Paths of a free quantum particle

[lavinia]

In continuous Brownian motion - while it is completely unpredictable - there are nevertheless many theorems about its behavior. For instance, continuous sample Brownian paths are almost surely nowhere differentiable.

Since the Schrödinger equation for a free particle is the heat equation with a factor of i tacked on, one might ask what the paths of free particles look like.

The experiment would maybe be to break up a fixed time interval into n increments and at each successive increment measure the position of the particle with a detecting screen. This will give a piecewise linear path with n segments. Now let the increments become increasingly smaller and take the limit. What are the properties of these paths?

One might expect some interesting statistics since amplitudes follow a Markov like process. Instead of conditional probabilities, one has conditional amplitudes. But otherwise, it is the same.

For a path following a Brownian motion, one can not predict exactly where it will be at a future time, but one can describe its probabilities. So then why can't one think of a quantum mechanical particle as following some path but whose statistics are determined by the Schrödinger equation rather than the Heat equation?

 

[atyy]

There are several ways to do it.

1) Feynman path integral. Here the paths are not real, since one has to rotate to get rid of the imaginary number, so that it becomes standard Wiener integrals. http://arxiv.org/abs/quant-ph/0501167, http://www.scholarpedia.org/article/Path_integral:_mathematical_aspects

2) If the particle has "real" paths without measurement, then one can introduce paths as in Bohmian Mechanics, There is more than one dynamics compatible with quantum mechanics, so even if we grant that the paths are real, we have many choices. http://arxiv.org/abs/quant-ph/9704021

3) If we don't introduce hidden variables, then there are no subensembles associated with a pure state, because a pure state is the complete description of a single physical system in the Copenhagen interpretation, or an extremal point of the space of density matrices, Here paths only emerge by continuous measurement, since the wave function is not real, but only the outcomes of measurements are. In the continuous measurement formalism, one does end up with honest stochastic processes: http://arxiv.org/abs/quant-ph/0611067.

 

[TrickyDicky]

In continuous Brownian motion - while it is completely unpredictable - there are nevertheless many theorems about its behavior. In particular, continuous sample Brownian paths are almost surely nowhere differentiable.

Since the Schrödinger equation for a free particle is the heat equation with a factor of i tacked on, one might ask what the paths of free particles look like.

The experiment would maybe be to break up a fixed time interval into n increments and at each successive increment measure the position of the particle with a detecting screen. This will give a piecewise linear path with n segments. Now let the increments become increasingly smaller and take the limit. What are the properties of these paths?

One might expect some interesting statistics since amplitudes follow a Markov like process. Instead of conditional probabilities, one has conditional amplitudes. But otherwise, it is the same.

For a path following a Brownian motion, one can not predict exactly where it will be at a future time, but one can describe its probabilities. So then why can't one think of a quantum mechanical particle as following some path but whose statistics are determined by the Schrödinger equation rather than the Heat equation?

Interesting question. As atyy explained it can be done in different ways. It is in fact also interesting to show how the Schrödinger equation gives deterministic paths(see Mott's problem). This suggests that deterministic(classical) paths and stochastic (Brownian also classical) paths are extremes of the same continuum.