San José State University
& Tornado Alley
Quantum Mechanical Analysis and the|
Correspondence Principle with Special
Emphasis for that of a Hydrogen Atom
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 Schroedinger 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 das 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.
The conclusion which is developed here is that the solutions to Schroedinger's equations pertain to the proportion of the time a system spends in its various allowable states. This is a probability density distribution that even deterministic classical systems possess. The proposition that the Correspondence Principle dictates that the probability density distributions of quantum mechanics pertain to time spent probability density of classical analysis is stunningly simple and unavoidable. As the energy of a system increases the result of quantum analysis is always a probability density distribution and even after averaging the result is a probability density distribution. If the limit of the quantum analysis is to be compatible with classical analysis it must correspond to some probability density distribution for deterministic classical analysis. What can that be other than the time spent probability distribution? Thus there is no possibility that the result of quantum analysis is some disembodied probabilities and that the basic particles exist only in terms of those probabilities as is maintained in the Copenhagen Interpretation of the wave function in Schroedinger's equations.
It is useful at this point to introduce the terminology of static versus dynamic appearances. The blurred disk of a rapidly rotating fan is its dynamic appearance; the solid, nonrotating fan is its static appearance. In its dynamic appearance the material of the fan appears to be smeared uniformly throughout an unchanging disk. It is not so much that the Copenhagen Interpretation is completely wrong as that it applies to the dynamic appearance of a physical system rather than its static reality. Effectively it is talking about blurred disks and not about fans per se. To realize how blurred are the systems are at the quantum level consider that the electron in a hydrogen atom is revolving about the proton quadrillions of times per second. Analysis based upon Schroedinger's equations has nothing to say about the reality of the particles of physical systems.
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/(4πε0).
For a planet orbiting a star the Hamiltonian is
where m is the mass of the planet and M the mass of the star. The variable p is the momentum of the planet, The parameter G is the gravitational constant.
Note that these two Hamiltonians are mathematically the same; they only differ in the names given to the parameters
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 Schroedinger's equation is then
where ψ, called the wave function, Schroedinger left unspecified but now is interpreted as such that |ψ|² is the probability density function for the system.
For physical systems such as a harmonic oscillator in which there is but a single variable x the Laplacian ∇² is equal to (∂²_/∂x²) and the models are solvable. The solutions involve rapid fluctuations in probability density over x and the spatial averaging eliminates the fluctuations leaving the time-spent probability density function of the classical analysis. Here is an illustration.
The heavy line is the classical time-spent probability density distribution.
The previous analysis was applied to the electron of a hydrogen atom. The conventional analysis 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 being separated into three ordinary differential equations for the different factors. This assumption is not required by the physics; it is only a mathematical convenience which allows the solution of the equations but it results in physical implications. In particular, the separation-of-variables assumption implies a certain symmetry of the PDF's. And, more importantly, for systems involving particles the separation-of-variables assumption implies the particles cannot have trajectories.
The conventional quantum analysis of the electron in a hydrogen atom 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 added to the set of three. The spin number could only be ±½.
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 analysis 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 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 conventional quantum analysis presumes that the energy is bounded from above by zero. Thus the limit to the conventional solution as energy increases without bound cannot be determined. However there are many macro systems with negative energy. For example, consider the planets of our solar system. It would take an enormous amount of energy to remove a planet from the gravitational field of the Sun. Thus the Correspondence Principle involves E→−∞ as well as E→+∞. Also there must be a correspondence between the limit of the quantum analysis and the classical analysis as m→∞.
There is no way the extension of the conventional quantum analysis as E→−∞ can be compatible with the classical analysis of a particle in an inverse distance squared field. Thus the conventional quantum analysis of an electron in a hydrogen atom is incompatible with the Correspondence Principle and thus invalid. The conventional quantum analysis is dictated by the separation-of-variables assumption rather than the Correspondence Principle. The separation-of-variables assumption is a mathematical convenience for getting a solution regardless if that solution has any physical validity. As stated previously the separation-of-variables assumption precludes there being any trajectory for the particles of a physical system. The notion that subatomic particles do not have trajectories does not come from analysis; it come plain and simply from the separation-of-variables assumption.
The above analysis does not mean results of the QM analysis of the electron of a hydrogen atom have 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 necessarily invalid. Some of the results derived from that conventional quantum analysis may still be valid; they just need to be derived in a different way.
The conventional quantum analysis presumes a hydrogen atom is stable; that it cannot be disassociated without a positive input of energy. However it is entire possible for an electron to be moving at some positive velocity at distance from the proton that makes its potential energy nearly zero.
In this case a moving electron is diverted from its straight line path toward the proton but it is not captured. The direction of its movement is changed as shown in the diagram below.
This interaction is described by the time-dependent Schroedinger equation
where i is the square root of negative one; i.e. the imaginary unit.
The time-independent Schroedinger equation for the electron in a hydrogen atom can be put into the form
where ∇² is the Laplacian for the coordinate system, k²(r) is equal to (2m/
h²)[E − α/r] and E is energy,
m is the mass of the electron, α/r is potential energy and h is Planck's constant
divided by 2π. The wave function φ is such that |φ|². is the probability density. This equation may be called a generalized Helmholtz equation.
For a polar coordinate system the Laplacian is
In cases where analytic solutions are not available, computational physics provides methods for numerical solution. Those methods require initial values and there are no conditions that constitute initial values in the solution of a time-independent Schroedinger equation.
The nature of the solution for it to be compatible with the Correspondence Principle can be specified. It involves fluctuations in the probability density as a function of θ as shown below.
A numerical simulation of the solution of the time-dependent Schroedinger equation for an electron with positive total energy in the field of a proton was carried out. The simulation did not use the separation-of-variables assumption and the results indicated the analysis was compatible with the Correspondence Principle.
The quantum analyses of one dimensional systems are compatible with the Correspondence Principle. The probability density distributions that come these analyes correspond to the classical time-spent probability density distributions.
The solutions to the time-independent Schroedinger equation for an electron in a hydrogen atom are entirely dependent upon the separation-of-variables assumption. As such they are incompatible with the Correspondence Principle.
Thus the valid quantum theoretic analysis corresponds to the time-spent probability density distributions.
The force on an electron due to a proton at a separation distance s is
where G is a constant and e is the absolute value of the charge. The change in potential energy ΔV due to a movement from a separation distance of s1 to 2 is given by
when the potential energy at s is given as − Ge²/s the potential energy at an infinite separation is presumed to be zero. This arbitrary assignment of a value to V(∞) is asserted to be of no consequence, is that really true. The potential energy V(s) more generally may be expressed as
The total energy E is then T
where K(v) is the kinetic energy of the electron as a function of its velocity. Thus total energy E contains the arbitrary component V(∞) and hence the term energy where employed in analysis is actually (E−V(∞)).
The energy levels for an electron in a hydrogen atom according to the conventional quantum mechanical analysis are given approximately by the formula
where me is the mass of the electron, c is the speed of light and α is the fine structure constant (≅1/137). The integer n is called the principal quantum number of the system.
There should be a correspondence of the quantum analysis with the classical analysis as the mass of the system increases without bound.
Suppose the particle be considered is the Earth with a mass of 6×10+24 kilograms. In Planet Energies the total energy for the Earth in its orbit is computed to be −-2.6×10+33 joules. This means that the principal quantum number for the Earth should be given by
But the probability density distribution for the Earth should look something like this:
Obviously the conventional quantum analysis for the electron in a hydrogen atom is incompatible with classical analysis and reality.
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