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Chapter 3. Wave optics
2 in the immediate vicinity of path 1 have approximately the same
phase for ϕ ba . The waves for these paths therefore interfere constructively.
For paths which are remote from the classical path, a small variation in path leads to a large variation in phase. This is represented
by paths 3 and 4 in Figure 3.2. The phase factor in the expression for ϕ ba oscillates rapidly for these paths. The sum over these
paths therefore is very close to zero on average. Only paths in the
immediate vicinity of the classical path contribute significantly to
the overall amplitude K(x b , t b ; x a , t a ). Quantitatively, the action
integrals S ba for the two nearby paths 1 and 2 can at most differ
by a reasonably small fraction of h ¯ for constructive interference to
occur.
Further physical insight can be gained by noticing that, for any
single path
ϕ ba = ϕ bc · ϕ ca ,
(3.135)
where (x c , t c ) is any intermediate space-time point along the particular path. This arises from (3.132), which leads directly to
S ba = S bc + S ca .
(3.136)
This is depicted for a hypothetical system in Figure 3.3. The motion can be decomposed into a path from (x a , t a ) to an intermediate point (x c , t c ), followed by a path from (x c , t c ) to the end point
at (x b , t b ). Since the kernel K(x b , t b ; x a , t a ) is the integral over all
possible paths, it follows that
∞
K(x b , t b ; x a , t a ) =
K(x b , t b ; x c , t c ) K(x c , t c ; x a , t a ) dx c .
−∞
(3.137)
This represents the motion between a specific starting point
(x a , t a ) and a specific end point (x b , t b ).
For many purposes it is sufficient to know the state of the system
at a given end point (x b , t b ), without regard for the prior history of
how the system got there. To this end we define the wave function
Chapter 3. Wave optics
2 in the immediate vicinity of path 1 have approximately the same
phase for ϕ ba . The waves for these paths therefore interfere constructively.
For paths which are remote from the classical path, a small variation in path leads to a large variation in phase. This is represented
by paths 3 and 4 in Figure 3.2. The phase factor in the expression for ϕ ba oscillates rapidly for these paths. The sum over these
paths therefore is very close to zero on average. Only paths in the
immediate vicinity of the classical path contribute significantly to
the overall amplitude K(x b , t b ; x a , t a ). Quantitatively, the action
integrals S ba for the two nearby paths 1 and 2 can at most differ
by a reasonably small fraction of h ¯ for constructive interference to
occur.
Further physical insight can be gained by noticing that, for any
single path
ϕ ba = ϕ bc · ϕ ca ,
(3.135)
where (x c , t c ) is any intermediate space-time point along the particular path. This arises from (3.132), which leads directly to
S ba = S bc + S ca .
(3.136)
This is depicted for a hypothetical system in Figure 3.3. The motion can be decomposed into a path from (x a , t a ) to an intermediate point (x c , t c ), followed by a path from (x c , t c ) to the end point
at (x b , t b ). Since the kernel K(x b , t b ; x a , t a ) is the integral over all
possible paths, it follows that
∞
K(x b , t b ; x a , t a ) =
K(x b , t b ; x c , t c ) K(x c , t c ; x a , t a ) dx c .
−∞
(3.137)
This represents the motion between a specific starting point
(x a , t a ) and a specific end point (x b , t b ).
For many purposes it is sufficient to know the state of the system
at a given end point (x b , t b ), without regard for the prior history of
how the system got there. To this end we define the wave function