San José State University

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Thayer Watkins
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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) = k1∇² + k2(∇²)² + k3(∇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 m0

K = ½m0

Therefore

v = (2K/m0)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 = m0/(1−β²)½

where β=v/c.

Relativistic kinetic energy is given by

K = mc² − m0
= m0c²(1/(1−β²)½ − 1)

Let K/(m0c²) be denoted as α.The above relationship is more conveniently expressed as

α + 1 = (1/(1−β²)½
and hence
1 − β² = 1/(α+1)²
and therefore
β² = 1 − 1/(α + 1)² = (α² + 2α)/(α + 1)²
and
β = (α² + 2α)½/(α + 1)

Thus

1/|v| = (1/c)(α + 1)/(α² + 2α)½
and
1/|v| = (1/c)[(α + 1)/(α + 2)½]/α½

As α→0 the fraction [(α + 1)/(α + 2)½] goes to 1/√2. This means the probability density asymptotically becomes inversely proportional to α½, Since α is K/(m0c²) probability density, as K→0, is asymptoticallyn inversely proportional to K½, just as for the nonrelativistic case as it should be.

For the case as K and α→∞ note that

1/v = (1/c)(1+1/α)/(1+2/α)½

So, as α→∞, (1/|v|)→(1/c); i.e. |v|→c.

(To be continued.)


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