A Basic Quantum Model

Consider a particle of mass m moving in space subject to a potential function V(z), such that V(0)=0 and V(−z)=V(z) where z is the location coordinates of a point and V"(z)≥0 for all z. The time-independent Schrödinger equation for the wave function φ(z) for this physical system is

(−h²/2m)∇²φ(z) + V(z)φ(z) = Eφ(z)

where h is Planck's constant divided by 2π and E is the energy of the system. It can be reduced to

∇²φ = −μK(z)φ(z)

where μ= (2m/h²) and K(z)=E−V(z), the kinetic energy of the system as a function of particle location.

It should be noted that m is dimensional and therefore by the proper choice of units m can take on any numerical value. It is not the numerical value which is important; it is the process of taking the asymptotic limit which is important.

Here is an example of what the K(r) might look like.

However, in the determination of probability distributions constant factors are irrelevant because in the normalization process they cancel out. Note that the above equation may also be expressed as

∇²φ = −μE(1−V(z)/E)φ(z)

This indicates that it is the variation in the energy E relative to the potential V(z) that is important. Let V(z)/E be denoted as U(z). Then instead of thinking of the issue being what happens to φ(z) as E increases without bound, it is what happens to φ(z) as U(z)→0 for all z. But first it is necessary to find a way to deal with the rapid oscillations in φ(z). Here is an example of φ²(z) for 1D space. It is for a harmonic oscillator, where V(z)=½kz².

What happens when E increases is not so much that the level of φ(z) increases but instead the density of the fluctuations increases. The range over which φ(z)² is nonzero also increases.

The equation for the wave function can be reduced to

∇²φ = −J(z)φ(z)

where J(z)=μK(z) and φ²(z) must be normalized.

The Classical Model

Consider again a particle of mass m moving in space whose position is denoted as z. The potential field given by V(z) where V(0)=0 and V(−z)=V(z). Let v be the velocity of the particle, p its momentum and E its total energy. Then

E = ½mv² + V(z)

Thus

v = (2/m)^½(E−V(z))^½

For a particle executing a periodic trajectory the time spent in an interval ds of the trajectory is ds/|v|, where |v| is the absolute value of the particle's velocity. Thus the probability density of finding the particle in that interval at a random time is

P(z) = 1/(T|v(z)|)

where T is the total time spent in executing a cycle of the trajectory; i.e., T=∫dt=∫dx/|v|. It can be called the normalization constant, the constant required to make the probability densities to sum to unity. This is the time-spent probability distribution for the particle. Thus

P(z) = [(m/2)^½/T]/(K(z))^½

The constant factor of (m/2)^½/T is irrelevant in determining P(z) because it is also a factor of T and thus cancels out.

The final result is that the time-spent probability distribution is thus inversely proportional to (J(z))^½.

J(x) is proportional to kinetic energy and particle velocity is proportional to (J(x))^½, as is also momentum p. Therefore the probability density function is inversely proportional to (J(z))^½.

Increasing the Mass
of the Particle

Now consider two harmoneeqic oscillators of the same mass m and the same stiffness coefficients k. Moving separately they would have the same oscillatory frequency

ω = (k/m)^½

If the particles are fastened together the system has a mass of 2m and a stiffness coefficient of 2k. Thus its oscillatory fquency would be

(2m/(2k))^½ = (k/m)^½ = ω

the same as when the particles are not fastened together.

Now consider the effect of the double particle on the coefficient function in the Helmholtz equation. That function is equal to

H(z) = (2m/h²)^½ (E−V(z))sup>½

When m→2m, V(z)→2V(z), and E→2E the resulting coefficient H₂(z) is

H₂(z) = (2(2m)/h²)^½ (2E−2V(z))^½
which reduces to
2H(z)

Trembling Motion

Just as the velocity of the Classical Model determines its time-spent probability distribution, the quantum probability distribution from Schrödinger's time-independent equation determines the velocity profile of the quantum particle. That profile consists of a sequence of intervals of slower movementby faster movement. The intervals of slower movement correspond to what the Copenhagen Interpretation erroneously calls allowed states and the intervals of faster movement correspond to what it erroneously identifies as instantaneous quantum jumps.

Erwin Schrödinger labeled this sequence of slower-faster movement as zitterbewegung (trembling motion). Zittering seems to be a good descriptive term for the phenomenon. It may ultimately stem from the discreteness of space and time.

Since the frequency of the oscillations is proportional to the square root of the coefficient, their wave length is inversely proportional to that quantity. Thus combining two particles results in a zittering 1/√2 in length.

More generally combing N particles results in a coefficient N times larger and a wave length 1/√N times smaller. Since mass and the number of particles are proportional the wave length of the zittering interval is inversely proportional to the square root of the mass of the system.

Conclusion

Increasing the mass of a physical system along with corresponding quantities such as physical scale and energy results in a more rapid zittering that is inversely proportional to the square root of that mass. The spatial averaging of the motion over the interval of that zittering gives a pattern that asymptotically approaches that of the classical model.