applet-magic.com Thayer Watkins
Silicon Valley & Tornado Alley
U.S.A.

The Motion of a Particle at the Quantum Level

The Classical Case

A particle in a potential field of V(x) executes a periodic trajectory x(t). Its velocity can be determined from
the equation for total energy; i.e.,

½mv² + V(x) = E
and hence
v = [(2/m)(E−V(x)]^{½}

where m is the mass of the particle. It is convenient to denote [E−V(x)] as K(x), the kinetic energy of the particle
as a function of its location. Thus

v = [(2/m)K(x)]^{½}

Although the motion of the particle is deterministic a probability density function can be constructed for it that gives the
probability density of finding the particle at any location x of its trajectory. That probability density function is based
upon the time spent in an interval dx of its trajectory. That time spent is dx/v and hence the probability density is proportional
to 1/v(x). The probability density must sum to unity so the probability density is given by

P_{C}(x) = (1/v(x))/T

where T is equal to ∫dx/v.

The Quantum Case

For a particle in a portential field V(x) the probability density function is given by φ² where φ is the
solution to the time independent Schrödinger equation. For the one dimensional case this is

−h²)(∂²φ/∂x²) + V(x)φ = Eφ
or, equivalently
(∂²φ/∂x²) = −(2m/h²)[E−V(x)]φ
which can be expressed as
(∂²φ/∂x²) = −(2m/h²)K(x)φ

where h is Planck's constant divided by 2π.

The solution is such that the probability density function P_{QM}(x) consists of a series of
lobes extending from zero to maxima.

Just as in the classical case the velocity was translated into a probability density function, in the quantum case
the probability density function can be translated into a velocity function. This velocity function indicates that
motion at the quantum level consists of a pattern of slow-slow-fast-slow-slow-fast…. In the Copenhagen Interpretation
of quantum mechanics a particle can exist in a number of allowed states and jumps from one allowed state to another.
In the above interpretation there are no allowed and disallowed states as such. The so-called allowed states are simply
the portions of the particle's path where it moves slowly. The disallowed states are the places where it moves
fast. The division of the path of the particle into portions where it moves slowy and where it moves fast is in the
nature of a quantization over time for the particle's motion.