Continuity equation for the probability density function in quantum mechanics ↩

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created: 2022-01-05 13:54:30
modified: 2022-01-10 04:13:04

Where is the probability density of the particle, is the divergence and is the probability current density.

Integral form

Let's integrate for a fixed volume . Using Gauss's theorem it's easy to see that:

Proof

Let's start from the Schrödinger equation:

And it's complex conjugate:

The Hamiltonian operator remains unchanged since it is Hermitian.

We use the following standard Hamiltonian operator:

The time derivative of the probability density then becomes:

The terms involving disappear, so we obtain:

And then from the definition of the probability current density, we see that: