The Basic Single-Particle Aleatric Field
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In 1929, in his Nobel Prize acceptance speech, Louis-Victor de Broglie (pronounced de Bröy) said, ADetermination of the stable motion of electrons in the atom introduces integers, and up to this point the only phenomena involving integers in physics were those of interference and of normal modes of vibration. This fact suggested to me the idea that electrons too could not be considered simply as particles, but that frequency (wave properties) must be assigned to them also.@ De Broglie expressed and developed that idea in his 1925 treatise, AOn the Theory of Quanta.@
He noted in that treatise that Max Planck=s quantum hypothesis associates with any given energy a characteristic frequency and he noted that Einstein=s mass-energy theorem associates with any given mass a characteristic energy, so in inferred that he could associate with the given mass of a particle a characteristic frequency through the energy relations. Thus de Broglie tacitly associated a particle with a standing wave. If the particle moves, it acquires linear momentum and the wave, moving with the particle, acquires a wave number (or propagation vector), which we associate with the linear momentum, which de Broglie called the necessary partner of energy. To clinch the association between waves and particles de Broglie reviewed the principle of least action, particularly in the form devised by Leonhard Euler and Pierre-Louis Moreau de Maupertuis, noting that it describes both the propagation of light and the motions of particles. In that way de Broglie extended the wave/particle duality of the photon model of light to the quantum description of matter.
Throughout his treatise de Broglie used relativistic dynamics, thereby presenting to physicists a straightforward path to the creation of a relativistic version of the quantum theory, smoothly combining both aspects of the epiNewtonian physics of the Twentieth Century. Based on de Broglie=s inspiration, Erwin Schrödinger produced a non-relativistic wave mechanics based on his famous equation of the Hamiltonian function describing a particle to the particle=s total energy. Dirac, Klein and Gordon, and Proca quickly corrected that oversight by producing relativistic versions of Schrödinger=s Equation, those relativistic equations differing from each other in the amount of spin that the particles they describe have to carry. But instead of constructing Lorentz-invariant analogues of Schrödinger= s Equation, as Dirac, Klein and Gordon, and Proca did, physicists could have applied the Lorentz Transformation directly to the wave function that solves those equations. After all, in developing relativistic mechanics from the Newtonian version, we apply the Lorentzian analysis to the descriptive functions (e.g. those representing linear momentum and energy) rather than to the constraining equations. Now I want to try that approach on the quantum mechanical state function, taking de Broglie=s hypothesis as our foundational postulate, and see what happens to the quantum theory in consequence.
According to de Broglie, we associate with a particle of rest mass m0 traveling with a speed v along the x-axis of our coordinate frame the wave function
which wave function encodes the particle=s state of motion. In three-dimensional space that formula represents a plane wave propagating in the positive x-direction: if we look at a given phase of the wave (say, a crest or a trough), then as t increases with the elapse of time x must also increase to keep the argument of the exponential unchanged and thus representing that particular phase. In accordance with de Broglie=s hypothesis the wave number in that formula corresponds to the particle=s linear momentum in the mathematical form
and the angular frequency corresponds to the particle=s kinetic energy in the mathematical form
(In those equations aitch-bar, also known as Dirac=s constant, equals Planck=s constant divided by two pi).
That wave function in Equation 1 satisfies Schrödinger=s Equation, HΨ = EΨ, in the form
for which physicists define the linear momentum and energy operators to have the mathematical form
The requirement that the wave function satisfy Schrödinger=s Equation, especially when we add in the potential energy of any forcefield the particle encounters, establishes the particle=s state of motion, just as Newton=s second law of motion establishes a body=s trajectory when we specify the form of the force acting on the body. Thus we say that psi represents the particle=s state function, whatever form it may take: in the de Broglie-inspired version of the quantum theory it takes the form of a wave function.
Once we have established mathematical form of the state function we want to use it to produce a description of the particle=s contingent properties, to map out its world-line (to use Minkowski=s term). In accordance with Max Born=s hypothesis, we interpret the wave function through the statement that, when we have properly normalized it, the product of the state function with its complex conjugate(Ψ*Ψ) represents the probability density that the particle in the given state exists in a given element of spatial volume. Because the particle must exist somewhere we know that
stands true to Reality when we evaluate the integral over all space. If the particle carries some property q that we can measure, then there exists a mathematical operator Q (usually differential, as in Equations 5 and 6) that will act on the state function to extract a numerical value just as if we had made a measurement on the particle, giving us
By using that equation to modify Equation 7 we obtain the formula for calculating the expectation value of the property
Thus we obtain a kind of average, as if we had made the same measurement on a large number of identical quantum systems, which gives us all that we can rightly expect from a system subject to Heisenberg=s indeterminacy principle.
