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A flow equation approach to semi-classical approximations : a comparison with the WKB method

Jacobus Daniel Thom

Published Dec 1, 2006• 0 citations

Abstract

The aim of this thesis is the semi-classical implementation of Wegner's flow equations and comparison with the well-established Wentzel-Kramers-Brillouin method.We do this by converting operators, in particular the Hamiltonian, into scalar functions, while an isomorphism with the operator product is maintained by the introduction of the Moyal product.A flow equation in terms of these scalar functions is set up and then approximated by expanding it to first order in .We apply this method to two potentials, namely the quartic anharmonic oscillator and the symmetric double-well potential.Results obtained via the flow equations are then compared with those obtained from the WKB method.OPSOMMING Die doel van hierdie tesis is die semi-klassieke implementering van Wegner se vloeivergelykings en vergelyking met die welbekende Wentzel-Kramers-Brillouin metode.Ons doen dit deur operatore, en spesifiek die Hamilton operator, met skalare funksies te vervang, terwyl 'n isomorfisme met die operator produk behou word deur die invoering van die Moyal produk.'n Vloei-vergelyking in terme van hierdie skalare funksies word opgestel en dan benader deur dit uit te brei tot eerste order in .Ons pas hierdie metode toe op twee potensiale, naamlik die kwarties nie-harmoniese ossillator en die simmetriese dubbelput potensiaal.Resultate wat ons verkry vanaf die vloei-vergelykings word dan vergelyk met dié verkry vanaf die WKB metode.

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The aim of this thesis is the semi-classical implementation of Wegner's flow equations and comparison with the well-established Wentzel-Kramers-Brillouin method.We do this by converting operators, in particular the Hamiltonian, into scalar functions, while an isomorphism with the operator product is maintained by the introduction of the Moyal product.A flow equation in terms of these scalar functions is set up and then approximated by expanding it to first order in .We apply this method to two potentials, namely the quartic anharmonic oscillator and the symmetric double-well potential.Results obtained via the flow equations are then compared with those obtained from the WKB method.OPSOMMING Die doel van hierdie tesis is die semi-klassieke implementering van Wegner se vloeivergelykings en vergelyking met die welbekende Wentzel-Kramers-Brillouin metode.Ons doen dit deur operatore, en spesifiek die Hamilton operator, met skalare funksies te vervang, terwyl 'n isomorfisme met die operator produk behou word deur die invoering van die Moyal produk.'n Vloei-vergelyking in terme van hierdie skalare funksies word opgestel en dan benader deur dit uit te brei tot eerste order in .Ons pas hierdie metode toe op twee potensiale, naamlik die kwarties nie-harmoniese ossillator en die simmetriese dubbelput potensiaal.Resultate wat ons verkry vanaf die vloei-vergelykings word dan vergelyk met dié verkry vanaf die WKB metode.

In this chapter we study another very important and powerful approximation method, the so-called eikonal approximation or Wentzel-Kramer-Brillouin (WKB) method, which is used in studying quantum mechanical systems that are subjected to complicated potentials. This method can be used to find approximate solutions to differential equations where the highest derivative is multiplied by a small parameter. The WKB method gives approximate solution to the Schrödinger equation irrespective of its complicated potential and is mostly used to study one-dimensional systems and also the radial equations for higher-dimensional systems with rotational symmetry.

The mutual consistency and structural interconnection between the Wentzel-Kramers-Brillouin (WKB) and Wigner-Kirkwood (WK) semiclassical approximations is investigated for nonrelativistic N-particle systems, with mutual scalar interactions and coupling to an external time-varying electromagnetic field. The generalized WK expansion of the propagator 〈x\ensuremath{\Vert}U(t,s)\ensuremath{\Vert}y〉 is obtained from a large-mass expansion of the higher-order WKB approximation. Two techniques are described for computing the WK coefficient functions. One relies on a large-mass expansion of classical paths and the transport representation of the WKB approximation; the other is recursive in nature. For time-independent Hamiltonians H the standard WK expansion of the heat kernel 〈x\ensuremath{\Vert}${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$\ensuremath{\Vert}y〉 is recovered.

Abstract This paper introduces a widely used approximation method, the Wentzel-Kramers-Brillouin (WKB) method, for solving the Schrodinger equation, sets up a theoretical framework to derive the general solutions of the method, and carries out the application of the approximation method in specific cases to verify the validity and practicality of the method. Throughout the study, numerical methods, such as the Simpsons method, and programming tools, such as Python, are used to process sophisticated calculations. The final approximated solutions for the Schrodinger equation are able to demonstrate the state of a particle in an isolated quantum system. All the presented studies are based upon the knowledge and skillsets of a high school student.

The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.

We show how Wegner's flow equations can be reformulated as ordinary differential equations through the use of the Moyal bracket. In finite-dimensional Hilbert spaces the introduction of the Moyal bracket leads naturally to the identification of a small expansion parameter, namely the inverse of the dimensionality of the space. This expansion corresponds to a non-perturbative treatment of the coupling constant. In the case of infinite-dimensional spaces ℏ plays the role of the small parameter and the Moyal formulation then allows for a semi-classical treatment of the flow equation. We demonstrate these statements for the Lipkin and Dicke models as well as for the symmetric x 4 and double-well potentials.