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The Asymptotic Limit of the
Probability Distribution of a
Physical System Operating
in a Potential Field in 2D Space

Consider a physical with a Hamiltonian function H(p,z),
where z is the polar coordinates (r, θ) for a location for the
system and p is its momentum vector. The Hamiltonian is composed
of the Kinetic energy function K(p) and the Potential energy function
V(z); i.e.;

H(p, z) = K(p) + V(z)

The Hamiltonian function H(p, z) is converted into the Hamiltonian operator H^(∇², z)
by a set of substitutions;
i.e., p → (h/i)∇ and z→ z.

The time independent Schrödinger equation is then

H^ψ(z) = Eψ(z)

where E is the total energy of the system and ψ(z) is its wave function. The squared
wave function |ψ(z)|² is the probability density function.

The Kinetic energy function K(p) has to have certain properties. If there is no momentum
there is no kinetic energy; i.e., K(0)=0. Also K(−p)=K(p). This means that the Maclaurin
series for the Kinetic energy operator is of the form

K^(∇²) = k_{1}∇² + k_{2}(∇²)² + k_{3}(∇²)³ + <>

The Potential energy function may also be assumed to satisfy the conditions V(0)=0 and V(−z)=V(z).

Kinetic Energy and Velocity

The crucial relationship is the one between Kinetic energy and velocities. Under nonrelavistic conditions for
a particle of mass m_{0}

K = ½m_{0}v²

Therefore

v = (2K/m_{0})^{½}

The time-spent probability density is proportional to 1/|v|
and hence inversely proportional to (K(z))^{½}.

On the other hand, under relativistic conditions,
mass is a function of relative speed; i.e.,

m = m_{0}/(1−β²)^{½}

where β=v/c.

Relativistic kinetic energy is given by

K = mc² − m_{0}c²
= m_{0}c²(1/(1−β²)^{½} − 1)

Let K/(m_{0}c²) be denoted as α.The above relationship is more conveniently expressed as

As α→0 the fraction [(α + 1)/(α + 2)^{½}] goes to 1/√2.
This means the probability density asymptotically becomes inversely proportional to α^{½},
Since α is K/(m_{0}c²) probability density, as K→0, is asymptoticallyn inversely
proportional to K^{½}, just as for the nonrelativistic case as it should be.