
which reduces to
(∂ψ/∂t) = (ih/(2m))[(∂²ψ/∂r²) + (1/r)(∂ψ/∂r)] + (i/h)(K/r)ψ

The derivatives with respect to r may be approximated with centered differences; i.e.,

(∂ψ/∂r) ≅ [ψ(r+Δr)−ψ(r−Δr)]/2Δr
and
(∂²ψ/∂r²) ≅ [ψ(r+Δr)−2ψ(r)+ψ(r−Δr)]/(Δr)²

The wave function at the next time point is then given by

ψ(r, t+Δt) = ψ(r, t) + (∂ψ/∂t)Δt

The initial condition and first iteration for the wave function are

(To be continued.)


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