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The Copenhagen Interpretation of the Wave
Function for Schrödinger's Equation Is Not Valid;
Instead the Wave Function is Related to the
Time Spent by the System in its Allowable States

Schroedinger's equation, which is the basis for Quantum Theory, is based on the Hamiltonian equations for a system. One inputs the static structure of a system and the solution to the Hamiltonian equations gives the dynamics of the system. What the Copenhagen Interpretation tries to say is that solution to Schroedinger's is giving a probabilistic static structure of the system instead of the dynamic appearance of the system.

One element of the Copenhagen Interpretation is that a particle exists simultaneously, sort of, in all of its allowable locations. Consider a rapidly rotating propellor. It appears to be a blurred disk. If it is illuminated by a stroboscopic light there appear to be multiple propellors existing simultaneously.

The alternative to the Copenhagen Interpretation is that the solution to the time independent Schrödinger equation represents the dynamic appearance of the system, the blurred disk of the rapidly rotating propeller, instead of any intrinsic uncertainty of the static structure of the system, the propeller. This not to say that results of Quantum Theory are wrong. It is only the conventional Copenhagen Interpretation of the results of them that is wrong.

When Erwin Schrödinger formulated the wave mechanics version of quantum physics in 1926 he did not specify what the wave function ψ represented. He thought its squared magnitude would represent something physical such as spatial charge density. Max Born suggested that its squared magnitude represented probability density of finding the particle near a particular location. Niels Bohr and his group in Copenhagen concurred and the notion that the wave function represents the intrinsic indeterminancy of the particle of the system came to be known as the Copenhagen Interpretation. What will be shown here is that the wave function relates not to any intrinsic indeterminancy of the particles but instead to the proportion of the time the system spends in its allowable states.

The Schrödinger equation is not derived in any meaningful sense of that word. Instead the equation is constructed according to definite rules. The justification for these rules is that the analysis so constructed seems to work empirically. The construction starts with the Hamiltonian function H for a system under consideration. For example, consider an electron moving in a potential energy field given by V(r), where r is the radial distance of the electron from the center of the potential field. The Hamiltonian function is then

H = p²/(2m) + V(r)

where p is momentum and m is the mass of the electron.

According to the prescription for generating the Schrödinger equation for the system, p² is replaced by −h²∇², where h is Planck's constant divided by 2π. Thus the time independent Schrödinger equation for the system is

−(h²/(2m))∇²ψ + V(r)ψ = Eψ
which can also be expressed as
−(h²/(2m))∇²ψ = (E−V(r))ψ = K(r)ψ

where E is the energy of the system and ψ is called the wave function. The symbol K(r) is the kinetic energy of the system expressed as a function of the radial distance.

Under the Copenhagen Interpretation |ψ|² is the probability density function for the system. Now the analysis will go to the classical Hamiltonian analysis.

Classical Hamiltonian Dynamics

The general form of Hamiltonian dynamics is that there is a set of generalized coordinates {q1, …, qn} and their canonic conjugate momenta {p1, …, pn} which obey the conditions

(dqj/dt) = (∂H/∂pj)
(dpj/dt) = −(∂H/∂qj)
for j = 1, …, n

where H is the total energy function. The generalized momenta are given by

pj = (∂E/∂(dqj/dt)) for j = 1, …, n

where E is the total energy of the system expressed in terms of velocities and the state variables.

Let the row vectors with components (q1, …, qn) and (p1, …, pn) be denoted as Q and P, respectively. The Hamiltonian equations can then be expressed as

(dQ/dt) = ∇PH
(dP/dt) = −∇QH

where ∇Q and ∇P are the gradient operators with respect to the variables of Q and P. (It is not clear that the other vector calculus operations besides the gradient can be defined for the momentum variables, but the gradient operation with respect to the momenta occurs naturally and it is only one that is used.)

