1.6 Probability Current

Consider a wire carrying some electric current $I$ . Mathematically, current is defined as the amount of charge passing a point per unit time. By analogy, we can also define a “probability current”, which denotes the amount of probability passing through a point per unit time.

1.6.1 Probability Conservation

The total probability of finding a particle for $x\in(-\infty,\infty)$ is one, and this does not change with time. This means that the total probability is conserved.

However, we can actually make a stronger claim than this. For the probability of finding a particle in a small interval $(x_{0},x_{0}+\operatorname{\mathrm{d}}\!x)$ to change, some probability must “flow” in or out of this interval. This is known as local conservation. We say that there is nonzero probability current $j$ in or out of the segment. The net change in probability current in the small interval is defined as

j(x_{0}+\operatorname{\mathrm{d}}\!x,t)-j(x_{0},t)=\operatorname{\mathrm{d}}\!% j(x_{0},t),

(1.59)

and the probability of finding the particle in the interval is

\lvert\psi(x_{0},t)\rvert^{2}\operatorname{\mathrm{d}}\!x,

(1.60)

so for probability to be conserved locally we must have

\operatorname{\mathrm{d}}\!j(x_{0},t)=-\frac{\partial}{\partial t}\lvert\psi(x% _{0},t)\rvert^{2}.

(1.61)

The minus sign is there because the left-hand side being positive implies that probability is leaving the region.

In the limit where the interval becomes infinitesimally small, we get

\frac{\partial\lvert\psi(x,t)^{2}\rvert}{\partial t}=-\frac{\partial j(x,t)}{% \partial x}.

(1.62)

This is known as the continuity equation, and it appears in all sorts of context in physics whenever we have local conservation (e.g. charge conservation in electromagnetism, mass conservation in fluid dynamics, etc.). The greater the magnitude of the slope of the probability current, the faster probability is flowing through a given point. Probability current $j$ has units ${\mathrm{s}}^{-1}$ , or “probability per unit time”.

1.6.2 Calculating Probability Current

To calculate the probability current $j$ for a given state $\psi$ , we can start from the continuity equation 1.62 and use the Schrodinger equation to write the temporal derivative in terms of a spatial derivative just like we did in section 1.4.2.

Starting with the left-hand side of equation 1.62 and using the product rule, we have

$\displaystyle\frac{\partial\lvert\psi\rvert^{2}}{\partial t}$	$\displaystyle=\frac{\partial\psi^{\ast}}{\partial t}\psi+\psi^{\ast}\frac{% \partial\psi}{\partial t}$	(1.63)
	$\displaystyle=\left(\frac{\hbar}{2mi}\frac{\partial^{2}\psi^{\ast}}{\partial x% ^{2}}-\frac{V}{i\hbar}\psi^{\ast}\right)\psi+\psi^{\ast}\left(-\frac{\hbar}{2% mi}\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{V}{i\hbar}\psi\right)$	(1.64)
	$\displaystyle=\frac{\hbar}{2mi}\left(\frac{\partial^{2}\psi^{\ast}}{\partial x% ^{2}}\psi-\psi^{\ast}\frac{\partial^{2}\psi}{\partial x^{2}}\right),$	(1.65)

where in the second line we have used relations 1.36.

It would be nice if we could write this expression as a derivative of something, because then we could just straightforwardly integrate the continuity equation to obtain $j$ . Luckily, by adding zero we can see that this expression has the form

	$\displaystyle\frac{\partial^{2}\psi^{\ast}}{\partial x^{2}}\psi-\psi^{\ast}% \frac{\partial^{2}\psi}{\partial x^{2}}$	$\displaystyle=\frac{\partial^{2}\psi^{\ast}}{\partial x^{2}}\psi+\frac{% \partial\psi^{\ast}}{\partial x}\frac{\partial\psi}{\partial x}-\psi^{\ast}% \frac{\partial^{2}\psi}{\partial x^{2}}-\frac{\partial\psi^{\ast}}{\partial x}% \frac{\partial\psi}{\partial x}$		(1.66)
		$\displaystyle=\frac{\partial}{\partial x}\left(\frac{\partial\psi^{\ast}}{% \partial x}\psi-\psi^{\ast}\frac{\partial\psi}{\partial x}\right),$		(1.67)

and hence we have

\frac{\partial\lvert\psi\rvert^{2}}{\partial t}=\frac{\hbar}{2mi}\frac{% \partial}{\partial x}\left(\frac{\partial\psi^{\ast}}{\partial x}\psi-\psi^{% \ast}\frac{\partial\psi}{\partial x}\right).

(1.68)

So by equation 1.62, $j$ takes the form

j(x,t)=\frac{h}{2mi}\left(\psi^{\ast}\frac{\partial\psi}{\partial x}-\frac{% \partial\psi^{\ast}}{\partial x}\psi\right).

(1.69)