The bootstrap method for quantum mechanics is a powerful tool for computing the energy eigenvalues of a Hamiltonian. However, we point out that this method suffers from an ambiguity problem: it fails to yield the correct spectrum when the potential contains different types of functions, such as polynomial and exponential terms. Similarly, the bootstrap method may break down when evaluating the expectation values of operators of different types. This issue can arise in a wide range of systems, including statistical models and matrix models. We propose three possible resolutions to this problem.
The bootstrap method for quantum mechanics is a powerful tool for computing the energy eigenvalues of a Hamiltonian. However, we point out that this method suffers from an ambiguity problem: it fails to yield the correct spectrum when the potential contains different types of functions, such as polynomial and exponential terms. Similarly, the bootstrap method may break down when evaluating the expectation values of operators of different types. This issue can arise in a wide range of systems, including statistical models and matrix models. We propose three possible resolutions to this problem.
Abstract The range of motion of a particle with certain energy E confined in a potential is determined from the energy conservation law in classical mechanics. The counterpart of this question in quantum mechanics can be regarded as what is the possible range of expectation values of the position operator 〈x〉 of a particle that satisfies E = 〈H〉. This range depends on the state of the particle, but the universal upper and lower bounds, which are independent of the state, must exist. In this study, we show that these bounds can be derived by using the bootstrap method. We also point out that the bootstrap method can be regarded as a generalization of the uncertainty relations, meaning that the bounds are determined by the uncertainty relations in a broad sense. Furthermore, the bounds on possible expectation values of various quantities other than position can be determined in the same way. However, in the case of multiple identical particles (bosons and fermions), we find some difficulty in the bootstrap method. Because of this issue, the predictive power of the bootstrap method in multi-particle systems is limited in the derivation of observables including energy eigenstates. In addition, we argue an application of the bootstrap method to thermal equilibrium states. We find serious issues that temperature and entropy cannot be handled. Although we have these issues, we can derive some quantities in micro-canonical ensembles of integrable systems governed by generalized Gibbs ensembles.
Recently, a novel numerical computation on quantum mechanics by using a bootstrap method was proposed by Han, Hartnoll, and Kruthoff. We consider whether this method works in systems with a $\ensuremath{\theta}$-term, where the standard Monte-Carlo computation may fail due to the sign problem. As a starting point, we study quantum mechanics of a charged particle on a circle in which a constant gauge potential is a counterpart of a $\ensuremath{\theta}$-term. We find that it is hard to determine physical quantities as functions of $\ensuremath{\theta}$ such as $E(\ensuremath{\theta})$, except at $\ensuremath{\theta}=0$ and $\ensuremath{\pi}$. On the other hand, the correlations among observables for energy eigenstates are correctly reproduced for any $\ensuremath{\theta}$. Our results suggest that the bootstrap method may work not perfectly but sufficiently well, even if a $\ensuremath{\theta}$-term exists in the system.
We present a semidefinite programming algorithm to find eigenvalues of Schrödinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasibility problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in one dimension.
Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.
We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the canonical energy-momentum tensor and the energy functional defined as the expectation value of the corresponding nonlinear Hamiltonian operator. Some examples of such ambiguity are presented for a particularly simple model and some known modifications. However, we point out that there exist a class of nonlinear modifications of the Schrödinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schrödinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schrödinger equation or its underlying Lagrangian density which is assumed to be ``bilinear'' in the wave function in some rather general sense. Yet, it is only for a particular form of this density that the ambiguity in question does not arise. A salient feature of this form is the presence of phase functionals. The present paper thus introduces a new class of modifications characterized by this desirable and rare property.
This paper explores the quantum bootstrap method, a promising numerical approach rooted in self-consistency conditions, and its application in microcanonical ensembles. Initially developed in the 1960s and ’70s, the bootstrap technique has found applications in various fields, including quantum mechanics, lattice theory, conformal field theory, and matrix models. While extensively studied in quantum mechanics, this paper introduces a classical correspondence to the quantum bootstrap method, expanding its applicability to microcanonical ensembles. The study incorporates a novel approach, demonstrating stronger constraints by integrating additional information in phase space. Numerical examples, including the double well potential, harmonic oscillator, Coulomb potential, and non-relativistic Toda model, illustrate the method’s effectiveness in resolving unphysical energy regions and improving computational precision. These results highlight its potential as a powerful tool in computational physics and quantum computing.