A Single Particle System

Consider again as an example a particle in a potential field V(r) where r is the radial distance from the center of the potential field to the particle. The coordinates for the system are (r, θ) where θ is the polar angle of the radial. The total energy funtion for this system is

E = ½m(dr/dt)² + ½mr²(dθ/dt)² − V(r)
and hence
pr = m(dr/dt)
pθ = mr²(dθ/dt) = mr(r(dθ/dt)

The Hamiltonian function for a system is its total energy expressed in terms of the momenta of the system

H = pr²/(2m) + pθ²/(2mr²) + V(r)

Thus the gradient of H with respect to the momenta is the vector (pr/m, pθ/(mr²)).

The components of the linear velocity vector V for this system are (dr/dt, r(dθ/dt)). In general, the velocity vector V is given by

V = (dQ/dt)G
and thus
V = (∇PH)G
or, equivalently
vj = (dqj/dt)gj
for j =1, … n

where G is a diagonal matrix Diag(g1, … gn). For the example system G=Diag(1, r).

The squared magnitude of particle velocity is then

V·V = ((∇PH)G)·((∇PH)G)

The time a particle spends in an interval ds of its path is ds/|v|. Thus the probability of finding the particle in the interval dx at any random time is inversely proportional its speed |v|. Stated the other way around the velocity is inversely proportional to the probability density, which will be expressed as φ². The constant of proportionality is given by

∫ds/|v| = ∫dt = T

Therefore

1/(φ²)² = 1/φ4 = T²(V·V)
= T²((∇PH)G)·((∇PH)G)
and hence
((∇PH)G)·((∇PH)G)φ4 = 1/T²

The term ((∇PH)G)·((∇PH)G) is a scalar function. For convenience let it be denoted by A. The above equation is then

4 = 1/T²

Now consider the gradient with respect to the state variables in Q of both sides of the above equation. The vector calculus operations henceforth are all with respect to the state variables and will not be subscripted with Q.

∇[Aφ4] = ∇(1/T²) = 0

where 0 is the zero vector.

Conclusion

The proportion of the time a classical system spends in its allowable states constitutes a probability density function in the sense that it is the probability densities of finding the system in those states at a randomly selected time. An equation can be derived for a wave function whose squared magnitude is the classical probability density function. This equation has a mathematical structure similar to that of a time independent Schrödinger equation. A Schrödinger equation does not have a derivation; it is simply created according to definite rules from the classical Hamiltonian of the system under consideration. As a result there is a question of the interpretation of its solution. The Copenhagen Interpretation takes the solution to be related to a probability distribution for the intrinsic indeterminacy of particles. The results found in the analysis given in this study, along with the Correspondence Principle for quantum physics, strongly indicate that the solutions to the Schrödinger equation correspond to the proportion of time a system spends in its allowable states.


φ²(r) = 1/(Tv(r))
or, equivalently
φ = 1/(T½v½)

The first and second derivatives of this equation are

(dφ/dr) = −(1/T½)v3/2)(dv/dr)
and
(dφ/dr)² + φ(d²φ/dr²) = (1/v³)(d²v/dr²) − (1/v²)(d²v/dr²)

2ln(φ) + ln(v) = 0
(2/φ)(dφ/dr) + (1/v)dv/dr = 0
or, equivalently
dφ/dr = −½(φ/v)(dv/dr)

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San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
U.S.A.

The Copenhagen Interpretation of the Wave
Function for Schrödinger's Equation Is Not Valid;
Instead the Wave Function is Related to the
Time Spent by the System in its Allowable States

The purpose of this webpage is to tie the probability density distribution derived from the time independent Schrödinger's (Schroedinger) equation to the probability density distribution derived from the time spent along a periodic path for a particle according to classical physics. According to the Correspondence Principle as articulated by Niels Bohr, for a quantum analysis to be valid it must betat when it is is scaled up by energy to a macroscopic level it must equal or asymptotically approach the results of classical analysis. What Bohr left out is that the macro level observation necessarly involves spatial and time averages, The correspondence is between the spatial and time average of the scaled up version of the quantum analysis.

