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and the Correspondence Principle |
In a previous study made the case that the use of the separation-of-variables assumption to solve the time-independent Schroedinger equation gives solutions that while mathematically correct are physically irrelevant because they are incompatible with the notion of particles and their trajectories and hence violate the Correspondence Principle.
The separation-of-variables assumption is that a function F(x, y) is equal to G(x)H(y) and likewise for functions of three or more 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.
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 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 Schroedinger'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
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 soy to the separation-of-variables assumption.
Strictly speaking there are three cases in which separation-of-variables is compatible with particles and their trajectories. These are
The first of these corresponds to circular orbits with r and ψ constant. The second corresponds to a straight line orbit in which the center of the electron passes through the proton. The third does not correspond to any physically meaningful case.
The second case will be considered first. There is a natural inclination to think that this case somehow involves a small electron impossibly passing through a larger proton. But this puzzling case is not a matter of a small electron passing "through" some doughnut shaped proton but instead that of a big spherical shell of an electron passing over a smallish proton. This eliminates the apparent impossibilities of the previous conception. It is still true that the center of the spherical shell of the electron passes through the proton.
When the proton is inside of the spherical shell of an electron there is no force between them so the possible "infinity" from 1/r² does not arise.
This case may be identified as the zero angular momentum case.
The potential energy function is then V(r)=−α/|r| for |r| greater than the radius of the electron and has the shape shown below
where α is a positive constant
The Hamiltonian function for the system is
where p is the momentum of the electron and m is its mass.
The Hamiltonian operator is then
From the total energy E=½mv² −α/|r| it is found that the velocity v is given by
This can be solved analytically but it is convenient to get the solution numerically i.e.,
Here are the results of the computation.
The electron virtually jumps where it passes the proton.
Consequently the relative probability density as a function of location is
A circular orbit of radius r requires a balance between the attraction of the electron for the proton and the centrifugal force on the electron due to the curvature of the orbit. Let e be the charge of the electron and proton and m the mass of the electron. The required balance is then given by
where G is the electrostatic (Coulomb) force constant.
Since |v| is constant for a fixed r the probability density function is also constant throughout the circular orbit and is thus equal to 1/(2πr).
The hydrogen atom is stable and hence its total energy E is negative. By the Virial Theorem the kinetic energy is one half the magnitude of the potential energy and thus potential energy is one half of total energy; i.e.,
According to the Old Quantum Theory of Niels Bohr the absolute value of the angular momentum m|v|r is an integral multiple of Planck's constant. This leads to a quantization of r, but for a fixed r the velocity is constant around the orbit. Thus the time spent probability density function is constant around the orbit as in the case of the classical analysis. Thus the Old Quantum Theory satisfies the Correspondence Principle. The quantum theory relying on the separation-of-variables assumption is a different matter.
The probability density function for an electron in a hydrogen atom which results from the solution of the time-independent Schroedinger's equation using the separation-of-variables assumption is of the form
The function Θ(θ) is the square of a trigonometric function, say A·cos²(nθ), where n is an integer. The higher the energy the larger is n and the more rapid the fluctuations over the range of θ from 0 to 2π. Consequently the spatial average of Θ(θ) will converge to a constant over a circle, just as in the case of the classical situation. But the behaviors of R(r) and Ψ(ψ) are another matter.
As the magnitude of the energy increases without bound, there is no tendency for spatial averages of R(r) and Ψ(ψ) to converge to point values corresponding to the ψ=0 and r=2α/|E| of the classical case. Therefore the separation-of-variables solutions in general are not compatible with the Correspondence Principle and hence are not physically valid. They are mathemically correct but physically irrelevant. It is only the two special cases of circular orbits and zero angular momentum orbits that satisfy the separation-of-variables assumption and are physical feasible and relevant.
The solution of the radial equation for R(r) is
where ρ is proportional to r, A is a constant and m, p and q are integers, called quantum numbers. The function L_{p}^{q}(ρ) is a Legendre polynomial. Here is what the radial component of probability density looks like for small values of the quantum numbers,
The radial component of probability density has a number of peaks which are suppressed by the negative exponential function. .l
If the probability density is plotted as a function of r and θ the picture would look something like the following. .
In the diagram the pink circles represent the peaks in probability density of the electron. The blue circle represents the proton The number of peaks in a circular orbit increases with increases in the principal quantum number. The radial function has a number of peaks and that number increases with the principal quantum number. Thus there is no tendency for the radial function to reduce to a spike function for a specific orbital radius as would have to occur if the quantum analysis is to be compatible with the Correspondence Principle. Therefore the conventional quantum analysis of an electron in a hydrogen atom based up the separation-of-variables assumption, while mathematically valid, is physically invalid.
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