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Some time ago I had an idea that, as the unitary evolution of the wavefunction is described by a deterministic equation (PDE, simply), could be the collapse of it be described by some kind of a stochastic differential equation? Was this question posed before (I suppose yes...), and what kind of an SDE could it be?

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This idea was suggested by Richard Feynman during his computational phase in the mid 1980s. Feynman asked whether you could simulate a decohering quantum system by stochastically projecting the quantum state whenever decoherence is sufficiently large, and produce a viable classical-quantum hybrid method, where you would only take into account quantum interference for non-decohered trajectories (I am paraphrasing from memory).

I don't think anyone has thought up such a scheme, and it is technically nontrivial. I used to think about it from time to time. None of this is in the literature, even Feynman's remark was off-the-cuff (I don't even remember where I read it), and if you come up with a way of doing it, I personally think it would be a very worthwhile thing.

Simulations people do

Nobody does honest simulations of a quantum system. They are always either stochastic simulations in imaginary time, designed to extract ground state correlation functions, or else density function calculations which are very far removed from the actual many-body wavefunction. The reason is that it is just plain impossible to simulate a quantum system classically, because it requires exponential resources in the general case.

But we know that we can understand most quantum systems at room temperature, because decoherence gets rid of the true exponential-resources quantum effects. Is there a way of actually projecting the wavefunction stochastically so that you get a hybrid quantum-classical-stochastic simulation?

The answer should be yes, but it has never been done to my knowledge. I had some ideas about this, which I will try to sketch out here later.

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Please do, this is very interesting. – quant_dev Sep 24 '11 at 9:18

For open systems you can always say that environment acts like an observer for your quantum subsystem and try to "collapse" the wave function of your subsystem. In this case, you can use so called Master equation, which is an equation for a reduced density matrix describing your quantum subsystem. There is nothing stochastic there (no random terms), but this equation is not easy to solve since it involves operators.

So people found a way out: they mapped the Master equation onto equivalent representation of various simpler equations, which involves only real or complex functions: Wigner representation, Positive-P representation, Generalized P-representation, Q-representation. All these functions (W,P,Q...) are ordinary functions (but can be singular) and they are easier to deal with. Once you know them, you can reconstruct your density matrix. These equation are still not easy to solve.

On this stage, to obtain W,P or can work in the configurational space (arguments of W,P or Q or phase spaces of these functions). Equations in the configurational space are stochastic, where the noise terms representing influence of the environment. It is like you solve Newton equation with some random forces. In these case phase space is spanned by coordinate x and momentum p. For W,P and Q these are some complex arguments instead.

So the procedure is like this:

  1. you solve stochastic Newton-type classical equations for trajectories in the phase space
  2. you reconstruct corresponding "probabilities" W,P or Q.
  3. Finally, you get back your density matrix and calculate observables.

So, stochasticity appears on the level of a phase space. One can show that using these methods, a quantum system may lose its coherence with time and becomes more like classical. I personally call this "collapse", but it is not sudden.

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Wavefunction collapse is most accurately treated through decoherence theory (see quantum decoherence, einselection, etc.) You calculate the evolution of the density matrix (instead of the wavefunction), and it's always a deterministic PDE, even during "measurement". You do not need any other kind of equation.

"Wavefunction collapse" (such as it is) is only relevant in how you describe the meaning of the final density matrix. It's not essential to the calculation. An instantaneous discontinuous collapse, like we teach to undergrads, is just a crude approximation to decoherence theory. (Not that there's anything wrong with crude approximations, they can be very useful! Just don't take them literally!). Some people even say there's no such thing as wavefunction collapse, like Sidney Coleman's lecture "Quantum Mechanics In Your Face". Others disagree. But the way to do the calculation (decoherence theory) is not controversial.

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