The auxiliary field method in quantum mechanics

Abstract

The auxiliary field method is a new technique to obtain closed formulae for the solutions of eigenequations in quantum mechanics. The idea is to replace a Hamiltonian $H$ for which analytical solutions are not known by another one $\tilde{H}$, including one or more auxiliary fields. For instance, a potential $V(r)$ not solvable is replaced by another one $P(r)$ more familiar, or a semirelativistic kinetic part is replaced by an equivalent nonrelativistic one. The approximation comes from the replacement of the auxiliary fields by pure real constants. The approximant solutions for $H$, eigenvalues and eigenfunctions, are then obtained by the solutions of $\tilde{H}$ in which the auxiliary parameters are eliminated by an extremization procedure for the eigenenergies. If $H=T(\mathbf{p})+V(r)$ and if $P(r)$ is a power law, the approximate eigenvalues can be written $T(p_0)+V(r_0)$, where the mean impulsion $p_0$ is a function of the mean distance $r_0$ and where $r_0$ is determined by an equation which is linked to the generalized virial theorem. The general properties of the method are studied and the connections with the envelope theory presented. This method is first applied to nonrelativistic and semirelativistic two-body systems, with a great variety of potentials. Closed formulae are produced for energies, eigenstates, various observables and critical constants, with sometimes a very good accuracy. The method is then used to solve nonrelativistic and semirelativistic many-body systems with one-body and two-body interactions. For such cases, analytical solutions can only be obtained for systems of identical particles, but several systems of interest for atomic and hadronic physics are studied. General results concerning the many-body critical constants are presented, as well as duality relations existing between approximate and exact eigenvalues.

Improved sect. 5.6 and new sect. 5.7

References in corpus (23)

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• Duality relations in the auxiliary field method
• Mass formula for strange baryons in large $N_c$ QCD versus quark model
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• Upper limit on the critical strength of central potentials in relativistic quantum mechanics
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• Bounds for Hamiltonians with arbitrary kinetic parts

Cited by in corpus (11)

• Numerical tests of the envelope theory for few-boson systems
• Baryon resonances in large $N_c$ QCD
• Improvement of the envelope theory with the dominantly orbital state method
• Approximate solutions for N-body Hamiltonians with identical particles in D dimensions
• Bound cyclic systems with the envelope theory
• Two- and three-body calculations within the dominantly orbital state method
• Improvement of the Envelope Theory for Systems with Different Particles
• The envelope theory as a pedagogical tool
• Compact equations for the envelope theory
• Tests of the envelope theory for three-body forces
• Accuracy Tests of the Envelope Theory