that gave us the correct interaction with the electron’s spin. We have applied this same momentum
operator relativistically, not much of a stretch. We have also written the equation in terms of four
components, but there was no new physics in that since everythingcould be computed from two
components, sayφ(L)since
φ(R)=1
mc(
i ̄h∂
∂x 0
−i ̄h~σ·∇~)
φ(L).Dirac’s paper did not follow the same line of reasoning. Historically, theSchr ̈odinger-Pauli Hamil-
tonian was derived from the Dirac equation. It was Dirac who produced the correct equation for
electrons and went on to interpret it to gain new insight into physics.
Dirac Biography
36.4 The Conserved Probability Current
We now return to the nagging problem of the probability density and current which prompted Dirac
to find an equation that is first order in the time derivative. We derived the equation showing con-
servation of probability (See section 7.5.2), for 1D Schr ̈odinger theory byusing the Schr ̈odinger
equation and its complex conjugateto get an equation of the form
∂P(x,t)
∂t+
∂j(x,t)
∂x= 0.
We also extended it to three dimensions in the same way.