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The Correspondence Principle
Reduced to a Single Parameter

In 1920 Niels Bohr articulated his Correspondence Principle; i.e., that since classical physics was well validated, valid quantum theory had to correspond to classical physics as the scale of the quantum analysis increased without limit. Later the Correspondence Principle was taken to mean that as the level of energy increased without bound the results of quantum analysis had to, at least asymptotically, approach the results of classical analysis. Still later other parameters were included in the requirement of the Correspondence Principle. As Planck's Constant goes to zero the results of quantum analysis should approach those of classical analysis. Likewise as the mass of a system increases without bound quantum results should approach the results of classical analysis.

This raises the question of whether there is a single parameter combining energy, Planck's Constant and mass which determines the results of quantum analysis and their asymptotic limit's correspondence with classical analysis.

Consider a particle of mass m moving in a potential field of V(r). The Hamiltonian function for such a system is

H = p²/(2m) + V(r)

where p is the momentum of the particle.

The Hamiltonian operator for the system is

H^ = −(h²/(2m))(∂²/∂r²) + V(r)

The time-independent Schroedinger equation for the system is then

h²/(2m)∇²ψ + V(r)ψ = Eψ

where E is system energy and ψ is called the wave function.

This can be rearranged to

h²/(2m)∇²ψ = (E − V(r))ψ = K(r)ψ

where K(r) is the kinetic energy of the system as a function of particle location.

A further rearrangement gives

∇²ψ = − (2mK(r)/h²)ψ

This equation can be represented as

∇²ψ = − Λ(r)²ψ


Λ(r) =(2mK(r))½/h
K(r) = E − V(r)

is the parameter sought after. If E or m →∞, or h →0, then Λ(r)→∞ for all r.

The above equation can also be represented as

∇²ψ = −((2mE)½/h)(1−V(r)/E)½ψ

The profile of the probability density function ψ² is determined by the function (1−V(r)/E)½.

(To be continued.)

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