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The Conventional Quantum Mechanical Analysis of a Hydrogen Atom Violates the Correspondence Principle 

Neils 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. In atomic physics the extension is in terms of scale and/or the level of energy. 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 thermodynamics.
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. This analysis was applied to the electron of a hydrogen atom. This analysis implied the existence of a set of three integers which determined the PDF for the electron. These integers were called the quantum numbers for the state of the electron. Later a fourth quantum number, called the spin, was addedl to the set of three. The spin number could only be ±½.
The conventional analyis is carried out utilizing an assumption called the separation of variables assumption. This assumes that the wavefunction ψ(r, θ, φ) in spherical coordinates is of the form R(r)Θ(θ)Φ(φ). This results in the partial differential equation for ψ to be separated into ordinary differential equations for the different factors. This assumption is not required by the physics; it is only a mathematical convenience but it results in physical implications. In particular, the separation of variables assumption implies a certain symmetry of the PDF's.
The classical version of an electron and proton revolving about their center of mass is deterministic but a PDF can be formulated in terms of the proportions of the time they spend in each state. This corresponds to the probability of finding a particle in a particular state at a randomly chosen point in time. This anaysis is carried out at Two Body Problem. The result is that the distance r of the electron from the center of mass is given by
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. The 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.
Here is a depiction of the conventional PDF for an electron in hydrogen 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 above analysis does not mean a QM analysis of the hydrogen atom has to violate the Correspondence Principle. It only means that the analysis based upon the separation of variables assumption is incompatible with the Correspondence Principle and is likely invalid. Some of the results derived from that conventional analysis may still be valid; they just need to be derived in a different way.
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