Following the convention among physicists I have used the Greek letter psi to represent the aleatric field of a particle or a quantum system. I derived the word aleatric from the Latin word for dice and note that it connotes chance, which we see most clearly in the quantum theory properly understood. The phrase wave function or state function, which we also represent with the Greek letter psi, denotes the mathematical description of the phenomenon that I call an aleatric field.
Thus we have a brief description of the classic quantum theory. But that theory, as presented, contains a flaw. In addition to not conforming to the theory of Relativity, it lacks an important element of de Broglie=s hypothesis; it takes no account of the frequency that we must associate with the energy manifested in the particle=s rest mass. So we need to modify the theory in a way that gives a stationary particle an oscillation at a frequency proportional to its rest mass and gives a moving particle a wave such as Equation 1 describes. And we must do so in a way that makes no inherent distinction between a moving particle and a stationary particle; that is, we must give the particle a description that differs between two observers by some aspect of the Lorentz Transformation between those observers= inertial frames of reference. The only entity that satisfies those requirements and still gives us something resembling Equation 1 associates with a stationary particle a standing wave, one that consists of the overlap of two waves, each like that in Equation 1, that are identical to each other in all respects except that they propagate in opposite directions. That is, we must have for our wave function
Further, we know that, in order for that description to conform to the requirements of Relativity, the component waves must propagate at a speed that no potential observer can ever achieve. If that proposition did not stand true to Reality, then some observer could travel as fast as one of the component waves and in that observer=s frame the Doppler shift would stretch that wave=s length to infinity (i.e. the wave number would have to zero out) and reduce the wave=s frequency to zero; in that frame, then, that component wave would simply not exist. Now we can imagine an observer moving with the particle accelerating into that frame in which the Doppler shift extinguishes one component of the wave function and then decelerating back into the frame occupied by the particle. The Doppler shift associated with the deceleration acting on the extinguished wave component will not restore it: multiplying zero by any number still yields zero. So this hypothesis gives us the means to change the fundamental description of the particle by moving ourselves out of and then back into one inertial frame. Reality will not conform to such a description of its behavior, so we must alter our description in a way that precludes such a thing from happening in our model of Reality; that is, we must add the proviso that the component waves propagate at the speed of light, the only speed that no observer can ever, even in theory, achieve. That proviso requires that the wave constants in Equation 10 give us
To modify the wave function to describe a particle moving in the positive x-direction at the speed u we can apply the Lorentz Transformation to Equation 10 via the relativistic Doppler shift. To that end we must conceive ourselves moving in the negative x-direction at the speed u relative to the particle. In that frame what initially appeared to us as a standing wave will propagate in the positive x-direction at the speed u. In that frame the first component of Equation 10, propagating in the negative x-direction, appears Doppler downshifted, having its wave number and frequency diminished (and its wavelength increased) and the second component, propagating in the positive x-direction, appears Doppler upshifted, having its wave number and frequency increased (and its wavelength diminished). As those waves beat against each other any point on the compound wave with a given amplitude will propagate in the positive x-direction at the speed u. For waves moving at the speed of light we have the Doppler factors
In that equation D+ represents the Doppler upshift and D- represents the Doppler downshift. We apply the Doppler shift to the wave parameters by simple multiplication, so in our moving frame we have the wave function of Equation 10 as
To see whether that wave function gives us a correct description of the particle we want to use it to calculate the linear momentum of the particle. Because we have two components in the state function, we must multiply the usual linear momentum operator by 2 to compensate. We thus apply the operator
to Equation 14 and incorporate the result into Equation 9. We thus obtain
But we already know from relativistic dynamics that we can describe the linear momentum of the particle as
so we must have
Combining that result with Equation 11 gives us
which looks like the relativistic version of Equation 3.
Now we can rewrite Equation 14 as
We can test the validity of that wave function by using it to calculate the expectation value of the total energy of the particle. To that end we apply the operator
which, again, we have modified by multiplying the standard energy operator by one half to account for the double-term form of the state function. We obtain the expectation value
which is exactly what Relativity leads us to expect.
