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A Reformulation of the Equations of Hamiltonian
Dynamics in Terms of Vector Calculus and the
Derivation of a Schrödinger-like Equation
for Time-Spent in the Allowable States

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)) 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.)

Single Particle System

Consider 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 velocity is inversely proportional to the probability density, which will be expressed as φ². The constant of proportionality is given by

∫dx/|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

Now consider the gradient with respect to the state variables in Q of both sides of the above equation

Q[((∇PH)G)·((∇PH)G)φ4] = ∇Q(1/T²) = 0

where 0 is the zero vector.

The LHS evaluates to

Q([(∇PH)G)·((∇PH)G)]φ4 + ((∇PH)G)·((∇PH)G)∇Q4) = 0
which reduces to
Q((∇PH)G)·((∇PH)G)φ4 + [((∇PH)G)·((∇PH)G)]4φ3Qφ = 0
and, upon division by φ3, to
 
Q[((∇PH)G)·((∇PH)G)]φ + 4[((∇PH)G)·((∇PH)G]∇Qφ = 0
or, equivalently
4[((∇PH)G)·((∇PH)G]∇Qφ = −∇Q[((∇PH)G)·((∇PH)G)]φ

The expression [((∇PH)G)·((∇PH)G)] is a scalar function. Let it be denoted by A. The above equation is then

4A∇Qφ = −φ(∇QA)

Now consider the divergence of both sides of the above equation

4A(∇Q·∇Qφ) + (∇Qφ)·(∇QA) = −φ(∇Q·∇QA) + (∇QA)·∇Qφ)

The divergence of a gradient is usually written as ∇²Q and called theLaplacian. Using that terminology the last equantion becomes;

4A(∇²Qφ) + (∇Qφ)·(∇QA) = −φ(∇²QA) + (∇QA)·(∇Qφ)

Since the dot product is commutative, the second terms on each side of the above equation are identical so the equation reduces to

4A(∇²Qφ) = −φ(∇²QA)
or, dropping the subscript Q
4A(∇²φ) = −φ(∇²A)
This is indeed a beautiful equation
which can be rearranged to the form
∇²φ = −(1/4)[(∇²A)/A]φ

This is an equation for which the solution φ as φ² gives the proportion of time spent in each location as the probability density. The corresponding Schrödinger equation is

∇²ψ = −κ²ψ

where κ² is equal to (2m/h²)K(r), h being Planck's constant divided by 2π and K(r) is the kinetic energy expressed as a function of r. The Laplacian is also with respect to the state variables. The two equations have the same structure but the difference would lie in the magnitude of the coefficient of the wave function. For the Schrödinger equation that coefficient is large because it is inversely proportional to h².

The primary result of the analysis is that an equation can be derived corresponding to the probability distribution for the time spent by a system in its allowable states. Furthermore that equation has a structure similar to that of the Schrödinger equation for the system.

However if one wants pursue the analysis further the next step in the analysis would be the evaluation of ∇PH and hence A and ∇²QA. For the example system

PH = (pr/m, pθ/(mr²))
and hence
(∇PH)G = (pr/m, pθ/(mr))
and therefore
A = pr²/m² + pθ²/(m²r2)
and thus
QA = (−2 pθ²/(m²r3))

For the example system with its polar coordinate system

∇² = (∂²/∂r²) + (1/r)(∂/∂r) + (1/r²)(∂²/∂θ²)

When this is applied to A the result is

∇²A = 24pθ²/(m²r5) + 6pθ²/(m²r5)
∇²A = 30pθ²/(m²r5)

Thus for the example system

∇²φ = −(1/2)(30pθ²/(m²r5)/[pr²/m² + pθ²/(m²r2)]φ
which reduces to
∇²φ = −(15pθ²/(m²r5)/[pr²/m² + pθ²/(m²r2)]φ
or upon multiplying the numerator and
denominator of the fraction
on the right by r5
∇²φ = −(15pθ²/m²)/[pr²r5/m² + pθ²r3/m²]φ
or, equivalently
∇²φ = −(15/r³)pθ²/[pr²r2 + pθ²]φ

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 to the proportion of time a system spends in its allowable states.

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