D.32 Integral conservation laws

This section derives the integral conservation laws given in addendum {A.14}.

The rules of engagement are as follows:

- The Cartesian axes are numbered using an index
, with1, 2, and 3 for , , andrespectively. - Also,
indicates the coordinate in the direction, , , or. - Derivatives with respect to a coordinate
are indicated by a simple subscript . - Time derivatives are indicated by a subscript t.
- A bare
integral sign is assumed to be an integration over all space, or over the entire box for particles in a box. The is normally omitted for brevity and to be understood. - A superscript
indicates a complex conjugate.

First it will be shown that according to the Schrödinger equation

Taking the right hand term to the other side and writing it in index notation gives

Multiply the left hand side by

To show that the integral

The constant is not important in showing that this is true, so just
examine for any

This equals

as can be seen by differentiating out the parenthetical expression with respect to

There is another way to see that

Here the

Now because of orthonormality of the eigenfunctions, the integration only produces a nonzero result when

That does not depend on time, and the normalization requirement makes it 1.

This also clarifies what goes wrong with the Klein-Gordon equation.
For the Klein-Gordon equation

The first sum are the particle states and the second sum the antiparticle states. That gives:

The final two terms in the sum oscillate in time. So the integral is no longer constant.

The exception is if there are only particle states (no

where the integrated square magnitudes of

Next it will be shown that the rearranged Klein-Gordon equation

preserves the sum of integrals

To do so it suffices to show that the sum of the time derivatives of the three integrals is zero. That can be done by multiplying the Klein-Gordon equation by

To check that, look at what each term in the Klein-Gordon equation
produces separately. The first term gives

or taking one time derivative outside the integral, that is

That is the first needed time derivative, since a number times its complex conjugate is the square magnitude of that number.

The second term in the Klein-Gordon equation produces

That equals

as can be seen by differentiating out the parenthetical expression in the first integral with respect to

The final of the three terms in the Klein-Gordon equation produces

That equals

as can be seen by bringing the time derivative inside the integral. This is the last of the three needed time derivatives.