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The Interpretation of a Solution to Schroedinger's|
Time-Independent Equation in Terms of Time Spent
in Various Portions of Its Trajectory:
The Two Dimensional Case
In 1926 Erwin Schrödinger formulated Wave Mechanics for quantum phenomena. Wave Mechanics involved the solution of partial differential equations which came to be known as Schrödinger's equations. These equations involved an unspecified variable called the wave function. Schrödinger came from a background in optics and thought the wave function was some sort of electromagnetic quantity. Very quickly however Max Born and Pascual Jordan at Göttingen University and Neils Bohr and Werner Heisenberg in Copenhagen began to interpret the wave function as being such that its squared magnitude is the probability density function for the particle. This interpretation of the wave function came to be known as the Copenhagen Interpretation (CI). The CI came to include more propositions, such as that a particle does not have a physical existence, but instead existed simultaneously in all of its allowed states as a probability distributiion until some interaction forces it to physically appear in one of those allowed states. When a particle moves it jumps instantaneously from one allowed state to another.
The purpose of this webwage is to demonstrate that the proper interpretation of the wave function in Schrödinger's time independent equation is in terms of a probability density function constructed from the time spent at the different points of its trajectory. The time that a particle spends in an infinitesimal interval ds of it trajectory is equal to ds/|v|, where v is the velocity of the particle. If T is the total time required for the particle to traverse its trajectory then the probability density D of finding it in the interval ds is
Thus, even though the classical analysis of the motion of a particle is deterministic there is a probability density distribution defined for its motion. This distribution will be referred to as the time-spent probability density distribution.
Let r be the lineara coordinate for a particle in a field with a potential function V(r). The total energy of the particle can be expressed as
where m is the mass of the particle and p is its momentum.
The function H(p, r) is known as the Hamiltonian function for the system.
For a one dimensional system the Hamiltonian function is converted into the Hamiltonian operator for the system by replacing the
momentum p with −i
h(∂/∂r), where i is the imaginary unit and h is
Planck's constant divided by 2π.
The time independent Schroedinger equation is
where H^ is the operator form of the Hamiltonian function for the system and E is the total energy of the sytem.
As noted above, for a particle in a potential field V(r) the Hamiltonian function is p²/(2m) + V(r) and converted to the operator form it is
Therefore the time-independent Schrödinger equation for the system is
The quantity K(r) is the kinetic energy of the particle expressed as a function of its position in its path.
The above equation can be expressed as
where k(r)² is equal to (2m/
h²)K(r). In this form the equation can be
characterized as a generalized Helmholtz equation. The solution to such an equation involves rapid
oscillations. This is because
Thus when φ is postive the slope (∂φ/∂r) is decreasing and goes to zero and becomes negative. Then φ goes to zero and becomes negative. When φ is negative the slope is increasing and comes back up to a positive level and subsequently φ becomes positive again. Thus φ fluctuates rapidly between positive and negative values. Since probability density is equal to φ² the probability density fluctuates between positive values and zero.
Below is the graph of the probability density for a harmonic oscillator. For a harmonic oscillator the force on the particle is proportional to its displacement from equilibrium, say −γr, and its potential function is ½γr².
The rapidity of the oscillations depends upon the energy of the oscillator, which is proportional to the the principal quantum number of the system. In this case the principal quantum number is 60.
The heavy line in the above graph is the probability density function for the harmonic oscillator of the same energy based on its classical analysis. What appears to be the case in this example is that a spatial average of the quantum mechanical probability density is equal to the classical time-spent probability density function. It is shown elsewhere that for the one dimensional case that as the energy E increases without limit a spatial average of the solution to the Schrödinger equation asymptotically approaches the classical time-spent progability density distribution. The question is whether that result holds for the two dimensional case.
Neils Bohr articulated the principle that for quantum level analysis to be valid its results when scaled up to the macro level must coincide with the classical analysis. Some quantum physicists, such as Werner Heisenberg, spoke of there being a boundary scale between the microscopic quantum level analysis and the macroscopic classical level at which the Correspondence Principle required the two analyses coincide.
It is asserted here that the above perception is wrong. It is asserted here that the quantum level dynamics prevails at all scales but at the macro level the quantum level results cannot be distinguished from the classical macro scale results. The classical anaysis involves a smoothing of the quantum level analysis over space and time. So asymptotically as the energy increases without bound the spatially smoothed quantum level results.
Qualitatively the Correspondence Principle imposes some stringent conditions. If the classical macro scale version of a system, such as a particle in a spherically symmetric potential field, has well defined orbits then the quantum level anaysis must show such periodic trajectories. approach the classical results.
