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The Separation-of-Variables Assumption and Its Effect on the Results of Quantum Analysis |
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The purpose of this material is to demonstrate that a standard mathematical assumption, called separation of variables, which is used in solving the differential equations arising in the quantum mechanical analysis of physical systems involving particles precludes the results applying to particles. The results of such analysis, although mathematically correct, are physically irrelevant. The solutions from such analysis cannot occur for physical particles and the disappearance of particality (particleness) is due to nothing other than the assumption of the separation of variables.
Consider the spherical coordinates, (r, ψ, θ), for the center of a particle that may be a point particle or a spatially distributed one. The variable r is the distance from the center of a force field, ψ is the azimuthal (latitude) angle and θ is the orbital (longitude) angle. Let S denote the column vector of these three coordinates.
Suppose some physical quantity is a function of the spherical coordinates of the center of a particle, say P(r, ψ, θ). In quantum analysis P could be the probability density for the particle. The separation of variables assumption is that the function P(r, ψ, θ) is the product of three functions, each depending on only one of the variables; i.e.,
When the separation-of-variables assumption is applied to the partial differential equation arising from Schrödinger's time-independent equation it leads to three ordinary differential equation that are more easily solved than the partial differential one. The problem is that the essential nature of a particle is that it has a trajectory that gives its coordinates as a function of time, say (r(t), ψ(t), θ(t)). If the trajectory is periodic with a period T then (r(t+T),ψ(t+T), θ(t+T))=(r(t), ψ(t), θ(t)). The time variable may be eliminated by solving for t as a function of θ and making the other coordinates functions of θ.
The classical analysis of a particle in an inverse distance squared force field is given in 1/r² force field. The full analysis of the two body problem for an inverse distance squared force is given in The Two Body Problem.
In the case of a particle in an inverse r² field the coordinate system can be chosen such the particle orbit has ψ=0 and r=A/(1+εcos(θ)), where A is a constant, ε is the eccentricity of the elliptical orbit and θ is the angle measured from the point of minimum r. This is not compatible with the separation-of-variables assumption in which the three coordinates are independent from each other and thus the whole notion of a trajectory disappears and along with it the notion of a particle. But that is not from the physical dynamics of the system; it is due solely to the separation-of-variables assumption.
The time-spent probability density is proportional to the time spent in an interval of path length; which is the reciprocal of particle speed (the absolute value of velocity). This reciprocal speed is given by
where B is a constant, which is eliminated when the probabilities are normalized; i.e., made to sum up to unity. The probability density has one point at which it is a maximum and another point at which it is a minimum. This PDF is nonzero only on an ellipse in one plane. Here is an example for the case of eccentricity equal to 0.25. The horizontal axis is in terms of θ measured in radians.
This PDF does not obey the separation-of-variables assumption and therefore no extension of the conventional QM PDF's which all obey the separation-of-variables assumption can be compatible with it.
Niels Bohr nearly a century ago observed that classical analysis for many areas of physics had been empirically verified. Therefore, for any quantum mechanical analysis its appropriate extension to the realm of classical analysis should agree with the classical analysis. This is called the Correspondence Principle. In atomic physics the extension is in terms of scale and/or the level of energy. What comes out of the solution to the time-independent Schrödinger equation for one dimensional systems is that with higher energy there are more and more rapid (dense) fluctuations in probability density. When these probability densities are averaged over time (and/or space) the result should be the time-spent probability density function of classical analysis. This is a modified version of the conventional Correspondence Principle in that it involves time-averaging as well as finding the limit as energy increases without bound. Classical means time-averaging as well as being macro with respect to scale and energy. The limit of the quantum theoretic solution is necessarily a probability density function. Classical analysis is deterministic but there is the proportion of the time the system spends in its various allowable states that makes sense as a probability density function.
In radiation physics the extension is the limit as h, Planck's constant, goes to zero. In statistical mechanics the limit as the number of molecules increases without bound should agree with classical thermodynamics. Although it is never mentioned there should be an extension of the analysis as the mass of a body and/or its charge increases without bound.
Schrödinger's wave mechanics is and has been the preferred formulation of quantum mechanics almost from its creation in the late 1920's. It generates a partial differential equation whose squared magnitude is the probability density function (PDF) for the particles of a system. The analysis starts with the Hamiltonian function for the system. For an electron in the field of a proton the Hamiltonian function is:
where p is the momentum of the electron, m is its mass and r is the distance from the proton. The parameter α is usually represented as 1/(4πε_{0}).
The Hamiltonian function H of a system is converted into a Hamiltonian operator H^ by replacing the momentum squared, p², with
−h²∇², where h is Planck's constant divided by 2π and ∇² is the Laplacian
for the coordinate system. The time-independent Schrödinger's equation is then
where Ω, called the wave function, which Schrödinger left unspecified but now is interpreted as such that |Ω|² is the probability density function (PDF) for the system. E is the energy of the system.
In the conventional quantum analysis Ω((r, ψ, θ) is assumed to satisfy the separation-of-variables assumption. This leads to three separate ordinary differential equations which can be solved when E is quantized. There are two quantum numbers involved in the solution.
Here is a depiction of the conventional PDF for an electron in a hydrogen atom with principal quantum number 4 and magnetic quantum number 3.
In this depiction shading, either white or black, indicates high probability density. It is not plausible that the conventional QM PDF for high principal quantum number would approach the classical PDF which is nonzero only on an ellipse.
The time-dependent Schrödinger equation for an electron in the field of a proton can be solved numerically and the asymptotic trajectory determined. The only difficult is what should be taken as the initial conditions. The asymptotic solution may well be independent of the initial condition. Preliminary results for an especially simple case indicate that the solutions satisfy the Correspondence Principle and thus are valid.
Since the solutions derived from the separation-of-variables assumption do not satisfy the Correspondence Principle they are not physically valid. The situation is like the old joke about a man searching under a street light for his watch. A passer-by helps him and concludes the watch is not there. He says, "Are you sure you lost your watch here?" The searching man replies, "No, I lost it over there." The passer-by then asked why he is searching for it under the street light, to which the searcher replied, "Because the light is better over here!"
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