16.3. Representation of quantum amplitudes
165
FIGURE 16.3
A ‘path’ from q
0
≡ q
i at τ i to q
N
≡ q
f at τ f , via the intermediate positions
q
1 , q
2 , . . . , q
N −1 at τ 1 , τ 2 , . . . , τ N 1 .
−
This can be evaluated by completing the square, shifting the integration variable, and using (16.62), to obtain (problem 16.5)
( )
[
]
1
π∈
1
2
I =
exp
−m (q
2
q
i )
2 .
(16.68)
m
4∈
−
Now the procedure may be repeated for the q
2 integral
∫
{
}
I
2
m
≡ exp − (q
2
− q
i )
2
m
− (q
3
− q
2 )
2
dq
2 ,
(16.69)
4∈
2∈
which yields (problem 16.5)
(
) 1
[
]
I
2
4π∈
2
=
exp
−m (q
3
3m
6
− q
i )
2 .
(16.70)
∈
As far as the exponential factors in (16.63) in (16.64) are concerned, the
pattern is now clear: after n − 1 steps we shall have an exponential factor
[
]
exp −m(q
n
− q
i )
2 /(2n∈) .
(16.71)