CP QM Ontology and Schroedinger's Equation

Posted by algore on January 01, 2000 at 00:00:03

There is no generally accepted ontology for small particles in the domain of Quantum Mechanics (QM). The experimental facts are beyond dispute, and the math (Schroedinger's equation, other QM math) is very accurate; but no one knows what the "particles" are actually doing. They don't behave like baseballs, that's for sure.

My proposed ontological picture, called "Conscious Particles" (CP), is somewhat like Bohm's. It has particles moving in discrete jumps with rudimentary volition, but that's not necessary for the math: we can just say the jump has a random component. One point in its favor: Schroedinger's equation can be derived from the basic CP assumptions, by quantizing Newton's law of motion (I hope).

it's not easy explaining this to an audience familiar with QM; here, I have to explain QM too .. I'll do my best. I'll discuss QM first for non-mathematicians ..


QM is a branch of physics that describes the motion of matter at the very small (atomic) level. The fundamental dynamical equation of QM is called Schroedinger's equation; it's based on Newton's equation (force equals mass times acceleration). Given the position and motion of something, dynamical equations predict where it will be in the future.

By playing with the equations (for instance, "integrating by x") you can derive many interesting quantities: velocity, position, momentum, energy (3 different types), and action. These math entities correspond pretty well with common usage. For instance if something is going faster, or is bigger, then it will have more momentum and (kinetic) energy. "Action" is the least intuitive; it's the product of energy and time. The more energy the thing has, and the longer the time, the more action.

Newton's equation gives a continuous solution: between any two positions, no matter how close, there's another one. Zeno's "paradox" points out a contradiction: if motion were really continuous, it would never happen, because there's no first step to take. This is not a paradox at all, instead it's a proof, by contradiction, that Newton's solution is wrong. The world really isn't, and can't be, continuous.

Time, position, energy, action, and everything else, occur in tiny little steps. They "don't exist" in between. Since they're all related, if you quantize one, you determine how all the others must step. I would have guessed time would be the fundamental quantum; but instead it's action. "Planck's constant" determines this "quantum of action". Schroedinger's equation is the result of quantizing Newton's equation.

Classical mechanics gives one solution trajectory for a given problem. Think of a ball thrown through the air: obviously, it can go along only one curve or track through space. However the discontinuous quantum solution takes little steps through the solution space. At each step it must be right in the track of the classical solution, but in-between it can be anywhere, or more accurately "nowhere". The body can "step off" the classical, pre-determined track, as long as it gets back to reality (any classical solution) at the end of the step. Thus particles spread out in forward directions on multiple trajectories. Since this is like wave motion, Schroedinger's is often referred to as the wave equation. When a particle has more than one way of getting to a given place, thy cancel each other out if they're out of step .. this is called "interference".

Multiple particles, traveling in a group or packet, can easily be imagined spreading out; but even one particle alone has its "wave". There's an ontological contradiction or puzzle here, because nobody ever observed a particle in more than one place, or with more than one speed. Most physicists follow Von Neumann's picture. Imagine a ghostly cloud spreading as prescribed by the wave equation, when nobody's looking. When you look, it suddenly and randomly becomes a specific single particle, and the cloud disappears: the "collapse of the waveform" caused by an observation.

I have a different way of picturing a single particle's quantum motion. You can't say say which picture is more accurate; they both fit the experimental facts. I imagine that the particle is always a single, whole thing. It's conscious and figures out its path according to certain rules. First it figures out where it could be at a given time, that is, calculates the wave form. The wave, then, exists only as knowledge or information internal to the particle. When forced to "collapse the waveform" by a physicist's observation, it randomly chooses one of these possibilities it's figured out.

Note that all other proposed "quantum ontologies" are as hard to accept as these two. "Many Worlds Theory" postulates infinite universes; Bohm postulates ghostly "pilot waves", etc. There's a very good reason they're weird: the experimental facts of QM are hard to believe.

Physicists generally don't worry about these questions. The equations give the right answer, that's all. They don't have to care about ontology, what the particles are actually doing, for one very good reason: scale. Quantum particles are very small and very quick. A typical electron whirring around a proton in a hydrogen atom takes about a billion billion steps in one second. Therefore physicists study only long-term, stable, "time-independent" solutions, sometimes referred to as "the amplitude equation" or "quantized energy levels". These aew due to interference ..

It's very hard to deal with these concepts non-mathematically. I haven't even gotten to linearity of the equations, spin, and many other things, and already it's out of control. I'll try a math explanation.


Newton's second equation is

force = mass *acceleration, or
m * a - F = 0.

define this function as "maf":
maf(x,t) = m * a(x,t) - F(x,t)

now integrate it once by x, then once by t, to get "action":
A(x,t) = action = int int maf(x,t) dx dt

In the simple case of a free particle, with constant momentum:

A = p*x - E*t

where p is momentum and E is total energy. However normally, A will be a messy integral.

The particle moves in little steps of a fixed amount of action. Planck's constant, the quantum of action, is

h = 6.547 x 10^-27 erg sec.

This cycle is represented mathematically as a complex wave traveling thru spacetime, which turns one cycle per quantum. Only the values at the ends, where the particle can actually exist, are real numbers. This complex wave is written as:

wave(A) == e^(2pi/hi * A), where e=2.718, pi = 3.1416, and hi = h * square root of -1.

Schrodinger's Equation can be derived from this, I think .. but I can't quite do it. For a free particle, it's easy. If A = (p*x - E*t), then the partials of the wave equation are

partial with respect to x, twice: (2pi/hi * p)^2 * wave(A)
partial with respect to time: 2pi/hi * E * wave(A)


p^2 * wave(A) = (hi / 2pi)^2 * (partial with respect to x, twice)
E * wave(A) = (hi / 2pi) * (partial with respect to time)

Now, multiply the energy equation by wave(A):

Energy equation: p^2/(2*mass) - V(x) = E
multiply by wave(A): p^2/(2*mass) * wave(A) - V(x) * wave(A) = E * wave(A)

Substitute the two terms derived from the partials above, and you get Schroedinger's, and of course you can add these solutions with a definite momentum and energy to get a general "psi" function, since the equation is linear. So that's fine for the free particle, but I can't yet make it work for the general case. This should give the idea, though.


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