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The Nature of the Probability Density
Function of Quantum Mechanical Analysis

In 1926-27 Erwin Schrödinger formulated wave mechanics which came to be the preferred formulation for quantum physics. Schrödinger cast his theory in terms of a wave function. This stemmed from his background in optics and his captivation by Louis de Brogliem's notion that particles have a wave aspect. The immediate question was what was the nature of Schrödinger's wave function. Max Born asserted that the squared magnitude of the wave function is the probability density function for the system under analysis. Neils Bohr and Werner Heisenberg concurred with this interpretation of the wave function and it became a key element of what came to be known as the Copenhagen Interpretation of quantum physics.

What is argued below is that the squared magnitude of the wave function is a probability density function but not of the nature it is given in the Copenhagen Interpretation. Instead Schrödinger's time independent equation gives the proportion of the time the system spends in the the various states in its periodic cycle. The system cycles through the allowed states moving relatively slowly through an allowed state and then relatively rapidly to the next allowed state.

The nature of the wave function must be of one sort for all quantum mechanical systems, so to establish the above alternative to the the Copenhagen Interpretation it suffices to establish it for one significant case. The easy case is for harmonic oscillators and this has been done in Harmonic Oscillators.

The rapidly fluctuating function is the quantum mechanical probability density and the heavy line is the corresponding classical concept. The classical concept is the proportion of time spent at each possible location. It represent the probability of finding the particle at the various possible locations at any randomly specified time. There is a close relationship between a spatial average of the QM probability density and the classical concept except at the end points for the oscillator.

However harmonic oscillators are not the most natural example and the case of two body interactions is used instead, but in the form of a particle moving in a potential energy function field.

A Particle Moving in a Central Field
with a Potential Energy Function V(r)

The Hamiltonian function for such a system is

H = p²/(2m) + V(r)

where p is the total momentum of the particle, m is its mass and V(r) is the potential energy of the particle as a function of its distance r from the center of the central field. The total momentum is made up of the radial momentum pr and the tangential momentum pθ. These momenta are orthogonal so

p² = pr² + pθ²

At a macroscopic level a particle in a central field revolves about the center of the field in an elliptical orbit. That orbit is entirely in a plane. In order for the quantum mechanical (QM) analysis to satisfly the Correspondence Principle it must also be limited to a plane. The Correspondence Principle says that in order for QM analysis to be valid it must be consistent with classical analysis as the scale or energy increases to macroscopic proportions.

The time-independent Schrödinger equation for the system is

−(h²/2m)∇²ψ + V(r)ψ = Eψ

where h is Planck's constant divided by 2π and ψ is the wave function for the particle.

For stable systems the potential energy V and the total energy E are negative. To emphasize their negativity they can be written as −|V(r)| and −|E|. Thus the above equation is

−(h²/2m)∇²ψ −|V(r)|ψ = −|E|ψ
or, equivalently
−(h²/2m)∇²ψ + (|E| −|V(r)|)ψ = 0
or, more succinctly as
−(h²/2m)∇²ψ − K(r)ψ = 0
or, better yet as
∇²ψ + (2m/h²)K(r)ψ = 0

where K(r) is the kinetic energy function (E− V(r)).

This last equation has a close mathematical relationship to the equation for a harmonic oscillator; i.e.,

(d²x/dt²) + (k/m)x = 0

where x is the displacement from equilibrium, t is time, k is the stiffness coefficient and m is mass.

The displacement x oscillates sinusoidally between two extremes. The frequency is equal to the square root of (k/m) and the wavelength is inversely proportional to that quantity.

The correspondences of the harmonic oscillator equation to the quantum mechanical equation are

Correspondences
Harmonic
Oscillator
QM Model
xψ
tx
(k/m)(2m/h²)

Just as the displacement oscillates back and forth between extremes over time so does the wave function ψ oscillate between extremes over space. The square of ψ oscillates between zero and maxima as seen in the previously displayed image.

The kinetic energy changes relatively slowly over space and to the approximation that it is constant the spatial frequency of ψ is equal to (2mK)½/h. The peak to peak spacing of the probability density is inversely this frequency and the higher the kinetic energy the more closely are the peaks spaced.

The equation for ψ can be multiplied by ψ to obtain

−(h²/2m)ψ∇²ψ − K(r)ψ² = 0
or, eliminating the negativity
(h²/2m)ψ∇²ψ + K(r)ψ² = 0

The quantity is ψ² is the probability density. For future reference the above equation will be referred to as the equation for probability density.

Note that the critical points of ψ² occur where

ψ∇ψ=0
and hence
where either
ψ = 0
or
∇ψ = 0

where 0 denotes the zero vector.

The points where ψ² is at or near a maximum correspond to an allowed state and where ψ² is zero or near zero correpond to a disallowed state. The particle moves relatively slowly through one allowed state and relatively quickly through an adjacent disallowed state to the next allowed state. This is not quantum jumping per se but it has similarities to that notion.

