Derivation of Continuity Equation in Quantum Mechanics
The continuity equation in orthodox quantum theory and Bohmian mechanics
The first time you encounter the continuity equation - an expression that belongs to a conserved quantity - is for sure in classical fluid dynamics. Though in fluid dynamics its meaning may be considered as intuitive, this kind of equation still holds for more abstract physical quantities in different theories, such as classical electrodynamics and quantum theory.
As you know quantum theory is a probabilistic theory, thus it is natural that the continuity equation in quantum theory describes the behaviour of the probability current: probability behaves like a fluid.
Starting point: Born rule
Let's start with the Born rule which says that the probability density of a wavefunction is ρ = | ψ |^2, the absolute square of the wavefunction - usually regarded as a mere probability amplitude. Integrating ρ over a specific volume gives us the probability of measuring the particle corresponding to this wave function inside of this volume. In order to arrive at the continuity equation we must have a look on the time evolution of ρ.
To express the time evolution of the wavefunction we need Schrödinger's equation (and its complex conjugate).
Now we plug this in the above formula:
Doing some further steps gives us:
Generalizing the last line to a three dimensional expression, we get our anticipated result, the continuity equation.
How to get a "classical" continuity equation
So far so good, we derived the continuity equation in terms of orthodox quantum theory. It is important to remark that the probability current density J we have derived does not seem to have the same structure as the current density j in the classical continuity equation. The quantum mechanical current density consists of wavefunctions, their complex conjugates and gradients, but there is no velocity vector field. Is it possible to formulate the quantum mechanical continuity equation in terms of a velocity vector field? Yes!Bohmian mechanics is the answer here.
While in Bohmian mechanics the Schrödinger equation describes the time evolution of the wavefunction there is a further equation: the guiding equation, that describes the time evolution of the positions of k particles. [2]
The guiding equation gives us the velocity vector field we need for the classical continuity equation. In the following we assume a probability current of the anticipated shape and will see that it is equivalent to the current we got above. [2][3]
Thus we get both with orthodox theory and Bohmian mechanics the same probability current: Assuming a classical continuity equation makes sense in the terms of Bohmian mechanics.
Equivariance and quantum equilibrium
What we've done here is closely related to two important concepts of Bohmian mechanics, that I want to mention without a lot of further explanation (this post is already long enough): equivariance and quantum equilibrium.
And how do we get Born rule from the ideas of Bohmian mechanics? It emerges from the quantum equilibrium hypothesis:
At first glance this doesn't seem very spectacular - but what does it actually mean? If the coordinates of particles a randomly distributed (at least they appear random), a system is in quantum equilibrium. Thus quantum equilibrium is in a way analogous to thermodynamic equilibrium. Although randomness appears in the Bohmian theory, it is not in the same sense fundamental as it is in orthodox quantum theory.
One last remark: Bohmian mechanics is a deterministic (though nonlocal) theory of the quantum realm that does not give up realism as easily as it is done in orthodox theory. Still it does not necessarily make 'more' predictions than orthodox quantum theory - it is limited by absolute uncertainty.
"In a universe governed by Bohmian Mechanics it is in principle impossible to know more about the configuration of any subsystem than what is expressed by ρ = | ψ |^2 - despite the fact that for Bohmian Mechanics the actual configuration is an objective property, beyond the wavefunction." ([1],p.20)
[1] D. Dürr, S. Goldstein, and N. Zanghì. Quantum Equilibrium and the Origin of Absolute Uncertainty. 1992 (p. 13-20)
[2] D. Dürr, S. Goldstein, and N. Zanghì. Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory. 2003 (p. 7-8)
[3] J. J. Sakurai, Modern Quantum Mechanics, Revised Edition, 1994 (p. 101, equation 2.4.16)
Source: https://mysteriousquantumphysics.tumblr.com/post/632242455587307520/the-continuity-equation-in-orthodox-quantum-theory
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