Schroedinger equation and random walks

[PatrickPowers]

I have a masters degree in probability and statistics but am new to quantum physics. I have been reading an elementary text about the Schroedinger equation and I keep thinking that the Heisenberg Uncertainty Principle could emerge from a random walk that has the characteristic that as the interval of time gets smaller the momentum becomes increasingly uncertain. The random walk would be like a fractal, except while a fractal has the same level of detail at all levels our random walk would have more detail and be more jagged the more closely one looks at it. It does this without bound, so it is divergent. So integrating in the dt way in which variation over tiny units of time is assumed to be smooth is going to work reasonably well but somewhat oddly, thus giving rise to the uncertainty principle and all this other phenomena.

My second thought is that someone else has thought of this before and worked it all out circa 1950. Norbert Wiener, perhaps. There are hints of this in the preface to the text, that when taking a more sophisticated look at the wave equation it is necessary to use measures and Lebesque integrals. The author is not going to do this, and writes that most physicists don't worry about it and consult an expert if necessary. So has anyone out there heard of a treatment of quantum mechanics in this way, based on a random walk that diverges inversely to dt, and could give me a few keywords to search on?

In the meanwhile I'll continue with the elementary text, which is working very well for me.

 

[Matterwave]

I have heard that the Schroedinger equation and Quantum Mechanics can be "derived" using a stochastic method. In this way, the Heisenberg Uncertainty relations become obvious. I don't know much about this myself (I don't remember the authors), my professor just mentioned it in passing...I just thought I'd bring up that such a derivation does exist...

 

[jacopo23]

You could have a look on a topic called stochastic quantization, you can also read the preface of a book by Mikio Namiki about these things.
Ciao.

 

[qsa]

There is a whole industry around this very important topic. I will give the details soon, but you can start with Ito's diff equation.google.

 

[PatrickPowers]

You could have a look on a topic called stochastic quantization, you can also read the preface of a book by Mikio Namiki about these things.
Ciao.

From Wikipedia:

The Wiener process forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process).

A Wiener process has a spectral representation as a sine series whose coefficients are independent N(0,1) random variables. This representation can be obtained using the Karhunen–Loève theorem.

---

Yes, this is what I had in mind. Mikio writes that the methods of calculation are different than they are in the path integral formulation.

 

[PatrickPowers]

On measuring the degrees of freedom of motion at a tiny scale.

Wikipedia: The Wiener process forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process).

A Wiener process represents Brownian motion. Brownian motion has two terms: viscosity and N-dimensional Gaussian noise. Viscosity we set to zero. The variance of the N-Gaussian we calculate from Planck's constant. Often something called drift is added in. This is a constant momentum and may be set to whatever you like.

A Wiener process with no drift has a spectral representation as a sine series whose coefficients are independent N(0,1) random variables. So that's where you get the "uncertainty." Now combine this random spectrum with the drift, the mean momentum term we added in. Take the inverse Fourier transform using the drift time as the variable of integration and you get the Schroedinger wave function. Cool!

For quantum mechanics we don't quite want a Wiener process because in Brownian motion there is a preferred frame of reference. Without the preferred frame we have perhaps a relativistic diffusion process. This is an active area of research. http://arxiv.org/abs/cond-mat/0608023*

It seems that the thing to do would be to find out what Feynman and Kac did. They may have solved the whole problem perfectly, but there is still some room for maneuver. The thing is that anyone is just guessing about the degrees of freedom of the relativistic diffusion process. But we don't have to guess it, it should be possible to measure it.

How to measure this, I don't know. The degrees of freedom makes a difference, but it shows up as a third order effect. In a way this is good: if the degrees of freedom differs from what Feynman and Kac assumed then that difference may be small enough to have gone unnoticed. There could be a situation in which these small differences are magnified, making it possible to measure them. Then we will have measured the number of degrees of freedom that our world has at the tiniest scale. 3? 4? 10? Eight and a half?----* Quoting from the final paragraph of that paper.

"Finally, it is worth emphasizing that ... in contrast with the Wiener process, the proposed diffusion model is non-Markovian. This is not a drawback, as it is generally impossible to construct nontrivial continuous relativistic Markov processes in position space ... In view ot the close analogy between diffusion and quantum propagators in the nonrelativistic case, it woud be quite imteresting to learn, if this also means that it is impossible to formulate a relativistic particle quantum mechanics on the basis of standard path integrals."

Hmm. It seems to me that we don't really need continuity. Even a standard Wiener process is not absolutely guaranteed to be continuous.