Consider the following vector calculus identity

∇·(ψ∇ψ) = ∇ψ·∇ψ + ψ∇·∇ψ = (∇ψ)² + ψ∇²ψ
or, equivalently
ψ∇²ψ = ∇·(ψ∇ψ) − (∇ψ)²

In words this is that the divergence of the function times the gradient of the function is equal to the dot product of the gradient of the function with itself plus the function times the divergence of the gradient of the function.

Thus the previous equation derived from the Schrödinger equation, the probability density equation, becomes

(h²/2m)(∇·(ψ∇ψ) − (∇ψ)²) + K(r)ψ² = 0

Consider the gradient of the probability density ψ²; i.e.,

∇ψ² = 2ψ∇ψ
and therefore
(∇ψ²)·(∇ψ²) = 4ψ²(∇ψ·∇ψ)
which can be expressed as
(∇ψ²)² = 4ψ²(∇ψ)²

Thus

(∇ψ)² = (∇ψ²)²/( 4ψ²)

When this expression is substituted into the probability density equation the result is

(h²/2m)(∇·(ψ∇ψ) − (∇ψ²)²)/( 4ψ²) + K(r)ψ² = 0

The term (∇·(ψ∇ψ) when integrated from a maximum to an adjacent minimum or from a minimum to an adjacent maximum is zero because at minimum ψ is zero and at a maximum the gradient ∇ψ is equal to the zero vector.

At a macroscopic classical level such a system as being considered would involve the particle traveling smoothly about an elliptical orbit. In order for the quantum mechanical system to asymptotically approach the classical behavior at the scale and/or energy increases it must have some semblance of an orbit. There is no dividing line between the quantum mechanical and the classical, what Werner Heisenberg called the Schnitt (cut).

A chain of alternating minima and maxima may be constructed labeled by an index j, say mj and Mj for j=1, 2, …, N. This chain would constitutes an orbit path. The intervals between minima and maxima can also be labeled by an index k, say sk. The interval from mj to Mj would be k=2j and from Mj to mj+1 would be k=2j+1. This chain of intervals is roughly the particle's path. When such integrations are carried out over the chain of intervals between maxima and minima and use made of the Extended Mean Value Theorem for Integrals the result is

(h²/2m)∫ds(∇ψ)²)/(4ψ²(s*) = K(r*)∫ds(ψ²)

where r* and s* are some values of r and s within the interval of integration.

The integral on the RHS can be represented as ψ²(s#)δk where δk is the length of the k-th interval and s# is some point in that interval.

When that substitution is made and the equation is multiplied by ψ²(s*) the result is

(h²/(8m))∫ds(∇ψ)²) = K(r*)ψ²(s#)ψ²(s*)δk
or, after division by δk
(h²/(8m))∫ds(∇ψ)²)/δk = K(r*)ψ²(s#)ψ²(s*)
which may be expressed as
(h²/(8m))(∇ψ(s+)²) = K(r*)(ψ(s^)²)²

where (∇ψ(s+))² is the average of (∇ψ)² in the k-th interval and (ψ(s^)²)² is the square of the probability density P at some point s^ in the k-th interval.

Thus

P(sj) = (h/(2m)½[∫ds(∇P(s))²)/4]½/(2K(r(sj))½

Over a wide range ∫ds(∇P(s))²) is relatively constant so the probability of being in interval sj is inversely proportional to K(r(sj))½. What is required to satisfy the Correspondence Principle is that the quantum mechanical probability function be asymptotically equivalent to the classical probability density function which is proportional to the time spent in the various locations as the scale or energy of the system increases. But QM distribution can be taken to be the corresponding quantity given the motion of the particle at the quantum level.

The Classical Path of a Particle in a Central Field

The energy function

E = ½mv² + V(r)

may be solved for the velocity v as

v = [2(E−V(r))/m]½ = (2/m)½K(r)½

The time spent by the particle in an interval ds of its path length s is ds/|v|. The probability density of finding the particle at that point at a random time is proportional to 1/|v(s)| and hence to 1/(E−V(r(s)))½ which is the same as 1/K(r(s))½. When the quantities which the probabilities are proportional to are normalized all constant factors are eliminated. Thus the quantities 1/K(r(s))½ are the only determinates of the probability. This is very close to what was found for the QM probability densities. Singularities arise at the end points where the kinetic energy is zero.

The Nature Of Probabilities

The concept of probability is a very useful construct for explaining statistical data. There is usually a subjective nature to probability; meaning the probabilities are conditional on what is known and thus not solely a property of the system under concideration. When there is some intrinsic component to probabilities such as for dice those probabilities are embpdied in the symmetry and uniformity of the dice. In the probability density functions considered above the probabilities are embodied in the periodic cycle of the system. Nowhere except in the Copenhagen Interpretation of Quantum Mechanics are there disembodied probabilities that exist like an electric field.

Conclusion

The squared magnitudes of the wave function which comes out of quantum mechanical analysis constitute a probability density function that represents the proportion of the time the system spends in various locations. The QM probability density function for a system does not represent some intrinsic uncertainty of the particles of the system.

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