130_notes.dvi

36 Dirac Equation

36.1 Dirac’s Motivation

TheSchr ̈odinger equation is simply the non-relativistic energy equationoperating on a
wavefunction.

E=

p^2

2 m

+V(~r)

The natural extension of this is therelativistic energy equation.

E^2 =p^2 c^2 + (mc^2 )^2

This is just theKlein-Gordon equationthat we derived for a scalar field. It did not take physicists
long to come up with this equation.

Because the Schr ̈odinger equation isfirst order in the time derivative, the initial conditions
needed to determine a solution to the equation are justψ(t= 0). In an equation that is second
order in the time derivative, we also need to specify some informationabout the time derivatives at
t= 0 to determine the solution at a later time. It seemed strange to give up the concept that all
information is contained in the wave function to go to the relativistically correct equation.

If we have a complex scalar field that satisfies the (Euler-Lagrange= Klein-Gordon) equations

✷φ−m^2 φ = 0

✷φ∗−m^2 φ∗ = 0,

it can be shown that the bilinear quantity

sμ=

̄h

2 mi

(

φ∗

∂φ

∂xμ

−

∂φ∗

∂xμ

φ

)

satisfies theflux conservation equation

∂sμ

∂xμ

=

̄h

2 mi

(

∂φ∗

∂xμ

∂φ

∂xμ

+φ∗✷φ−(✷φ∗)φ−

∂φ∗

∂xμ

∂φ

∂xμ

)

=

̄h

2 mi

m^2 (φ∗φ−φ∗φ) = 0

and reduces to the probability flux we used with the Schr ̈odinger equation, in the non-relativistic
limit. The fourth component of the vector is justctimes the probability density, so that’s fine too
(usingeimc

(^2) t/ ̄h
as the time dependence.).
The perceivedproblem with this probability is that it is not always positive. Because the
energy operator appears squared in the equation, both positive energies and negative energies are
solutions. Both solutions are needed to form a complete set. With negative energies, the probability
density is negative. Dirac thought this was a problem. Later, the vectorsμwas reinterpreted
as the electric current and charge density, rather than probability. The Klein-Gordon equation
was indicating that particles ofboth positive and negative chargeare present in the complex
scalar field. The “negative energy solutions” are needed to form a complete set, so they cannot be
discarded.
Dirac sought to solve the perceived problem by finding an equation that was somehow linear in the
time derivative as is the Schr ̈odinger equation. He managed to do this but still found “negative
energy solutions” which he eventually interpreted to predict antimatter. We may also be motivated
to naturally describe particles with spin one-half.