# Anharmonic Oscillator Spectrum via Diagonalization of Amplitudes

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The energy spectrum of a quantum system can be accurately calculated by the numerical diagonalization of the space-discretized matrix of its evolution operator, that is, the matrix of its transition amplitudes. Here we calculate the spectrum of a one-dimensional anharmonic oscillator with the potential , using level effective action. 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 calculated as , with the transition amplitude . In a recently developed effective action approach, the amplitude is expressed in terms of the effective potential. Then a set of recursive relations allows systematic analytic derivation of terms in the expansion of the effective potential in the time . The effective action thus obtained is characterized by a chosen level corresponding to the maximal order occurring in its expansion.

Contributed by: Antun Balaz (March 2011)

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

Open content licensed under CC BY-NC-SA

## Snapshots

## 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(18), 2005.

[2] A. Bogojevic, A. Balaz, and A. Belic, "Systematic Speedup of Path Integrals of a Generic N-Fold Discretized Theory," *Phys. Rev. B*, 72(6), 2005.

[3] A. Bogojevic, A. Balaz, and A. Belic, "Generalization of Euler's Summation Formula to Path Integrals," *Phys. Lett. A*, 344(2–4), 2005 pp. 84–90.

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(19), 2008 pp. 3341–3349.

[5] A. Balaz, A. Bogojevic, I. Vidanovic, and A. Pelster, "Recursive Schrödinger Equation Approach to Faster Converging Path Integrals," *Phys. Rev. E*, 79(3), 2009.

This method has been successfully applied to 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(6), 2009.

[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(6), 2009.

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," arXiv, 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 were developed at the Scientific Computing Laboratory of the Institute of Physics — Belgrade.