# Schrodinger Equation as Flow Equation

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• lagrangman
In summary: In my Mathematical Methods for Physics book by Wyld from 1976 it was referred to as a diffusion equation with an imaginary diffusion constant. Baym (1969) mentioned it in his Lectures on Quantum Mechanics assuming that is what you meant by a flow equation.
lagrangman
I was playing with the Schrodinger equation and realized that it can be interpreted as a flow equation.

If we set $$\psi = A e^{i \theta}$$

We can put the Schrodinger in the form ∂ψ∂t=(−∇ψ)⋅v+iEψ

If v=ℏθm and E=ℏ2m(−∇2AA+∇2θ)+ρV

I find this intuitive personally as it shows that the wavefunction flows. Is there any book that mentions this?

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Delta2
lagrangman said:
I was playing with the Schrodinger equation and realized that it can be interpreted as a flow equation.

If we set $$\psi = A e^{i \theta}$$

We can put the Schrodinger in the form ∂ψ∂t=(−∇ψ)⋅v+iEψ

If v=ℏθm and E=ℏ2m(−∇2AA+∇2θ)+ρV

I find this intuitive personally as it shows that the wavefunction flows. Is there any book that mentions this?
In my Mathematical Methods for Physics book by Wyld from 1976 it was referred to as a diffusion equation with an imaginary diffusion constant. Baym (1969) mentioned it in his Lectures on Quantum Mechanics assuming that is what you meant by a flow equation.

In his Lectures on Physics (Volume 3, chapter 16, section 1) Feynman speaking of the Schrödinger equation says;

In fact, the equation looks something like the diffusion equations which we have used in Volume I. But there is one main difference: the imaginary coefficient in front of the time derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of Eq. (16.13) are complex waves.

Last edited:
vanhees71, PeterDonis and Delta2
lagrangman said:
I was playing with the Schrodinger equation and realized that it can be interpreted as a flow equation.
There is a more important presentation of the Schrödinger equation as a flow equation which goes back to Madelung. His variables are often misleadingly named "hydrodynamic variables". For the probability distribution in the configuration space ##\rho(q,t) = |\psi(q,t)|^2## the Schrödinger equation gives a probability flow equation
$$\partial_t \rho + \partial_i \left(\rho v^i\right) = 0.$$

The velocity is defined by the phase ##\phi(q) = \hbar \Im \ln \psi(q)## by ##m v^i(q,t)=\partial_i \phi(q,t) ##
The Schrödinger equation gives also a quantum generalization of the Hamilton-Jacobi equation:

$$\partial_t \phi + \frac{1}{2} (\nabla \phi)^2 + V -\frac{\hbar^2}{2} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} = 0.$$

These variables are used in almost all realistic interpretations of quantum theory, in particular in de Brogllie-Bohm theory (also known as Bohmian mechanics), in Nelsonian stochastics, and (my favorite) Caticha's entropic dynamics. For the discussion of those interpretations there is a separate subforum.

dextercioby

## 1. What is the Schrodinger Equation as Flow Equation?

The Schrodinger Equation as Flow Equation is a mathematical equation that describes the behavior of quantum particles, such as electrons, in a quantum system. It is used to determine the probability of finding a particle at a specific location and time.

## 2. Who developed the Schrodinger Equation as Flow Equation?

The Schrodinger Equation as Flow Equation was developed by Austrian physicist Erwin Schrodinger in 1926. He was one of the pioneers of quantum mechanics and his equation is considered one of the fundamental principles of the theory.

## 3. How is the Schrodinger Equation as Flow Equation different from the classical Newtonian equations?

The Schrodinger Equation as Flow Equation is a quantum mechanical equation that describes the behavior of particles on a microscopic level, while classical Newtonian equations describe the behavior of larger objects in the macroscopic world. The Schrodinger Equation takes into account the wave-like nature of particles and the uncertainty principle, which are not accounted for in classical mechanics.

## 4. What is the significance of the Schrodinger Equation as Flow Equation in quantum mechanics?

The Schrodinger Equation as Flow Equation is a fundamental equation in quantum mechanics that allows us to calculate the probability of finding a particle at a specific location and time. It has been used to make predictions and explain the behavior of quantum systems, leading to advancements in fields such as chemistry, materials science, and technology.

## 5. Can the Schrodinger Equation as Flow Equation be solved analytically?

In some simple cases, the Schrodinger Equation as Flow Equation can be solved analytically, meaning a closed-form solution can be obtained. However, in most cases, it requires numerical methods to solve, which involves using computers to calculate the probability of finding a particle at a specific location and time.

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