162
Chapter 3. Wave optics
In words, (3.139) states that the wave function at any given point
in space-time represents the summation over all possible prior histories. In addition, it follows from (3.134) that the probability
density P (x b , t b ) for finding the particle at position x b at time t b
is given by
P (x b , t b ) = | ψ(x b , t b ) |
2 .
(3.140)
Given this, it is of great interest to explore the evolution of the
wave function in a differential sense, where the end time t b differs
from the initial time t a by a differential time interval f. By this
method we set out to derive a differential equation which describes
the evolution of the wave function ψ(x, t) in space-time. Applying
(3.139) it follows that
∞
ψ(x, t + f) =
K(x, t + f; x a , t) ψ(x a , t) dx a .
(3.141)
−∞
Because this represents an infinitesimal increment in space-time, it
follows that virtually all of the contribution is due to paths in the
immediate vicinity of (x, t). We therefore make the substitution
x a = x + η, where η is a small increment in position relative to x.
Substituting into (3.141) we obtain
(x, t +
∞
ψ
f) =
K(x, t + f; x + η, t) ψ(x + η, t) dη.
(3.142)
−∞
The kernel K is given to good approximation by
1
i t+E
K(x, t + f; x + η, t) =
exp
�
L(x, v; t
� ) dt
�
�
, (3.143)
A
h ¯ t
where A is a normalization constant, yet to be determined. For
the infinitesimal integration interval this in turn reduces to
1
i
η η
K(x, t + f, x + η, t) ≈
exp
�
f L x + ,
,
(3.144)
A
h ¯
2 f
�
where the first argument of L is the position and the second argument is the velocity, both averaged over the infinitesimal integration interval. The nonrelativistic approximation for the Lagrangian
(2.7) is
L(x, v; t) =
1 mv
2 + qv · A(x, t)
2
− qφ(x, t).
(3.145)