Quotes:
I think that I can safely say that nobody understands quantum mechanics.... I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, "But how can it be like that?" because you will get "down the drain," into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
- R. Feynman
Quantum mechanics, that mysterious, confusing discipline, which none of us really understands but which we know how to use.
- M. Gell-Mann
What is most basic in physics is not the mathematics but rather the set of underlying concepts.
- A. Einstein
So an angel can be in one place in one instant and in another in the next instant, without any time intervening. And he will be partly in one place and partly in another, not as though his substance were divisible into parts; but simply because his power will be applying itself, at any given moment, in part to one place and in part to the place next to it.
- Thomas Aquinas
The goal of "mechanics" - classical mechanics or quantum mechanics - is to describe the motion of something.
In classical mechanics, to describe the motion of an object - a ball, a planet, etc. - one needs to know the following:
1. The starting position of the object.
2. The initial momentum (or speed) of the object.
3. The forces acting on the object or,
- equivalently, the energy as a function of position and momentum, or
- also equivalently, the Lagrangian (essentially the kinetic energy minus the potential energy) as a function of position and momentum.
Classical mechanics doesn't do so well when (a) the object is moving very fast or (b) the object is very small. Describing motion when the object is moving very fast is what relativity is all about, and we won't concern ourselves with that here. Describing the motion of very small objects is the field of quantum mechanics.
In quantum mechanics, as in classical mechanics, one works with the variables position and momentum, and with the energy (or the Lagrangian) as a function of position and momentum. And as in classical mechanics, one wants to be able to postulate equations that interrelate position, momentum, and energy, and will let us know where something will be when.
The complication is that for very small "objects" (an electron, a photon, etc.) one cannot think of the object as a small localized ball that moves from place to place. The best that one can do is to assign probabilities for finding an object at a particular place at a particular time. Consequently, the best that quantum mechanics can do is to predict the probability of finding objects at particular places at particular times.
So how does one postulate the equations of quantum mechanics? There are several alternative formulations. What we look for is the formulation that's the simplest and, of course, gives correct predictions.
The standard set of postulates for quantum mechanics is given as well by Levine as by anybody else. This set of postulates is HERE (note: the links may open directly, or you may have to download them; they're all in pdf format). There are six postulates. One grows to love them more the more one is accustomed to them - there's no love at first sight!
There is a simpler formulation, called the phase-space dynamics formulation. At present, there's no experimental test of which is correct. One can take one's choice: a set of six postulates in the conventional formulation or two postulates in the phase-space dynamics formulation (there's Occam's Razor if one wants a basis to make a choice, given no experimental tests).
For the phase-space dynamics formulation, we have:
Given: The dynamics of a physical system is determined by functions of the two variables q and p, which we identify as position and conjugate momentum; for example, by the Lagrangian (the kinetic energy minus the potential energy)
L(p,q) = ½ p2/m - V(q)
The dynamics of such a system can then be completely described by a normalized (distribution) function D(q,p,t) which depends on q and p, and which may also change in time. ["Normalized" means that when the distribution function is integrated with respect to both q and p, the result is 1 - the total probability of finding the particle somewhere is 1. Also, for example, when the distribution function is integrated with respect to p, the result is the probability for finding the particle as a function of position and time.]
The Problem: Find a dynamical equation for the distribution function D(q,p,t) which leads to and/or is consistent with quantum mechanics. The equation appears reasonably complicated, so it won't be given here. Read one of the references in the links below for one such equation and its consequences.
The phase-space dynamics formulation was initially published in Theoretica Chimica Acta. The article is available HERE. A pamphlet giving the description in simpler language is available HERE. And as an added bonus, an article that deals with the quantum mechanics of photons is HERE. Enjoy.
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