Now we come to the next problem. The state function of Equations 10, 14, and 20 does not satisfy Schrödinger=s Equation. The relationship between the total energy (Equation 22) and the linear momentum (Equation 17) of the particle comes out as
which also comes to us from relativistic dynamics. So now we must do as Schrödinger did, identify the right side of that equation as the square of the relativistic Hamiltonian function, and write the quantum wave equation in relativistic form as
That equation actually looks to us more like a wave equation than the standard Schrödinger Equation does, but that thought comes to us only because the equation has the same form as does the electromagnetic wave equation. Indeed, if we let m0=0, we get exactly that.
On an historical note, we find that, although Erwin Schrödinger devised a version of Equation 24 in 1925, physicists actually call it the Klein-Gordon equation. Schrödinger found the equation unsatisfying and worked out his better-known equation to replace it. In 1927 Oskar Benjamin Klein (1894 Sep 15 B 1977 Feb 05) and Walter Gordon (1893 B 1940), inspired by Vladimir Aleksandrovich Foch (1898 Dec 22 B 1974 Dec 27), published their own version of it. Originally believing that it describes the electron, they discovered later that it describes particles that carry no spin at all. Thus it gives us the perfect equation for our initial effort to produce a relativistic quantum description of matter.
So far our description of a particle=s aleatric field yields a probability density that has the same value at every point for which we calculate it. Equation 20 describes a particle that exists effectively smeared uniformly over all of space. We like to think of particles as being a little more localized, so now we want to amend Equation 20 to describe a particle that, at any given instant, occupies a unique location in space, in the manner of a classical point particle, albeit with the indeterminacy required by Heisenberg=s principle. I will tackle that problem in another essay. For now, though, we have a good basis for describing a single particle in the quantum realm.
As a bit of a teaser I will note that up to now I have tacitly treated the amplitude A of the wave function as a constant. That amplitude represents an envelope around the wave function that localizes the particle, putting it at or near a specific point in space. We assume that the particle can exist at only one location (or in its immediate vicinity, given the inherent fuzziness of quantum descriptions), so we must describe the amplitude with a non-repeating function of x-ut for a particle moving in the positive x-direction at the speed u. In solving that problem we will have as our primary constraint the fact that if we were to change the value of Planck=s constant toward zero, converting our quantum system into the equivalent classical Einsteinian-Newtonian one, that envelope would have to shift into a shape described by the Dirac delta,δ(x-ut). A number of functions correspond to the Dirac delta in the limit as some parameter goes to zero. We will want to discern which of those functions correctly describes a real particle when that parameter corresponds to the quantum of action.
Finally, I can say that we have two ways to bring Relativity into the quantum theory. The conventional way leads us to replace Schrödinger=s Equation with a relativistic equivalent Hamiltonian function to obtain the Dirac Equation, the Klein-Gordon Equation, or the Proca Equation and then to solve the equation for the state function. The new way, the way that I have just followed, comprises the steps of modifying the state function to make it Lorentz covariant and of then determining the corresponding Hamiltonian function in order to devise the correct constraining equation.
With classical physics, the physics of 1900, physicists culminated a process begun about 300 years earlier by Galileo when he devised the law of accelerated motion from his experiments with balls rolling down grooved planks. We accept readily that the variables in the mathematical expressions of physical laws represent quantities that we measure more or less directly and that the equations expressing the laws of physics represent to us a kind of picture of how those quantities change, (of how matter moves).
In the quantum theory we find a radical departure from that pattern. At the center of the theory we have a mathematical object that does not represent a measurable quantity, but contains all of them. Mathematical operations applied to that state function represent interactions (e.g. measurements) with stochastic outcomes acting on the particle or particles that the state function represents. We represent the stochastic nature of this entity by the square of the state function denoting a probability density. We find that we must go from classical physics to quantum physics by way of the principle of least action and the calculus of variations.
How do we make the transition from deterministic physics to stochastic physics? In one we solve an equation, apply boundary conditions, and calculate the value of the measurable quantity and in the other we apply an operator to the state function and calculate an expectation value for the measurable quantity. In order that our physics accurately represent Reality those two different approaches must segue, one into the other, at some appropriate scale.
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