3
Relativistic Quantum Mechanics
It is clear that the non-relativistic Schr¨ odinger equation is quite inadequate
to analyse the results of experiments at energies far higher than the rest
mass energies of the particles involved. Besides, the quarks and leptons have
spin1 , a degree of freedom absent from the Schr¨ odinger wavefunction. We
2
therefore need two generalizations – from non-relativistic to relativistic for
spin-0 particles, and from spin-0 to spin1 . The first step is to the Klein–
2
Gordon equation (section 3.1), the second to the Dirac equation (section 3.2).
Then after some further work on solutions of the Dirac equation (sections 3.3–
3.4), we shall consider (section 3.5) some simple consequences of including the
electromagnetic interaction via the gauge principle replacement (2.44).
3.1 The Klein–Gordon equation
The non-relativistic Schr¨ odinger equation may be put into correspondence
with the non-relativistic energy–momentum relation
E = p
2 /2m
(3.1)
by means of the operator replacements
1
E → i∂/∂t
(3.2)
p → −i∇,
(3.3)
these differential operators being understood to act on the Schr¨ odinger wavefunction.
For a relativistic wave equation we must start with the correct relativistic
energy–momentum relation. Energy and momentum appear as the ‘time’ and
‘space’ components of the momentum 4-vector
p
μ = (E, p)
(3.4)
which satisfy the mass-shell condition
2
μ
2
2
p = p μ p = E
2
− p = m .
(3.5)
1 Recall ħ = c = 1 throughout (see appendix B).
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