WOLFRAM|DEMONSTRATIONS PROJECT

Short-Time Expansion of Quantum Amplitudes

​
harmonic oscillator
anharmonic oscillator
double-well potential
initial position of the particle a
-0.05
final position of the particle b
-0.05
level of the effective action p
1
compare with another p
1
show exact amplitude
In this Demonstration we show that short-time quantum-mechanical transition amplitudes can be very accurately calculated if their expansion in the time of propagation is known to high orders. We consider imaginary-time amplitudes of the one-dimensional harmonic oscillator
V(x)=
1
2
2
x
, anharmonic oscillator
V(x)=
1
2
2
x
+
g
24
4
x
, and double-well potential
V(x)=-
1
2
2
x
+
g
24
4
x
. For a general quantum system described by the Hamiltonian
H
, the probability for a transition from an initial state
b〉
to a final state
b〉
in time
t
is equal to
A(ab;t)
2
|
, where
A(ab;t)=〈b
-tH/ℏ
e
a〉
is the transition amplitude. In a recently developed effective action approach, the amplitude is expressed in terms of the effective potential and a set of recursive relations allows systematic analytic derivation of terms in an expansion of the effective potential in time t. The effective action thus obtained is characterized by a chosen level
p
, corresponding to the maximal order
p
t
in its expansion. If level
p
effective action is used, errors in the calculation of the transition amplitudes are proportional to
p+1/2
t
.