3.1. Quantum mechanical description of particle motion
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3.1 Quantum mechanical description
of particle motion
We seek a dynamical equation to describe the motion of a single
particle of charge q and rest mass m in the presence of a general
electromagnetic potential. To this end, we begin with a review of
basic quantum mechanics. For clarity, we will do this deductively,
beginning with the fundamental postulates of quantum mechanics,
and proceeding to the motion of a single charged particle in a
general electromagnetic potential. The reader can refer to any of a
number of excellent textbooks on basic quantum mechanics. [59],
[79].
3.1.1 The postulates of quantum mechanics
We begin with a fundamental postulate as follows:
Every measurable dynamical variable C has a corresponding opˆ
erator C, which satisfies a linear operator equation
ˆ
C ϕ = c ϕ.
(3.2)
The dynamical variable C can be any measurable physical quantity. Examples include position, momentum, and energy, to name
ˆ
a few. The operator C acts on the function ϕ, which is called an
eigenfunction. The multiplicative constant c is called an eigenvalue. In the following we will always denote an operator by a
hat over the letter, to distinguish it from an ordinary variable or
function. The definiton of a linear operator is explored further in
Problem 1.
The eigenfunction and eigenvalue are not necessarily unique, but
can take on various values. The number and character of possible values depends on the physical situation being described. For