Quantum Mechanics for Engineers 

© Leon van Dommelen 

A.36 Alternate Dirac equations
If you look in advanced books on quantum mechanics, you will likely
find the Dirac equation written in a different form than given in
chapter 12.12.
The Hamiltonian eigenvalue problem as given in that section was
where was a vector with four components.
Now assume for a moment that is a state of definite momentum.
Then the above equation can be rewritten in the form
The motivation for doing so is that the coefficients of the
matrices are the components of the relativistic momentum fourvector,
chapter 1.3.1.
It is easy to check that the only difference between the and
matrices is that through get a minus
sign in front of their bottom element. (Just multiply the original
equation by and rearrange.)
The parenthetical expression above is essentially a fourvector dot
product between the gamma matrices and the momentum fourvector.
Especially if you give the dot product the wrong sign, as many
physicists do. In particular, in the index notation of chapter
1.2.5, the parenthetical expression is then
. Feynman hit upon the bright idea to
indicate dot products with matrices by a slash through the
name. So you are likely to find the above equation as
Isn’t it beautifully concise? Isn’t it completely
incomprehensible?
Also consider the case that is not an energy and momentum
eigenfunction. In that case, the equation of interest is found from
the usual quantum substitutions that becomes
and becomes
. So the rewritten Dirac
equation is then:
In index notation, the parenthetical expression reads
. So following Feynman
Now all that the typical physics book wants to add to that is a
suitable nonSI system of units. If you use the electron mass as
your unit of mass instead of the kg, as unit of velocity instead
of m/s, and as your unit of angular momentum
instead of kg m/s, you get
No outsider will ever guess what that stands for!