In the following, we shall use
,
,
to represent the Cartesian coordinates
,
,
, respectively, and
to represent
.
The time dependent wavefunction then takes the form
. Adopting
standard relativistic notation, we write the four
's as
, for
. A space-time
vector with four components that transforms under Lorentz transformation in an analogous manner to the four space-time coordinates
is termed a *4-vector*, and its components are written like
(i.e., with an upper
Greek suffix). We can lower the suffix according to the rules

(1098) | ||

(1099) | ||

(1100) | ||

(1101) |

Here, the are called the

a summation being implied over a repeated letter suffix. The metric tenor is defined

(1103) | ||

(1104) | ||

(1105) | ||

(1106) |

with all other components zero. Thus,

(1107) |

Likewise,

(1108) |

where , , with all other components zero. Finally, if , and otherwise.

In the Schrödinger representation, the momentum of a particle, whose components are written , , , or , , , is represented by the operators

for . Now, the four operators form the covariant components of a 4-vector whose contravariant components are written . So, to make expression (1109) consistent with relativistic theory, we must first write it with its suffixes balanced,

(1110) |

and then extend it to the complete 4-vector equation

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 , where is the velocity of light in vacuum.