We temporarily change
variable from [t,x] time and space to [r,
]
which are number of steps to the right or left on the space time diagram:
Next, we count the number
of paths with r and
right and left moving steps respectively with c
corners. We distinguish the following.cases
NRR:
path starts out moving to the right ends up moving to the left,
NLL: path starts out moving to the left ends
up moving to the right
NRL: path
starts out moving to the right, ends up moving to the left etc.:
To count the NRL paths for example, refer to this diagrambased on paper by Louis Kauffman:
In
this example, for NRL 
.
The blue(vertical) arrows are locations of the corners where the particle
changes direction from left moving to right moving. Since, by hypothesis
the particle starts out moving to the right, and ends up moving left the
first and last locations are excluded. Since each right to left transition
matches with a left to right transition except for the final right to left,
there are (c-1)/2
of these blue arrows and they are distributed among r
–1 locations. The usual combinatorial formula applies. Similar
analysis apples to the right to left corners. Main difference is that the
hypothesis of the particle starting out to the right fixes the location
of the first right to left corner(light red horizontal arrow).
Unnormalized Wave Functions
Apply
the weighing rule “a
factor of i for each
corner”:
Normalized
Wave Function
To
get anormalized wave function, divide the un-normalized wave function by
the following normalizing constant:
where
The
probability and current functions are:
Add these together
to get the unnormalized weighed complex probability:
(Treat the initial
condition as a special case the functions Nxx
are all zero there)
We can show for large t:
Animated GIF of the process:
Compare with Dirac Equation
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