Stochastic Shrodinger equations.

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In summary, the conversation discusses references for works that consider the Schrodinger equation with a stochastic Gaussian delta-correlated potential that is time and space-dependent and has a zero average. The average wave function is equal to U-potential and the density of probability of existence is denoted by P(F). The potential is described by a delta-function of Dirack and follows a Gaussian law. The conversation also mentions quantum Brownian motion as a possible topic related to this equation.
  • #1
Alexey
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Dear frends!
Prompt please references to works in which it was considered the Schrodinger equation with stochastic (random) Gaussian delta-correlated potential which
time-dependent and spaces-dependent and with zero average (gaussian delta-correlated noise). I am interesting what average wave function is equal.

U - potential.
<> - simbol of average.

P(F) - density of probability of existence of size F.

Delta-correlated potential which
time-dependent and spaces-dependent:
<U(x,t)U(x`,t`)>=A*delta(x-x`) *delta(t-t`)
delta - delta-function of Dirack.
A - const.

Zero average:
<U(x,t)>=0

Gaussian potential (existence of probability is distributed on Gauss law):
P(U)=C*exp(U^2/delU^2)

C - normalizing constant.
delU - root-mean-square fluctuation of U.
 
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  • #2
Are you talking about quantum Brownian motion??
 
  • #3
Originally posted by arcnets
Are you talking about quantum Brownian motion??

Yes it is.
 
  • #4
Google gives some hits on "quantum Brownian motion", maybe there's what you're looking for.
 
  • #5
Originally posted by arcnets
Google gives some hits on "quantum Brownian motion", maybe there's what you're looking for.

Thanck you! I find any-thing.
 

1. What is a Stochastic Schrodinger equation?

A Stochastic Schrodinger equation is a mathematical equation that describes the time evolution of a quantum state in a system that is subject to random fluctuations or noise. It is a generalization of the deterministic Schrodinger equation that is used to describe the behavior of quantum particles in a non-random environment.

2. How is a Stochastic Schrodinger equation different from a regular Schrodinger equation?

A Stochastic Schrodinger equation includes an additional term that accounts for the effects of random fluctuations on the quantum state. This term is known as the noise term and is typically represented by a stochastic process. In contrast, the regular Schrodinger equation does not include any terms related to random fluctuations.

3. What are the applications of Stochastic Schrodinger equations?

Stochastic Schrodinger equations have a wide range of applications in various fields, including quantum optics, condensed matter physics, and quantum information science. They are particularly useful in studying systems where the behavior of quantum particles is affected by random fluctuations, such as in quantum transport and quantum measurement processes.

4. How do scientists solve Stochastic Schrodinger equations?

Solving Stochastic Schrodinger equations often involves using numerical methods, such as Monte Carlo simulations or stochastic differential equations. These methods allow scientists to simulate the time evolution of a quantum state under the influence of random fluctuations and obtain statistical averages that can be compared to experimental data.

5. What are the challenges associated with Stochastic Schrodinger equations?

One of the main challenges of Stochastic Schrodinger equations is that they can be mathematically complex and difficult to solve analytically. This means that numerical methods are often necessary, which can be computationally intensive and time-consuming. Additionally, the interpretation of the results obtained from Stochastic Schrodinger equations can be challenging, as they may differ from the predictions of the regular Schrodinger equation.

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