ShortTime Expansion of Quantum Amplitudes
ShortTime Expansion of Quantum Amplitudes
In this Demonstration we show that shorttime quantummechanical transition amplitudes can be very accurately calculated if their expansion in the time of propagation is known to high orders. We consider imaginarytime amplitudes of the onedimensional harmonic oscillator , anharmonic oscillator , and doublewell potential . For a general quantum system described by the Hamiltonian , the probability for a transition from an initial state to a final state in time is equal to , where 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 , corresponding to the maximal order in its expansion. If level effective action is used, errors in the calculation of the transition amplitudes are proportional to .
V(x)=
1
2
2
x
V(x)=+
1
2
2
x
g
24
4
x
V(x)=+
1
2
2
x
g
24
4
x
H
b〉
b〉
t
A(ab;t)
2

A(ab;t)=〈ba〉
tH/ℏ
e
p
p
t
p
p+1/2
t