This Quantum World/Implications and applications/Probability flux

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Probability flux[edit]

The time rate of change of the probability density (at a fixed location ) is given by


With the help of the Schrödinger equation and its complex conjugate,



one obtains



The terms containing cancel out, so we are left with



Next, we calculate the divergence of :


The upshot:

Integrated over a spatial region  with unchanging boundary

According to Gauss's law, the outward flux of through equals the integral of the divergence of over 

We thus have that

If is the continuous density of some kind of stuff (stuff per unit volume) and is its flux (stuff per unit area per unit time), then on the left-hand side we have the rate at which the stuff inside  increases, and on the right-hand side we have the rate at which stuff enters through the surface of  So if some stuff moves from place A to place B, it crosses the boundary of any region that contains either A or B. This is why the framed equation is known as a continuity equation.


In the quantum world, however, there is no such thing as continuously distributed and/or continuously moving stuff.  and respectively, are a density (something per unit volume) and a flux (something per unit area per unit time) only in a formal sense. If is the wave function associated with a particle, then the integral gives the probability of finding the particle in if the appropriate measurement is made, and the framed equation tells us this: if the probability of finding the particle inside  as a function of the time at which the measurement is made, increases, then the probability of finding the particle outside  as a function of the same time, decreases by the same amount. (Much the same holds if is associated with a system having  degrees of freedom and is a region of the system's configuration space.) This is sometimes expressed by saying that "probability is (locally) conserved." When you hear this, then remember that the probability for something to happen in a given place at a given time isn't anything that is situated at that place or that exists at that time.