Math Infinitum

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Intuition

Probability can move but cannot vanish: whatever leaves one region must flow into a neighbouring one. That flow is the probability current, built from the wavefunction and its slope. With it, the conservation of total probability becomes a local statement, the continuity equation — the same equation that says charge or water is conserved in a pipe.

The water in a stretch of river changes only by what flows in at one end and out at the other. The probability in a stretch of the line changes only by the current at its two ends.

The continuity equation

The probability current is built from the wavefunction and its derivative. Together with the density $\rho=|\psi|^{2}$ it satisfies the continuity equation, which follows from the Schrödinger equation with a real potential.

j=\frac{\hbar}{m}\operatorname{Im}\Bigl(\psi^{*}\frac{\partial\psi}{\partial x}\Bigr),\qquad \frac{\partial\rho}{\partial t}+\frac{\partial j}{\partial x}=0

Reading the current

For a plane wave $Ae^{ikx}$ the current is $|A|^{2}\hbar k/m$ : density times velocity, as for a flowing fluid.
A real wavefunction, or a real one times a constant phase, carries no current.
Integrating the continuity equation over a stretch: the probability inside changes by the current in at one end minus the current out at the other.

The continuity equation

\frac{\partial|\psi|^{2}}{\partial t}=-\frac{\partial j}{\partial x}

Differentiate the density with the product rule, and replace each time derivative by the Schrödinger equation and its conjugate. The potential terms cancel because $V$ is real, and what remains is the derivative in $x$ of the current.

Proof steps

\partial_{t}|\psi|^{2}=\psi^{*}\partial_{t}\psi+\psi\,\partial_{t}\psi^{*}

The product rule.

\partial_{t}\psi=\frac{i\hbar}{2m}\partial_{x}^{2}\psi-\frac{i}{\hbar}V\psi

The Schrödinger equation, divided by $i\hbar$ ; the conjugate has $i\to-i$ .

\partial_{t}|\psi|^{2}=\frac{i\hbar}{2m}\bigl(\psi^{*}\partial_{x}^{2}\psi-\psi\,\partial_{x}^{2}\psi^{*}\bigr)

The terms with $V$ cancel, since $V$ is real.

\psi^{*}\partial_{x}^{2}\psi-\psi\,\partial_{x}^{2}\psi^{*}=\partial_{x}\bigl(\psi^{*}\partial_{x}\psi-\psi\,\partial_{x}\psi^{*}\bigr)

The cross terms $\partial_{x}\psi^{*}\partial_{x}\psi$ cancel inside the derivative.

\partial_{t}|\psi|^{2}=-\partial_{x}\Bigl[\frac{\hbar}{m}\operatorname{Im}\bigl(\psi^{*}\partial_{x}\psi\bigr)\Bigr]

Since $z-z^{*}=2i\operatorname{Im}z$ .

Applications

Practice

Density Times Velocity

The current of a plane wave $Ae^{ikx}$ is the density, $|A|^{2}$ , times the velocity $\hbar k/m$ .

j=|A|^{2}\,\frac{\hbar k}{m}

Try it

With $\hbar=m=1$ , what is the probability current of the plane wave $2e^{3ix}$ ?