Let #\varphi(x,t)# be a (normalizable) solution of the Schrödinger equation:

#iħ\frac{\partial}{\partial t}\varphi(x,t) = -\frac{ħ^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}\varphi(x,t) + V(x)\varphi(x,t),#

and let #\bar{\varphi}(x,t)# denote it's complex conjugate. Then, taking the complex conjugate of the Schrödinger equation (check all the calculations!), the following equation is satisfied for #\bar{\varphi}(x,t)#:

#-iħ\frac{\partial}{\partial t}\bar{\varphi}(x,t) = -\frac{ħ^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}\bar{\varphi}(x,t) + V(x)\bar{\varphi}(x,t).#

We also have the standard result

#\varphi(x,t)\bar{\varphi}(x,t) = |\varphi(x,t)|^{2}.#

Taking the time derivative and multiplying by #iħ#, we get:

#iħ\frac{\partial}{\partial t}|\varphi(x,t)|^{2} = iħ\frac{\partial}{\partial t}[\varphi(x,t)\bar{\varphi}(x,t)]#

#=\bar{\varphi}(x,t)iħ\frac{\partial}{\partial t}\varphi(x,t) + \varphi(x,t)iħ\frac{\partial}{\partial t}\bar{\varphi}(x,t).#

Plugging in the two versions of the Schrödinger equation presented above:

#iħ\frac{\partial}{\partial t}|\varphi(x,t)|^{2} = -\frac{ħ^{2}}{2m}[\bar{\varphi}(x,t)(\frac{\partial^{2}}{\partial x^{2}}+V(x))\varphi(x,t)#

#- \varphi(x,t)(\frac{\partial^{2}}{\partial x^{2}}+V(x))\bar{\varphi}(x,t)].#

The terms with the potential #V(x)# clearly cancel. Therefore,

#iħ\frac{\partial}{\partial t}|\varphi(x,t)|^{2} = -\bar{\varphi}(x,t)\frac{ħ^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}\varphi(x,t)#

#+ \varphi(x,t)\frac{ħ^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}\bar{\varphi}(x,t).#

Let's define a convenient quantity:

#j(x,t) = \frac{iħ}{2m}[\varphi(x,t)\frac{\partial}{\partial x}\bar{\varphi}(x,t) - \bar{\varphi}(x,t)\frac{\partial}{\partial x}\varphi(x,t)].#

We'll call #j# the *probability current*, in analogy with the *probability density* #\rho(x,t) = |\varphi(x,t)|^{2}#. The motivation behind this definition will be clear in what follows.

Using the chain rule once again,

#\frac{\partial}{\partial x}j(x,t) = \frac{iħ}{2m}[-\bar{\varphi}(x,t)\frac{\partial^{2}}{\partial x^{2}}\varphi(x,t) + \varphi(x,t)\frac{\partial^{2}}{\partial x^{2}}\bar{\varphi}(x,t)#

#- \frac{\partial}{\partial x}\bar{\varphi}(x,t)\frac{\partial}{\partial x}\varphi(x,t) + \frac{\partial}{\partial x}\varphi(x,t)\frac{\partial}{\partial x}\bar{\varphi}(x,t)]#

#=\frac{iħ}{2m}[-\bar{\varphi}(x,t)\frac{\partial^{2}}{\partial x^{2}}\varphi(x,t) + \varphi(x,t)\frac{\partial^{2}}{\partial x^{2}}\bar{\varphi}(x,t)].#

Now we can see, from our previous result, that

#iħ\frac{\partial}{\partial t}|\varphi(x,t)|^{2} = -iħ\frac{\partial}{\partial x}j(x,t).#

Dividing this equation by #iħ# and writing it in terms of #\rho# and #j#, we get

#\frac{\partial}{\partial t}\rho(x,t) + \frac{\partial}{\partial x}j(x,t) = 0,#

wich is just the continuity equation. This justifies our definition of probability current.

This allows us to write:

#\frac{d}{dt}\int_{-oo}^{+oo}|\varphi(x,t)|^{2}dx = \int_{-oo}^{+oo}\frac{\partial}{\partial t}|\varphi(x,t)|^{2}dx#

#= \int_{-oo}^{+oo}\frac{\partial}{\partial t}\rho(x,t) dx = -\int_{-oo}^{+oo}\frac{\partial}{\partial x}j(x,t) dx#

#= \lim_{x \to +\oo}j(x,t) - \lim_{x \to -\oo}j(x,t).#

A normalizable wavefunction must clearly vanish at infinity. Therefore,

#\lim_{x \to +\oo}j(x,t) = \lim_{x \to -\oo}j(x,t) = 0,#

and we have proved that

#\frac{d}{dt}\int_{-oo}^{+oo}|\varphi(x,t)|^{2}dx = 0,#

that is, the norm of the wavefunction does not vary with time. For this reason, we say that, in Quantum Mechanics, time evolution is *unitary*.

This result can be generalized to three (in fact, any number of) dimensions:

#\frac{\partial}{\partial t}\rho(\vec{r},t) + \grad * \vec{j}(\vec{r},t) = 0,#

where

#\rho(\vec{r},t) =|\varphi(\vec{r},t)|^{2},#

#j(\vec{r},t) = \frac{iħ}{2m}[\varphi(\vec{r},t)\grad\bar{\varphi}(\vec{r},t) - \bar{\varphi}(\vec{r},t)\grad\varphi(\vec{r},t)].#

The probability current can also be writen in terms of the momentum operator #\hat{P}=-iħ\grad# (in the position representation):

#j(\vec{r},t) = \frac{1}{2m}[\varphi(\vec{r},t)\hat{P}\bar{\varphi}(\vec{r},t) - \bar{\varphi}(\vec{r},t)\hat{P}\varphi(\vec{r},t)]#

#=\frac{1}{m}"Re"[\varphi(\vec{r},t)\hat{P}\bar{\varphi}(\vec{r},t)]#.

Note also that we have assumed that the potential #V# is a real function, so that #\bar{V} = V#. If we remove this hypothesis (that is, we let #V# be a complex function), the result does not hold. This is useful when considering particle decay, a situation where we expect the probability density to decrease exponentially with time (this decrease depends on the imaginary part of #V#).

A further generalization can be obtained with the Dirac (or bra-ket) notation:

#\frac{d}{dt}||\varphi||^{2} = \frac{d}{dt}\langle\varphi|\varphi\rangle = 0,#

wich holds for very general quantum mechanical systems.