# Short-Time Expansion of Quantum Amplitudes

Short-Time Expansion of Quantum Amplitudes

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 , anharmonic oscillator , and double-well 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)=x

1

2

2

V(x)=x+x

1

2

2

g

24

4

V(x)=-x+x

1

2

2

g

24

4

H

b〉

b〉

t

|A(ab;t)|

2

A(ab;t)=〈bea〉

-tH/ℏ

p

t

p

p

t

p+1/2