This Quantum World/Implications and applications/Probability flux

Probability fluxEdit

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.