A particle immersed in a given environment (a fluid, for instance) undergoes a jittering movement dubbed Brownian motion. As far as quantum effects are concerned, could we describe the quantum Brownian motion of this particle through a Schrödinger function? In other words, is the wavefunction description of quantum phenomena universal?
What you are describing sounds like a particle going through a random walk function. So if that assumption is correct no wave function would adequately describe a random path through a three dimensional space filled with whatever else is there.
There might be a possibility of modeling a wave for a finite length (say between two collisions with other particles), redirecting randomly, and then modeling another wave. But this would mean a calculation intense exercise to model something which is effectively a randomly generated path.
So I would wager an answer to your question would be yes, the wavefunction description is universal; but technological limitations require that we "simulate" Brownian motion instead of actually "modeling" it. In other words, accuracy must sometimes be sacrificed for practicality.
Experts in Brownian motion may have more enlightening commentary.
Dear Jimmy Davis,
What are the differences between both processes of modelling and simulating Brownian movement?
From what I remember, a simulation is a method of replicating observable behavior without having an actual solution. A model is the use of a known solution to replicate observable behavior.
So for Brownian motion for example, I might have a mathematical solution f(x). I can code this into a program (ie: MATLAB) and watch how the solution behaves as I tweak certain variables. That would be a model. If you have a solution (or even a potential solution) you model that solution.
On the other hand, if I were to use a random number generator to obtain a picture of something very similar to Brownian motion just so I can observe what is happening (in a broad sense), that would be a simulation. In other words, it's not really Brownian motion, but it looks close enough that I might be able to gather some insight into whatever phenomenon I am researching.
So I can model the two-body system, but I can only simulate (or approximate) the behavior of a three-body system (unless somebody solved it and I didn't hear about it).
A lot of times the words are used interchangeably so this really just boils down to semantics. To keep it straight in my head I try and remember what my computational physics professor once told me: "Simulated sugar isn't real sugar." Admittedly, I still mix it up when I communicate verbally.
Oh yeah and I should add that often, a simulation usually evolves into a model once enough insight is obtained to get a solution and start working with it.
The path of a particle in Brownian motion is erratic and frequently changes direction.
I presume it's due to gas particles the object hits with. Does this eccentric behavior of the object due to the motion engages itself with the theory of chaos?
Quantum particles in interaction with an environment decohere or "classicalize"
quite rapidly. In fact, the Brownian trajectory of a quantum
particle is easily observable, for instance, in a bubble chamber.
Brownian motion by itself is a classical concept since one can unambiguously
affiliate with a particle in a given point and at a given time a probability density function
(which is the normal distribution) without any need for a wave function description.
Entirely different concept is a quantum Brownian motion, where
one a particle goes simultaneously over all possible histories along Brownian-like paths.
This is an essence of a Feynman sum over histories approach to quantum mechanics.
From your original question I, however, reckon that you reasoning was aimed
at the classical Brownian motion.
Schrödinger function is completely certain for the particles in solid. Fluid is also equivalent to less solid. Difference exist in particle motion.
A quantum Brownian particle, or QBP, is a generalization of a usual quantum particle. Such a usual particle satisfies the Copenhagen Interpretation, for instance, and thus it has a wave function which is a solution of the usual Schroedinger equations. On the other hand, a QBP - being a generalization - it does not correspond to the usual Schroedinger equation, but to a generalization of that equation. Hence, even if a QBP may have its own so called wave function, that is not the usual wave function which, as such, would have to satisfy the usual Schroedinger equation.
In short, physicists like to have all kind of particles, and like to have corresponding equations describing the dynamics of such particles.
Related to quanta, the usual Schroedinger equation is supposed to define what is, and what is not, a quantum particle. And the solutions of this equation are called wave functions.
If now one modifies that equation, for instance, by some generalization, then its solutions are no longer usual wave functions. And of course, they do no longer correspond to usual quantum particles.
Thus, you can mess with quantum particles or with the Schroedinger equation, but then you are outside of the usual quantum world, thus in particular, outside of the usual quantum particles, or usual wave, functions as well ...
Schrodinger equation can give the wave function only if the potential in which it moves is known. In the case of the Brownian motion, the potential in which a particle moves can be calculated only if the wave function of other particles are known. So one has to use Self consistent methods like Hartee- Fock methods. It is quite a challenge even for a super computer.
Classical Brownian motion
In the classical realm the Brownian motion of a particle in a fluid is described by both the Langevin equation and the Fokker-Planck equation. The Langevin dynamics of this Brownian particle (our open system) is given by the time evolution of random variables such as its position x and momentum p, for instance, while the Fokker-Planck dynamics yields the time evolution of the probability distribution function F(x,p,t). As far as the fluid’s influence is neglected, the Brownian particle turns into a Newtonian particle, that is, an isolated system to be described by the Newton equation and the Liouville equation.
Quantum Brownian motion
In the quantum realm, what are the quantum-mechanical equations of motion describing a quantum Brownian particle? In my recent studies, I have been addressing this issue as follows. I start with the Klein—Kramers equation (a sort of Fokker—Planck equation in classical phase space). Then making use of a non-canonical method of quantization, I can derive a Caldeira—Leggett quantum master equation describing the quantum Brownian motion in terms of a generalized von Neumann function (the so-called density matrix) expressed in terms of two coordinates in configuration space: x and x’. As this quantum open system renders isolated, the Liouville-von Neumann equation of motion is obtained. In addition, it is readily to check that the Schrödinger equation can be derived since the von Neumann function is factorized in terms of a Schrödinger function at point x and another one at point x’.
