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158
Chapter 3. Wave optics
[28]. The reader is referred to an emended version by Styer of the
original text by Feynman and Hibbs [29] for a detailed, comprehensive, and highly readable description. Our present goal is to
summarize the highlights of the text, with particular application
to the motion of a single charged particle moving in a general electromagnetic potential.
We begin by studying the motion of a particle in one dimension,
where the position x is a function of time t. Classically, the motion
from an initial time t a to a later time t b is along that path which
represents an extremum of the action integral S ba given by
t b
S ba =
L(x, v; t) dt,
(3.132)
ta
where L is the Lagrangian given by (2.9). The beginning point
charcterized by position and time (x a , t a ), and the end point characterized by position and time (x b , t b ) are assumed to be fixed.
In a quantum-mechanical description we seek a probability amplitude ϕ ba for the particle to propagate from an initial position
and time (x a , t a ) to a final position and time (x b , t b ). Again we
assume that the end points (x a , t a ) and (x b , t b ) are fixed.
At this point we form a key hypothesis, namely, the amplitude
ϕ ba can be written for a given path of motion as
i
ϕ ba = const × exp
S ba ,
(3.133)
h ¯
where S ba is the action integral. An infinite number of possible
paths exist, each path having a distinct value of S ba . This is illustrated for a hypothetical system in Figure 3.2. The general rule
in quantum mechanics is that the amplitudes for all alternative
paths must be added to form the resultant amplitude. The absolute square of this resultant amplitude then represents the probability density for finding the system at a given coordinate x. The
amplitudes ϕ ba for all possible paths must therefore be summed to
form the overall probability amplitude K(x b , t b ; x a , t a ). Following
�
158
Chapter 3. Wave optics
[28]. The reader is referred to an emended version by Styer of the
original text by Feynman and Hibbs [29] for a detailed, comprehensive, and highly readable description. Our present goal is to
summarize the highlights of the text, with particular application
to the motion of a single charged particle moving in a general electromagnetic potential.
We begin by studying the motion of a particle in one dimension,
where the position x is a function of time t. Classically, the motion
from an initial time t a to a later time t b is along that path which
represents an extremum of the action integral S ba given by
t b
S ba =
L(x, v; t) dt,
(3.132)
ta
where L is the Lagrangian given by (2.9). The beginning point
charcterized by position and time (x a , t a ), and the end point characterized by position and time (x b , t b ) are assumed to be fixed.
In a quantum-mechanical description we seek a probability amplitude ϕ ba for the particle to propagate from an initial position
and time (x a , t a ) to a final position and time (x b , t b ). Again we
assume that the end points (x a , t a ) and (x b , t b ) are fixed.
At this point we form a key hypothesis, namely, the amplitude
ϕ ba can be written for a given path of motion as
i
ϕ ba = const × exp
S ba ,
(3.133)
h ¯
where S ba is the action integral. An infinite number of possible
paths exist, each path having a distinct value of S ba . This is illustrated for a hypothetical system in Figure 3.2. The general rule
in quantum mechanics is that the amplitudes for all alternative
paths must be added to form the resultant amplitude. The absolute square of this resultant amplitude then represents the probability density for finding the system at a given coordinate x. The
amplitudes ϕ ba for all possible paths must therefore be summed to
form the overall probability amplitude K(x b , t b ; x a , t a ). Following