Nonlinear equation and state superposition

[kroni]

Hi,

In quantum physics, solution of Shrodinger equation live in a Hilbert space which is a vector space. The state superposition is obtained by mixing solution of the équation which is LINEAR so a linear combination of solution is a solution.

Now i have a non-linear equation of a scalar field named Q. So a linear combination of solution of Q is not a solution. To obtain a state superposition, i define a probability density P over the space of all scalar field which is a functionnal space. Is it a good way ? did you know similar approach ?

Because, now at the oposite of Shrodinger equation, the time evolution is a modification of density P over time. I have strong difficulties to write the equation which control P over time because it need to derivate a scalar field defined over a function space.

Thanks for your answer.

 

[eljose79]

Hi,

Thank you for your question. It is interesting that you have a non-linear equation for a scalar field and are trying to find a way to obtain a state superposition. In quantum mechanics, the Schrödinger equation is indeed linear, but there are also non-linear equations used in other areas of physics, such as in fluid dynamics.

In terms of your approach, using a probability density over the function space may be a good way to handle the non-linearity of the equation. This is similar to how the wave function in quantum mechanics represents the probability of finding a particle in a certain state.

As for the time evolution, it is indeed a challenge to write an equation that can control the probability density over time. One possible approach could be to use the Hamiltonian formalism, where the time evolution is described by the Hamiltonian operator acting on the probability density. This is similar to how the Schrödinger equation describes the time evolution of the wave function.

I am not familiar with a similar approach being used in quantum mechanics, but there may be some research on this topic that you could look into. I suggest consulting with other physicists or conducting further research to see if there are any existing methods that could be applied to your problem.

I hope this helps and good luck with your research!