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# Klein-Gordon Equation in Schrödinger Form

For some calculations it is useful to write the Klein-Gordon equation in the two-component form

 (4.84) (4.85)

We see that and .

The charge density for these state is (setting )

 (4.86)

which is rather simple and somewhat similar to the nonrelativistic case.

obey the coupled equation

 (4.87)

and

 (4.88)

If we define the 2-component spinor

 (4.89)

we can combine equations 4.87 and 4.88 to be

 (4.90)

 (4.91)

Using the Pauli matrices, ,

 (4.92)

and we can write

 (4.93)

This is a first order Schrödinger equation (cf. ). The quantity in square brackets is .

Also in this notation, the charge density can be written as

 (4.94)

The normalization condition becomes

 (4.95)

It will be shown later that the sign is determined by whether we start with particles () or antiparticles ().

The Klein-Gordon Hamiltonian is

 (4.96)

The Hamiltonian appears to be non-hermitian, since

 (4.97)

However

 (4.98)

and

 (4.99)

We notice that

 (4.100)

and

 (4.101)

gives

 (4.102)

Therefore

 (4.103)

Because of the normalization condition, the Hamiltonian is effectively hermitian.

Consider the free particle solutions

 (4.104)

A positive-energy plane-wave solution normalized to unit density (equation 4.41) is

 (4.105)

Since ,

 (4.106)

We can write

 (4.107)

where

 (4.108)

A corresponding negative-energy plane-wave solution is

 (4.109)

giving

 (4.110)

Orthogonality shows

 (4.111)

and it can also be shown that

 (4.112)

 (4.113)

 (4.114)

In the nonrelativistic limit we have

 (4.115)

The components of equation 4.108 are

 (4.116) (4.117)

Equation 4.108 in the nonrelativistic limit is

 (4.118)

which holds to second order in the velocity. Similarly

 (4.119)

By completeness, any wavepacket can be expanded in terms of a linear combination of positive- and negative-energy solutions.

 (4.120) (4.121)

where only depends on the magnitude of , and is a function of time and the magnitude of .

If the wave function is normalized to unity we have

 (4.122)

Next: Zitterbewegung Up: Klein-Gordon Equation Previous: Charge Conjugation
Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18