Bootstrapping Simple QM Systems

Abstract

We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians, following the approach of Han, Hartnoll, and Kruthoff. We focus on comparing the bootstrap method data to known analytical predictions for the hydrogen atom and the harmonic oscillator. We resolve many energy levels for each, and more levels are resolved as the size of the matrices used to solve the problem increases. Using the bootstrap approach we find the spectrum of the Coulomb and harmonic potentials converge exponentially fast.

Numerical Bootstrap in Quantum Mechanics

“…Note added: While this work was near completion, the preprint [9] appeared which has some overlap with our work.…”

Bootstrapping the deuteron

“…[28], the sizes of the allowable regions decrease quickly as K increases and shrink to a number of points approximately at large Ks. Similar strategies are also used in many other quantum mechanical bootstrap studies [29][30][31][32][33][34][35]. However, as we mention before, for general potentials such as those encountered in lowenergy nuclear physics, it is not easy to work out useful recursive relations.…”

“…Mathematically, this is done by requiring the so-called bootstrap matrix to be always positive semidefinite, with its matrix elements obtained via some recursive relations (see Section II A for details). Their bootstrap method is soon applied to other quantum mechanical problems, such as harmonic oscillator, hydrogen, double-well potential, and Bloch bands [29][30][31][32][33][34][35]. Besides intelligential novelty, it is possible that these studies may eventually lead to an alternative formulation of quantum mechanics that will be important in the future.…”

Bootstrapping microcanonical ensemble in classical system

“…It may be related to the observation that the quantum mechanical bootstrap (in particular, the recursive technique) requires more creative thinking when we want to study more than one (effective) dynamical degrees of freedom. 4 As a simple demonstration of the above statement, let us consider the double-well potential: H = p 2 − x 2 + gx 4 at g = 1. The recursion relation becomes…”

“…In the globe, the classical limit of the archipelago could have been a peninsula in the ice age, and so might be in the bootstrap. 4 A bootstrap approach to thermodynamic quantities will be found in [13], 5 Our bootstrap analysis is chosen to be the classical limit of [5][7]. When E < 0, the lower boundary of the allowed region in figure 3 seems to be saturated by a "tunneling solution" with an imaginary momentum.…”