Summarizing, a quantum Brownian particle cannot be described by any Schrödinger function at all. Put it differently, a quantum Brownian particle does not have a wavefunction because it is a non-isolated system. This conclusion has been reached without resorting to decoherence, entanglement, superposition, Hartree-Fock methods, quantum time or mixed states as advocated by the colleagues.
In conclusion, the wavefunction description of the quantum phenomena is not a universal one. Therefore, the existence of the quantum Brownian movement does imply the breakdown of the universality of the Schrödinger function for describing open quantum systems.
In addition, I strongly recommend the reading of my paper:
http://www.researchgate.net/publication/220000766_The_dynamical-quantization_approach_to_open_quantum_systems
Article The dynamical-quantization approach to open quantum systems
Lo que no entiendo es que, si el movimiento browniano fue la observación de que pequeñísimas partículas de polen se movían aleatoriamente en un líquido, cómo podemos suponer que con funciones de onda cuánticas podríamos modelar su movimiento. El movimiento de esas partículas en el líquido, se ve condicionado por las características físicas macroscópicas del líquido y de las partículas de polen, así que, aunque su movimiento sea alterado por los movimientos subatómicos, ya no guardan relación directa con ellos, pertenecen a otra escala de fenómenos. Digamos que el movimiento browniano es una `traducción´ a otra escala de lo que ocurre en el interior de las moléculas, en concreto en el átomo. ¿Me estoy confundiendo o voy bien? ¿Cómo podríamos modelar el movimiento browniano con pautas mecánico-cuánticas, si en realidad ya no está en un medio cuántico? Quizá debiéramos rendirnos entonces a lo aleatorio, o al análisis de `todos los caminos de Feynman´, que no es más que una técnica comprometida y limitada de previsión de resultados, no la observación directa del fenómeno.
What happens when particle described properly by QM (like spins properties measured by NMR) are diffusing ? Correct quantum descriptions of those spin assembly are given by density-operator, whose matrix trace represent classical magnetization. As the density matrix comes from wavefunctions ( Ô=|a>
If the Universe could be imagined as an isolated system, then we may resort to a "wave function of the Universe"!
Nevertheless, we cannot ascribe a wave function to a quantum Brownian particle.
If we envisage the Universe as a collection of a myriad of many, many, many universes and pinpoint only the Hawking-Hartle universe undergoing a quantum Brownian movement, then the wave wave function description is thoroughly unnecessary!!!
In 2001 I and my collaborators wrote a Phys. Rev. Lett. paper [vol. 86, 5092 (2001)] on how to handle a quantum Brownian particle using the Monte Carlo Wave Function method. We even did numerical calculations for a particle (H) diffusing on a metal surface, with results in good agreement with relevant experiments. The MCWF method does include a wave function description of the Brownian particle in terms of its energy eigenstates.
Thanks to A.O. Bolivar for posting his paper. Dr. Bolivar may be overstating his case, though. I suspect that a wave function description of Brownian motion is possible (with rapidly vanishing off-diagonal elements in the state operator), but worthless because it is so much simpler to treat all the randomness in the problem statistically! I suppose we can use quantum mechanical scattering theory to describe in great QM detail the behavior of a gas of, say, 100000 helium atoms, but why bother when statistical mechanics is there to help with the job. Still, there are aspects of Brownian motion where quantum behavior is relevant--perhaps Taplo Ala-Nisslla's paper referenced above.
David Bohm has an interesting footnote in his book "Causality and Chance in Modern Physics," Ch. 4, Section 2, showing that a particle undergoing ordinary Brownian motion obeys an indeterminacy relationship for momentum and position uncertainties analogous to the Heisenberg uncertainty principle.
Dear Tapio,
I have read your PRL paper.
In fact, the MCWF method makes use of a sort of wave function description of open quantum systems. Yet such a wave function cannot be identified with the Schrödinger function.
Averaging the stochastic wave function [Eq. (1) in PRL paper], which actually relies on two random quantities, leads to a numerical description of a quantum Brownian particle (QBP). This averaging procedure does not imply that a QBP has any Schrödinger function.
Read please the Rosinger’s clarifying comments above:
“…even if a QBP may have its own so called wave function, that is not the usual wave function which, as such, would have to satisfy the usual Schroedinger equation”.
My paper quoted above upholds the view that a QBP cannot be described by a Schrödinger function. This upshot raises some complaints against the universality of the Schrödinger picture of quantum phenomena.
P.S.: Is it possible to derive the Schrödinger time evolution starting from your stochastic wave function [Eq. (1)]?
Dear Frank,
Tapio’s PRL paper is about a stochastic wave function description of QBM. However, such a wave function has not to do with any Schrödinger function (see please my reply to Tapio’s comments above).
As to the allusion to Bohm’s theory, I would like to comment the following.
Just as there exists a deterministic approach explaining the classical theory of Brownian motion, Bohm tries to make an analogy between the Copenhagen interpretation and his theory based on sub-quantum phenomena.
Bohm’s theory is based on isolated systems in which the quantum potential is built up from the Schrödinger function. For open quantum system there is no Schrödinger function. Accordingly, quantum potential cannot exist too!!
For details see please my book:
http://pt.scribd.com/doc/21963130/A-O-Bolivar-Quantum-Classical-Correspondence
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