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arXiv:quant-ph/0202067 · Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

Ultraviolet analysis of one dimensional quantum systems

Marco Frasca

Published Feb 12, 2002• 0 citations
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Abstract

Starting from the study of one-dimensional potentials in quantum mechanics having a small distance behavior described by a harmonic oscillator, we extend this way of analysis to models where such a behavior is not generally expected. In order to obtain significant results we approach the problem by a renormalization group method that can give a fixed point Hamiltonian that has the shape of a harmonic oscillator. In this way, good approximations are obtained for the ground state both for the eigenfunction and the eigenvalue for problems like the quartic oscillator, the one-dimensional Coulomb potential having a not normalizable ground state solution and for the one-dimensional Kramers-Henneberger potential. We keep a coupling constant in the potential and take it running with a generic cut-off that goes to infinity. The solution of the Callan-Symanzik equation for the coupling constant generates the harmonic oscillator Hamiltonian describing the behavior of the model at very small distances (ultraviolet behavior). This approach, although algorithmic in its very nature, does not appear to have a simple extension to obtain excited state behavior. Rather, it appears as a straightforward non-perturbative method.

DOI
10.48550/arxiv.quant-ph/0202067

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Starting from the study of one-dimensional potentials in quantum mechanics having a small distance behavior described by a harmonic oscillator, we extend this way of analysis to models where such a behavior is not generally expected. In order to obtain significant results we approach the problem by a renormalization group method that can give a fixed point Hamiltonian that has the shape of a harmonic oscillator. In this way, good approximations are obtained for the ground state both for the eigenfunction and the eigenvalue for problems like the quartic oscillator, the one-dimensional Coulomb potential having a not normalizable ground state solution and for the one-dimensional Kramers-Henneberger potential. We keep a coupling constant in the potential and take it running with a generic cut-off that goes to infinity. The solution of the Callan-Symanzik equation for the coupling constant generates the harmonic oscillator Hamiltonian describing the behavior of the model at very small distances (ultraviolet behavior). This approach, although algorithmic in its very nature, does not appear to have a simple extension to obtain excited state behavior. Rather, it appears as a straightforward non-perturbative method.
With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H ( x , p ) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, E i , turns out to be zero for all potentials V ( x ) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.
A renormalization-group scheme is developed for the three-dimensional O(2N)O(2N)O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a 1/N1/N1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at z=0,z=0,z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at z=0.z=0.z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of lny.\mathrm{ln}y.lny.
We have derived some excitation states of the one-dimensional Gross-Pitaevskii equation coupled by the gravitational potential. The methods that we have used here are taken by pursuing the recent work of Kivshar et. al. by considering the equation as a macroscopic quantum oscillator. To obtain the states, we have made the appropriate transformation to reduce the three-dimensional Gross-Pitaevskii equation into the one-dimensional Gross-Pitaevskii equation and applying the time-independent perturbation theory in the general solution of the one-dimensional Gross-Pitaevskii equation as a linear superposition of the normalized eigenfunctions of the Schrödinger equation for the harmonic oscillator potential. Moreover, we also impose the condition by assuming that some terms in the equation should be so small in order to preserve the use of the perturbation method.
We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.