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The Asymptotic Equality of a Spatial Average of the Probability Density from Schroedinger's Equation for a Particle in a Potential Field and the Classical Time-Spent Probability Density for that System |
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Let V(x) be the potential energy function for the particle with V(0)=0. If m is the mass of the particle and v is its velocity then the total energy E is
Therefore
where K(x) is the kinetic energy of the system expressed as a function of the location of the particle.
The time spent in an interval dx at x is then
Thus the probability density of finding the particle in an interval about x is
where T the time required by the particle to complete its orbit; i.e.,
But the factor (m/2)^{½} cancels out in the normalization process and is thus irrelevant and can be discarded. Thus
where T_{1}=∫dz/K(z))^{½})
The crucial result is that the classical time-spent probability density distribution is inversely proportional to (K(z))^{½}.
If a probability density function P(z) is proportional to a function g(z), say
then the constant of proportionality is irrelevant because
But
Thus the coefficient α is irrelevant in determining P(z).
The Hamiltonian function for the system is
where p is the momentum of the particle.
The Hamiltonian operator H^{^} for the system is created by substituting
−ih(∂/∂x)
for p in the Hamiltonian function, where h is Planck's constant divided by 2π and
i is the imaginary unit.
The time-independent Schroedinger's
equation for the system is then
where E is the total energy and φ is the wave function.
This equation can be expressed as
with K(x)=E−V(x).
This may be considered the very simple case of a generalized Helmholtz equation
with k²(x) being equal to (2m/h²)K(x).
The graph below shows the probability density distributions for a harmonic oscillator.
The thin line is the quantum mechanical solution and the thick line is the time-spent probability density distribution based upon classical analysis.
If k(x) is a constant over some interval then the solution over that interval is
Let φ(0)=0 so the solution is
Then
The maxima of φ are equal to
This occurs at kx=π/2 and hence x=π/(2k).
The average value of φ²(x) over the interval from 0 to π/(2k) is ½B² since the average value of sin²(x) over that interval is ½. Thus the probability of the particle being in the interval [0, π/(2k)] is πB²/(4k). If there are N equal intervals over the range of x then
N and k are related to the energy of the system by the relations
Hence
Thus the spatial average of the probability density from Schroedinger's equation is
where γ is the irrelevant constant factor of [2(2m)^{½}h/π].
Any empirical observation of a physical system involves a time average. Strictly instantaneous values cannot be observed because in an instant no energy can be transferred. For a particle executing periodic motion a time average is closely related to a spatial average. Thus any empirical observation of a particle system involves spatial averages instead of instantaneous values.
Solutions to Schroedinger's equation involve rapid oscillations and the higher the energy the more rapid the oscillations. Spatial averages eliminate the oscillations and what is left is the time spent probability density distribution of classical mechanics.
Now consider the limit as E→∞ at a fixed x and the ratio R of P_{Sch} to P_{C}
where T_{C} and T_{Sch} are the normalization factors for the two distributions and are closely related to the time required to execute an orbit and thus are equal. Therefore the above formula reduces to
As E increases without bound so does K(x) and hence V(x)/K(x) goes to zero. Thus asymptotically at each x, P_{Sch}(x) goes to P_{C}(x). Thus the probability density distribution from Schroedinger's equation is tied to the classical time-spent probability distribution rather than, as in the Copenhagen Interpretation, to some intrinsic uncertainty of the particle.
As articulated by Niels Bohr the Correspondence Principle is that for Quantum theory to be valid its solution must asymptotically approach that of the classical analysis. It is often thought that there are two regimes: the microscopic quantum regime and the macroscopic classical regime with a boundary scale where the two match. This is most likely erroneous. A better characterization of the situation is that the quantum theory applies at all scales but there is a scale at which the smoothed version of the quantum theoretic behavior is indistinguishable from the results of classical analysis. This is analogous to relativistic effects applying at all velocities but the difference between them and classical analysis being infinitesimal at low velocities.
In order for the quantum theoretic analysis of a particle in a potential field to be valid according to the Correspondence Principle it must involve a particle trajectory. Whether the setting is for such a system is one dimensional, two dimensional or three dimensional the existence of an orbit trajectory reduces the analysis to the one dimensional. The analysis presented here would then apply with path length being the coordinate. Thus in a 1D, 2D or 3D setting the spatial average of the probability density distribution from Schroedinger's equation for a particle in a potential field is asymptotically equal to the classical time-spent probability distribution.
The Copenhagen Interpretation holds that particles do not have a physical existence until they are subjected to measurement, which causes their probability density functions to collapse to a spiked one involving a definite location.
It cannot be that the nature of reality is one thing for solvable models and yet another thing for more complex models that cannot be analytically solved. Thus the Copenhagen Interpretation applies for both the simple and complex models or for neither. Since it does not apply for a particle in a potential field it does not apply anywhere.
Bohr's Correspondence Principle is satisfied in a way different from the original formulation but entirely satisfactorily intuitively. Quantum mechanics prevails at all scales but observations necessarily involve time and hence spatial averages. Those observed spatial averages of the quantum mechanical probability densities are equal asymptotically to the classical time spent probability densities. Thus quantum mechanics can be valid without there being any intrinsic uncertainty about the material existence of the particles.
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