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, …, q_n} and their canonic conjugate momenta {p₁, …, p_n} which obey the conditions

(dq_j/dt) = (∂H/∂p_j)
(dp_j/dt) = −(∂H/∂q_j)
for j = 1, …, n

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

p_j = (∂E/∂(dq_j/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, …, q_n) and (p₁, …, p_n) 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
p_r = m(dr/dt)
p_θ = 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 = p_r²/(2m) + p_θ²/(2mr²) + V(r)

Thus the gradient of H with respect to the momenta is the vector (p_r/m, p_θ/(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
v_j = (dq_j/dt)g_j
for j =1, … n

where G is a diagonal matrix Diag(g₁, … g_n). 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

Aφ⁴ = 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/(T^½v^½)

The first and second derivatives of this equation are

(dφ/dr) = −(1/T^½)v^3/2)(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)