If we know the wave function at time , then it is give at
some other time,
, using
where
and
is a Green function.
Because
satisfies Klein-Gordon equation, so does
the Green function
:
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(4.171) |
The most general solution with this property is
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(4.172) |
where is a scalar function of
alone.
Therefore
is a constant as it must be a function of
only,
which is a constant.
To see this, we use plane waves with momentum
and write
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|
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(4.173) |
Therefore
.
It is also useful to write
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(4.174) |
so that
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(4.175) |
is the Green theorem for any solution of the Klein-Gordon equation.
Since the Schrödinger equation is first order, this means that
given the state vector at time , one can determine the state at
all future times.
On the other hand, since the Klein-Gordon equation is second order,
one requires two initial conditions - both the wave function and its
time derivative.
These problems are resolved by the realization that a properly
relativistic wave equation involves of necessity both particle and
antiparticle degrees of freedom.
The existence of both particle and antiparticle degrees of freedom in
the wave function is the reason that one needs two boundary conditions
at time
in order to predict the future behaviour.