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In the following, we shall employ
,
,
to represent the Cartesian coordinates
,
,
, respectively, and
to represent
, where
is the velocity of light in vacuum.
The time-dependent wavefunction then takes the form
. Adopting
standard relativistic notation, we write the four
's as
, for
[93]. A space-time
vector with four components that transform under Lorentz transformation in an analogous manner to the four space-time coordinates,
, is termed a 4-vector [93], and its components are written like
(i.e., with an upper
Greek suffix). We can lower the suffix according to the rules
![$\displaystyle a_{0}$](img3798.png) |
![$\displaystyle =+ a^{\,0},$](img3799.png) |
(11.1) |
![$\displaystyle a_{1}$](img3800.png) |
![$\displaystyle =-a^{\,1},$](img3801.png) |
(11.2) |
![$\displaystyle a_{2}$](img3802.png) |
![$\displaystyle = -a^{\,2},$](img3803.png) |
(11.3) |
![$\displaystyle a_{3}$](img3804.png) |
![$\displaystyle = -a^{\,3}.$](img3805.png) |
(11.4) |
Here, the
are called the contravariant components of the 4-vector
, whereas the
are termed the covariant components. Two 4-vectors
and
have the Lorentz-invariant scalar product
![$\displaystyle a^{\,0}\,b^{\,0}-a^{\,1}\,b^{\,1}-a^{\,2}\,a^{\,2}-a^{\,3}\,b^{\,3} = a^{\,\mu}\,b_{\mu}= a_{\mu}\,b^{\,\mu},$](img3808.png) |
(11.5) |
a summation being implied over a repeated letter suffix [93]. The metric tenor,
, is defined
![$\displaystyle g_{\,00}$](img3810.png) |
![$\displaystyle =+1,$](img3811.png) |
(11.6) |
![$\displaystyle g_{\,11}$](img3812.png) |
![$\displaystyle =-1,$](img3813.png) |
(11.7) |
![$\displaystyle g_{\,22}$](img3814.png) |
![$\displaystyle =-1,$](img3813.png) |
(11.8) |
![$\displaystyle g_{\,33}$](img3815.png) |
![$\displaystyle =-1,$](img3813.png) |
(11.9) |
with all other components zero [93].
Thus,
![$\displaystyle a_{\mu} = g_{\,\mu\,\nu}\,a^{\,\nu}.$](img3816.png) |
(11.10) |
Likewise,
![$\displaystyle a^{\,\mu} = g^{\,\mu\,\nu}\,a_{\,\nu},$](img3817.png) |
(11.11) |
where
,
, with all other components zero.
Finally,
if
, and
otherwise.
In the Schrödinger representation (see Section 2.4), the momentum of a particle, whose Cartesian components are written
,
,
, or
,
,
,
is represented by the operators
![$\displaystyle p^{\,i} = -{\rm i}\,\hbar\,\frac{\partial}{\partial x^{\,i}},$](img3828.png) |
(11.12) |
for
. Now, the four operators
form the covariant components of a
4-vector whose contravariant components are written
. (See Exercise 1.) So, to make
expression (11.12) consistent with relativistic theory, we must first write it with its
suffixes balanced,
![$\displaystyle p^{\,i} = {\rm i}\,\hbar\,\frac{\partial}{\partial x_{\,i}},$](img3832.png) |
(11.13) |
and then extend it to the complete 4-vector equation,
![$\displaystyle p^{\,\mu} = {\rm i}\,\hbar\,\partial^{\,\mu},$](img3833.png) |
(11.14) |
where
.
According to standard relativistic theory, the new operator
, which forms a 4-vector when
combined with the momenta
, is interpreted as the energy of the particle divided by
[93].
Next: Dirac Equation
Up: Relativistic Electron Theory
Previous: Introduction
Richard Fitzpatrick
2016-01-22