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Instantons in quantum mechanics (QM)

Jean Zinn‐Justin

Published Apr 15, 2021• 0 citations
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Abstract

Abstract Perturbative expansion can be generated by calculating Euclidean functional integrals by the steepest descent method always looking, in the absence of external sources, for saddle points in the form of constant solutions to the classical field equations. However, classical field equations may have non-constant solutions. In Euclidean stable field theories, non-constant solutions have always a larger action than minimal constant solutions, because the gradient term gives an additional positive contribution. The non-constant solutions whose action is finite, are called instanton solutions and are the saddle points relevant for a calculation, by the steepest descent method, of barrier penetration effects. This chapter is devoted to simple examples of non-relativistic quantum mechanics (QM), where instanton calculus is an alternative to the semi-classical Wentzel–Kramers–Brillouin (WKB) method. The role of instantons in some metastable systems in QM is explained. In particular, instantons determine the decay rate of metastable states in the semi-classical limit initially confined in a relative minimum of a potential and decaying through barrier penetration. The contributions of instantons at leading order for the quartic anharmonic oscillator with negative coupling are calculate explicitly. The method is generalized to a large class of analytic potentials, and explicit expressions, at leading order, for one-dimensional systems are obtained.

DOI
10.1093/oso/9780198834625.003.0037

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Abstract Perturbative expansion can be generated by calculating Euclidean functional integrals by the steepest descent method always looking, in the absence of external sources, for saddle points in the form of constant solutions to the classical field equations. However, classical field equations may have non-constant solutions. In Euclidean stable field theories, non-constant solutions have always a larger action than minimal constant solutions, because the gradient term gives an additional positive contribution. The non-constant solutions whose action is finite, are called instanton solutions and are the saddle points relevant for a calculation, by the steepest descent method, of barrier penetration effects. This chapter is devoted to simple examples of non-relativistic quantum mechanics (QM), where instanton calculus is an alternative to the semi-classical Wentzel–Kramers–Brillouin (WKB) method. The role of instantons in some metastable systems in QM is explained. In particular, instantons determine the decay rate of metastable states in the semi-classical limit initially confined in a relative minimum of a potential and decaying through barrier penetration. The contributions of instantons at leading order for the quartic anharmonic oscillator with negative coupling are calculate explicitly. The method is generalized to a large class of analytic potentials, and explicit expressions, at leading order, for one-dimensional systems are obtained.
We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow to construct a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist on an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant. The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property. Accordingly we resort to the zeta-function method to compute the functional determinants in a systematic way. The effect of the multi-instantons configurations is also carefully considered. To illustrate the instanton calculus in a relevant model we go to the double-well potential. The second popular case is the periodic-potential where the initial levels split into bands. The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.
We present a formalism for semiclassical time evolution in quantum mechanics, building on a century of work. We identify complex saddle points in real time, real saddle points in complex time, and complex saddle points in complex time that reproduce the known answers in classic problems. For the decay of a metastable state, we find finite time and finite energy analogs of the "bounce" which do not have strict zero or negative modes. The one-loop phase of the wave function and the multiplicity of bounce solutions at late times are discussed. The motivation of this work is to learn how to compute decay rates in quantum field theory in situations with non-trivial time dependence, by first taking a humble step backwards to the fascinating world of quantum mechanics.
We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle.The configurations which allow construction of a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions.The quantum amplitudes consist of an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant.The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property.Accordingly, we resort to the zeta-function method to compute the functional determinants in a systematic way.The effect of the multi-instantons configurations is also carefully considered.To illustrate the instanton calculus in a relevant model, we go to the double-well potential.The second popular case is the periodic-potential where the initial levels split into bands.The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.
Within the framework of the instanton approach, we present analytical results for the following model problems: (i) particle penetration through a parabolic potential barrier, where the instanton solution practically coincides with the exact (quantum) one; (ii) descriptions of highly excited states in anharmonic potentials of two types: double-well X 4 and decay X 3 . For the former potential, the instanton method accurately reproduces not only single-well and double-well quantization, but also a crossover region (in contrast to the standard WKB approach that fails to describe the crossover behavior); for the latter potential, the instanton method allows studying the resonance broadening and collapse phenomena. We also investigate resonance tunneling that plays a relevant role in many semiconducting devices. We show that the instanton approach gives exact (quantum) results in a broad range of energies. Applications of the method and of the results are applicable to various systems in physics, chemistry, and biology exhibiting double-level behavior and resonance tunneling.
This chapter focuses on Quantum Mechanics and Quantum Field Theory in a euclidean formulation. This means that, in general, it discusses the matrix elements of the quantum statistical operator eβH (the density matrix at thermal equilibrium), where H is the hamiltonian and β is the inverse temperature. The chapter begins by first deriving the path integral representation of matrix elements of the quantum statistical operator for hamiltonians of the simple form p 2/2m + V (q). Comparing classical statistical physics in one space dimension and quantum statistical physics of the particle, it introduces statistical correlation functions and discusses their quantum interpretation. It then explicitly calculates the path integral corresponding to a harmonic oscillator in a time-dependent external force. This result can be used to reduce the evaluation of path integrals in the case of analytic potentials to perturbation theory. The chapter shows on a first example that path integrals are especially well suited to the study of the classical limit, by relating a quantum and classical partition function. The appendix explains some general properties of the two-point function, and use the semi-classical approximation of the partition function to derive Bohr–Sommerfeld's quantization condition.
This chapter addresses a range of problems for which functional integrals provide unique insights and approximations. It shows how to generate a variety of alternative functional integrals for the evolution operator which gives rise to different physical approximations in the stationary-phase approximation. The chapter aims to perform the general saddle point approximation around a static mean field solution and derive an expansion in terms of the fundamental vibrational excitations of the system. It also shows how time-dependent mean field solutions may be used to calculate transition amplitudes between specific states, approximate quantum eigenstates of large amplitude collective motion, study barrier penetration and spontaneous fission. The chapter evaluates the asymptotic behavior of large orders of perturbation theory and explores the freedom available in formulating functional integrals for the evolution operator and how to exploit this freedom in making physical approximations.