Here we repeat the arguments from the wave equation of a particle in a scalar potential
and extend it to a more general case where the potential
is an explicit function
of time, specifically a particle with charge
in an electromagnetic field derived from a vector potential
and a scalar
potential
. In the latter case, the classical relation
![]() |
(1) |
must be replaced by the relation
![]() |
(2) |
Considerations of the behavior of wave packets on the "geometrical optics" approximation lead us to the wave equation
![]() |
(3) |
It is the Schrödinger equation of a charged particle in an electromagnetic field. On the right hand side of equation (3), the operator
designates the scalar product
of the vector
operator
by itself; in other words, the function which results from its action on
is the sum of the expression
and of two other expressions which are obtained from it by substituting
and
for
, namely
In all of this one must realize that the components of the operator
and those of the operator
do not in general commute
with each other.
The Schrödinger equation for a particle in a potential
,
![]() |
(4) |
and equation (3) are the generalizations of the wave equation of a free particle and the same remarks apply to them. They are indeed linear, homogeneous, partial differential equations of the first order in the time. Furthermore, they can be deduced from the classical relations by the correspondence relation
[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].
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