Short-Time Expansion of Amplitudes for Time-Dependent Potentials

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. Here we consider imaginary-time amplitudes for the one-dimensional forced harmonic oscillator, with the time-dependent potential V(x,t)=(1/2)(x^2)-x sin t. For a quantum system in a time-dependent potential, the probability for a transition from an initial state |(Subscript[t, a]),a> to a final state |(Subscript[t, b]), b⟩ in time t=(Subscript[t, b])-(Subscript[t, a]) is equal to |A((Subscript[t, a]), a; (Subscript[t, b]), b)((|)^2), where A((Subscript[t, a]), a; (Subscript[t, b]), b)=⟨(Subscript[t, b]),b|U(t)|(Subscript[t, a]),a⟩ is the transition amplitude. The evolution operator has to take into account explicit time-dependence of the potential. In a recently developed effective action approach, the amplitude is expressed in terms of an effective potential, and a set of recursive relations allows systematic analytic derivation of terms in the 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 (t^p) in its expansion. If level p effective action is used, errors in the calculation of transition amplitudes are proportional to (t^(p+1/2)).

THINGS TO TRY

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DETAILS

The effective action approach was originally developed in a series of papers:

[1] A. Bogojevic, A. Balaz, and A. Belic, "Systematically Accelerated Convergence of Path Integrals," Phys. Rev. Lett. 94, 2005 pp. 180403.

[2] A. Bogojevic, A. Balaz, and A. Belic, "Systematic Speedup of Path Integrals of a Generic N-Fold Discretized Theory," Phys. Rev. B 72, 2005 pp. 064302.

[3] A. Bogojevic, A. Balaz, and A. Belic, "Generalization of Euler's Summation Formula to Path Integrals," Phys. Lett. A 344, 2005 pp. 84.

The extension of the approach to many-body systems is presented in the following papers:

[4] A. Bogojevic, I. Vidanovic, A. Balaz, and A. Belic, "Fast Convergence of Path Integrals for Many-Body Systems," Phys. Lett. A 372, 2008 pp. 3341.

[5] A. Balaz, A. Bogojevic, I. Vidanovic, and A. Pelster, "Recursive Schrödinger Equation Approach to Faster Converging Path Integrals," Phys. Rev. E 79, 2009 pp. 036701.

This method has been successfully applied for numerical studies of properties of various quantum systems:

[6] I. Vidanovic, A. Bogojevic, and A. Belic, "Properties of Quantum Systems via Diagonalization of Transition Amplitudes I: Discretization Effects," Phys. Rev. E 80, 2009 pp. 066705.

[7] I. Vidanovic, A. Bogojevic, A. Balaz, and A. Belic, "Properties of Quantum Systems via Diagonalization of Transition Amplitudes II: Systematic Improvements of Short-Time Propagation," Phys. Rev. E 80, 2009 pp. 066706.

Recently, this approach has been extended to systems in time-dependent potentials:

[8] A. Balaz, I. Vidanovic, A. Bogojevic, and A. Pelster, "Fast Converging Path Integrals for Time-Dependent Potentials" (2009).

Mathematica programs developed for symbolic derivation of higher-order effective actions, as well as the C programs developed and used in numerical simulations in the above papers, can be found at http://www.scl.rs/speedup/

All programs are developed at the Scientific Computing Laboratory of the Institute of Physics Belgrade.

Contributed by: Antun Balaz

After work by: Antun Balaz, Ivana Vidanovic, Aleksandar Bogojevic, Aleksandar Belic, and Axel Pelster

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