Since the result from Schrödinger's Equation is a probability density distribution, scaled up it is still a probability density distribution. What probability density distribution is there for classical analysis that the quantum analysis could correspond to? The answer is the time spent probability density distribution. This is the proportion of the time a system spends in its allowable states while traversing a periodic path. For a particle traveling at a velocity v in a spatial interval the time spent in that interval is its length divided by the magnitude of the velicity. If the length of the interval is ds then the time spent dt is ds/|v|. If the total time spent traversing the periodic path is T then the proportion of the time spent in the interval is ds/(T|v|). This is the probability of finding the particle in the specified interval at a randomly chosen time.

The Classical Analysis

The system being considered is a point particle of mass m in a one dimensional potential field given by the function V(x). The total energy E is given by

E = ½mv² + V(x)

Therefore

|v| = [(2/m)(E − V(x)]½

The quantity (E−V(x)] can be represented as K(x), the kinetic energy of the particle as a function of location.

Thus

|v| = (2/m)½K(x)½

The probability density function PC(x) for the particle is then

PC(x) = (1/T)K(x)−½

where T = ∫K(x)−½dx

The proposition has been established for harmonic oscillators, an illustration of which is show below. What is sought here is the establishment of the proposition for a more general type of potential function.

The light lines are for the probability density functtion according to the quantum analysis of a harmonic oscillator. The dark line is the result for the corresponding classical analysis. The spatial average of the quantum analysis closely approximates the results from the classical analysis of a harmonic oscillator.

The Quantum Analysis

Quantum analysis is based on theSchrödinger's equationfor the system, which is the based for it Hamiltonian function. For the particle in the potential field the Hamiltonian is just the total energy expressed in terms of its momentum p=mv; i.e., .,

H = ½p²/m + V(x)

This is converted into the Hamiltonian operator H^ by replacing the squared momentum p² with −h²(d²/dx²), where h is Planck's constant divided by 2π. Thus the time independent Schrödinger equation for the particle is

H^φ = −h²(d²/dx²) + V(x)φ = Eφ

where φ is such that its squared magnitude is equal to the probability density function for the particle.

It is convenient to convert this second order equation into a system of two first order equations by letting ψ=dφ/dx; i.e.,

dφ/dx = ψ
 
dψ/dx = −(K(x)/h²)φ

Let dz equal (i/h)K½(x)dx. Then

dψ/dx = (dφ/dz)(dz/dx) = (dφ/dz)(i/h)K½(x) = K(x)φ
and thus
(dψ/dz) = −ihK½(z)φ

On the other hand

dφ/dx = ψ
becomes
dφ/dz = −(i/h)K−½(z)ψ

This can be represented in matrix form as

dΦ/dz = M(z)Φ

where

| φ |
Φ =
| ψ |

and

  | 0 −(i/h)K−½ |
M =
  | −ihK½  0    |

The solution to the matrix system of equations is

Φ(z) = exp(∫0zMdζ)Φ(0)

This means that

φ(z) = (−½(i/h)∫0zK(ζ)−1/2dζ)ψ(0)

It is φ² which is the probability density function; so

PQM(z) = (½/h²)(∫0zK(ζ)−1/2dζ)²ψ(0)²

(To be continued.)

The Effect of a Change of Scale

Since K(z)=E−V(z) a change in E affects Φ(z) through K(z). Therefore

(∂Φ/∂E) = (∂(∫0zMdζ)/∂E)exp(∫0zMdζ)Φ(0)
= (∂(∫0zMdζ)/∂E)Φ(z)
= (∫0zNdζ)Φ(z)

where

| 0 ½(i/h)K−3/2 |
N =
| −(½ih)K−1/2       0       |

This means that

(∂φ/∂E) = ½(i/h)(∫0zK(ζ)−3/2dζ)ψ(0)

Conclusion

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