158
10 Quantum Finance and Path Integrals
It is instructive to calculate the propagator ˆ
K for a free particle (potential V (x) =
0), which turns the propagator into
ˆ
K (x b , t b ; x a , t a ) = lim
ε→0
1
A n
. . .
exp
im
2ε
n
k=1
(x k − x k−1 )
2
dx 1 . . . dx n−1 ,
(10.47)
where we see that the integrals are convolutions of Gaussian leading to a new Gaussian. In this way we obtain
ˆ
K (x b , t b ; x a , t a ) =
m
2πi(t b − t a )
1/2
exp
im(x b − x a )
2
2(t b − t a )
,
(10.48)
provided we somewhat ruthlessly convolute Gaussians with complex argument and
observe that the width of each individual Gaussian is proportional to
√ ε. Recall from
(9.14) that the sum of the squared widths determines the widths of the convolution,
which is N ε = t b − t a . We refer to Chap. 3.1 in [4] for the details of the calculation.
It remains to be shown that the formulation of quantum mechanics using path
integrals is actually equivalent to the formulation using the Schrödinger equation.
We follow Sect. 4.1 of [4] and write (10.42) for an infinitesimal temporal increment
ε
ψ(x, t + ε) =
1
A
∞
−∞
exp
iε
L
x − y
ε
, x
ψ(y, t)dy
(10.49)
=
1
A
∞
−∞
exp
im(x − y)
2
ε
exp
−
iεV (x)
ψ(y, t)dy .
As discussed above, if the original point y and the new point x are vastly different, the
velocity (x − y)/ε makes the action large and the phase varies wildly. We therefore
assume that y and x only differ by a small amount ξ = y − x. Inserting ξ in the
previous equation and changing the integration variable from y to ξ, we arrive at
ψ(x, t + ε) =
1
A
∞
−∞
exp
imξ
2
ε
exp
−
iεV (x)
ψ(x + ξ, t)dξ .
(10.50)
To proceed, we now utilize that the time step ε is small and expand the previous
expression to first order in ε with the result
10 Quantum Finance and Path Integrals
It is instructive to calculate the propagator ˆ
K for a free particle (potential V (x) =
0), which turns the propagator into
ˆ
K (x b , t b ; x a , t a ) = lim
ε→0
1
A n
. . .
exp
im
2ε
n
k=1
(x k − x k−1 )
2
dx 1 . . . dx n−1 ,
(10.47)
where we see that the integrals are convolutions of Gaussian leading to a new Gaussian. In this way we obtain
ˆ
K (x b , t b ; x a , t a ) =
m
2πi(t b − t a )
1/2
exp
im(x b − x a )
2
2(t b − t a )
,
(10.48)
provided we somewhat ruthlessly convolute Gaussians with complex argument and
observe that the width of each individual Gaussian is proportional to
√ ε. Recall from
(9.14) that the sum of the squared widths determines the widths of the convolution,
which is N ε = t b − t a . We refer to Chap. 3.1 in [4] for the details of the calculation.
It remains to be shown that the formulation of quantum mechanics using path
integrals is actually equivalent to the formulation using the Schrödinger equation.
We follow Sect. 4.1 of [4] and write (10.42) for an infinitesimal temporal increment
ε
ψ(x, t + ε) =
1
A
∞
−∞
exp
iε
L
x − y
ε
, x
ψ(y, t)dy
(10.49)
=
1
A
∞
−∞
exp
im(x − y)
2
ε
exp
−
iεV (x)
ψ(y, t)dy .
As discussed above, if the original point y and the new point x are vastly different, the
velocity (x − y)/ε makes the action large and the phase varies wildly. We therefore
assume that y and x only differ by a small amount ξ = y − x. Inserting ξ in the
previous equation and changing the integration variable from y to ξ, we arrive at
ψ(x, t + ε) =
1
A
∞
−∞
exp
imξ
2
ε
exp
−
iεV (x)
ψ(x + ξ, t)dξ .
(10.50)
To proceed, we now utilize that the time step ε is small and expand the previous
expression to first order in ε with the result