| dbp:proof
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- The time dependent Schrödinger equation and its complex conjugate are respectively:
where is the potential function. The partial derivative of with respect to is:
Multiplying the Schrödinger equation by then solving for , and similarly multiplying the complex conjugated Schrödinger equation by then solving for ;
substituting into the time derivative of :
The Laplacian operators in the above result suggest that the right hand side is the divergence of , and the reversed order of terms imply this is the negative of , altogether:
so the continuity equation is:
The integral form follows as for the general equation. (en)
- One of Maxwell's equations, Ampère's law (with Maxwell's correction), states that
Taking the divergence of both sides results in
but the divergence of a curl is zero, so that
But Gauss's law , states that
which can be substituted in the previous equation to yield the continuity equation (en)
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