As stated previously, even though the trajectory of the particle is deterministic and so there is no probability involved in its determination, a probability density function can be defined in terms of the probability of the particle being found in an interval at any randomly chosen time. The probability of the particle being found in an interval [x-½Δx, x+½Δx] is proportional to the amount of time the particle spends in the interval. That probability density is represented as
For a particle with energy E equal to ½mv² + V(r)
The constant factor (2m)½ appears in each value of Pc and in T and therefore cancels out and thus
Note that this means the classical probability density function is independent of the mass of the particle.
As noted before, the Hamiltonian function for the system of a particle of mass m in a potential field of V(r) is
where p is the momentum of the particle.
In general the Hamiltonian operator is obtained from the Hamiltonian function by
h²/(2m)]∇² for p².
Let r and θ denote the polar coordinates of a particle in a spherically symmetric potential field V(r). The Laplacian operator for polar coordinates is
But for a solution that satisfies the Correspondence Principle there must be a trajectory path for which r is a function of θ. For example, when the potential function is derived from a strictly inverse distance squared force, V(r)=−γ/r, the distance r is given by
where A and ε are constants; ε being called the eccentricity of the orbit.
Here is the classical time-spent probability density function for a particle subject to an inverse distance squared force. Probability density is shown as a function of the angle θ.
Here is the solution to Schroedinger's equation in terms of the angle θ
Furthermore there is a relationship between the angle θ and the distance along the particle path, the path length s. The angle θ may be expressed as a function of path length s and hence also orbit distance r is a function of s. Thus the potential energy V and wave function φ are functions of s.
The gradient of φ(s) is then simply (∂φ/∂s) and the Laplacian is
The time-independent Schrödinger equation for the system is then
where E is total energy.
This equation can be put into the form
If K(x) were constant then k(x) would be constant and the solution to the equation would be
The wavelength λ of such a solution is given by
Thus as E increases without bound so does k and hence λ goes to zero.
The equation for the wave function was put into the form
Now multiply each side of the equation by φ(s) to put the equation into the form
Note that the probability density function φ²(s) has its extremes where
The minima occur where φ(s)=0 and the maxima where φ'(s)=0.
Now note that
Let the minima and maxima be numbered and denoted as mj and Mj, respectively. When the above equation is integrated from mj to Mj the first term is zero. (The following analysis works equally for integrations from a maximum to a minimum, or from a minimum to a minimum or a maximum to a maximum.) The integration from a minimum to a maximum leaves
Thus the previous equation reduces to
The quantity k²(s) varies relatively slowly compared to φ(s) so by an extension of the Mean Value Theorem
where s is somewhere in the interval [mj, Mj].
is the probability that the particle is in the interval [mj, Mj].
Likewise, by the extension of the Mean Value Theorem used above, the integral on the left of the previous equation can be replaced by
where s̅j is a value of s in the interval [mj, Mj].
where δj is the length of the integration interval (Mj-mj) and sj* is a value of s in that interval.
The values s̅j, sj and sj* are approximately equal. Also
Let the average value of (dφ²/dz)² in the interval be denoted as μj².
Note that this φ(sj)² is an average over the interval of integration.
The graph below shows the quantum mechanical probability density function for a harmonic oscillator with V(s) equal to ½s² and a principal quantum number of 60.
Over a wide range of s the probability density function is nearly the same and thus μ is nearly the same.
Presume that the μj's are approximately equal for all j, say μ. Then in the normalization of the probability density function the constant factor of 2μ is eliminated.
Since k(s) is proportional to K(s)½ then
This is identical to the expression found for the classical probability density function. In the derivation it was presumed that two values of s in the interval [mj, Mj] were the same. As the energy of the system increases without bound the length of the interval goes to zero and thus the two values of s must converge to the same value. In other words, the spatial average of the quantum mechanical probability density function is asymptotically equal to the classical probability density function for the same system based upon the time averaged location of the particle.
This means that the quantum mechanical probability density functions do not represent some pure indeterminacy of the particle as in the Copenhagen Interpretation but instead the proportion of the time a moving particle spends near the various points.
In a sense the quantum mechanical probability density distribution represents a quantization of allowable locations for the particle. The particle still moves but spastically from allowable location to allowable location. The time average of the particle velocity looks like the graph shown below.
The unaveraged velocity would varying between infinity and zero.
This is in contrast to the smoothly varying velocity according to classical mechanics.
The implication of the above is not just that the spatial average of the quantum mechanical probability density function is asymptotically equal to the classical probability density function. The classical probability density function is derived from a particle moving in a cycle. This means that the quantum mechanical probability density function also describes the time averaged behavior of a particle in motion. The quantum mechanical motion of the particle involves near infinitesimal pauses at discrete locations and then near instantaneous shifts to the next location. Even quantum mechanically the particle is not at all of the discrete locations simultaneously.
The proper interpretation of the wave function of Schrödinger's equation is then the time-spent probability density function. The analysis was limited to the one dimensional case, but it is rather implausible that there could be a drastic difference in the nature of reality for the one dimensional case in contrast to cases of more than one dimension.
(To be continued.)
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