Fokker-Planck + Modified Hamilton-Jacobi → Schrödinger

— Is Nelson's assumption arbitrary? —

1. Location of the problem

In Nelson's stochastic mechanics,

• Fokker-Planck equation (time evolution of probability density)

• Modified Hamilton-Jacobi equation (phase information of motion)

from these two pillars,

ψ = √P · exp(i S / ħ)

the Schrödinger equation is obtained using the following variable transformation.

The question that naturally arises at this point is:

2. Conclusion (preview)

3. Starting point: Two nonlinear equations

In Nelson's framework, there are two unknown functions:

• P(x,t): Probability density

• S(x,t): Action (phase)

These satisfy the following:

(1) Continuity equation (probability conservation)

∂P/∂t + ∇·(P v) = 0

(2) Modified Hamilton-Jacobi

∂S/∂t + (∇S)^2/(2m) + V + Q(P) = 0

Nonlinear system of equations

4. Why one wants to 'integrate' them

As it stands:

• The equations are nonlinear and difficult to handle

• P and S are separated

• The linear structure (superposition) is not visible

Therefore

This motivation arises.

5. Requirements for variable transformation

Conditions required for the new function ψ:

1. Can represent probability density

   1. |ψ|^2 = P

2. Velocity is derived from phase

   1. v = (1/m) ∇S

3. Want to make it a linear time-evolution equation

4. Must be a local differential equation

6. The resulting form

The simplest form that satisfies these is:

ψ = √P · exp(i S / ħ)

Amplitude (√P) + Phase (S)

7. What actually happens

Using this ψ:

• Real part → Modified Hamilton-Jacobi

• Imaginary part → Continuity equation

are reproduced simultaneously.

And as a whole:

iħ ∂ψ/∂t = -(ħ^2/2m) ∇^2ψ + Vψ

Schrödinger equation

8. Where does the arbitrariness lie?

The following points involve choices:

• The velocity field can be written as a potential (v = ∇S/m)

• Identification of the diffusion coefficient (D = ħ/2m)

• Adoption of complex exponential functions

Not completely inevitable

9. Reasons why it is still not arbitrary

On the other hand, when conditions are imposed, the degrees of freedom almost disappear:

• The need to satisfy |ψ|² = P

• Representing motion by phase gradient

• Requirement of linearity

• Locality

If these are satisfied simultaneously

10. Mathematical identity

This transformation is:

and,

• Cole-Hopf transformation

• conversion to polar form

has the same structure as.

11. Organizing Perspectives

Nelson side

• diffusion

• density

• mechanics

explaining 'why it becomes quantum'

Schrödinger side

• linearity

• superposition

• eigenvalue problem

calculating 'what happens'

12. Final Conclusion

In one sentence