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 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 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.
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 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 consider the quantum tunneling phenomenon in a well-behaved triple-well potential. As required by the semiclassical approximation we take into account the quadratic fluctuations over the instanton which represents as usual the localised finite-action solution of the euclidean equation of motion. The determinants of the quadratic differential operators at issue are evaluated by means of the Gelfang-Yaglom method. In doing so the explicit computation of the conventional ratio of determinants takes as reference the harmonic oscillator whose frequency is the average of the individual frequencies derived from the non-equivalent minima of the potential. Eventually the physical effects of the multi-instanton configurations are included in this approach. As a matter of fact we obtain information about the energies of the ground-state and the two first excited levels of the discrete spectrum at issue.
EnglishThe path integral formulation, developed by Richard Feynman, provides an alternative yet equivalent approach to operator-based quantum mechanics, proving particularly effective for the study of semiclassical and non-perturbative phenomena. The main goal of this bachelor's thesis is to introduce the theoretical foundations of the path integral and apply them to the study of a particle in a one-dimensional double-well potential. This system represents a paradigmatic example of the tunneling phenomenon, for which perturbative expansions constructed separately around the two minima fail to reproduce the exponentially small energy splitting between the ground state and the first excited state. Through analytic continuation to Euclidean time, quantum tunneling is described via specific classical configurations known as instantons, which connect the two potential minima. In this framework, summing multi-instanton contributions within the dilute gas approximation allows one to derive the energy splitting between the two lowest-lying levels as well as the parity of the corresponding states. ItalianoIl formalismo degli integrali di cammino, sviluppato da Richard Feynman, fornisce una formulazione della meccanica quantistica alternativa ma equivalente a quella operatoriale, particolarmente efficace per lo studio dei fenomeni semiclassici e non perturbativi. L'obiettivo principale di questo lavoro di tesi è introdurre i fondamenti teorici dell'integrale di cammino e applicarli allo studio di una particella in una doppia buca di potenziale unidimensionale, sistema paradigmatico del fenomeno di tunneling per il quale gli sviluppi perturbativi costruiti separatamente attorno ai due minimi non sono in grado di riprodurre la separazione esponenzialmente piccola tra il livello energetico dello stato fondamentale e quello del primo stato eccitato. Attraverso la continuazione analitica al tempo euclideo, l'effetto tunnel viene descritto mediante particolari configurazioni classiche note come istantoni, che collegano i due minimi del potenziale. In questo modo, sommando i contributi multi-istantonici in regime di gas diluito, si ottiene infine lo splitting energetico dei due livelli più bassi e la parità dei relativi stati.
Abstract The one-sided bouncer (1SB) and the symmetric bouncer (SB) involve a one-dimensional particle in a piecewise linear potential. For such problems, the time-dependent quantum mechanical propagator cannot be found in closed form. The semiclassical Feynman path integral (PI) is a very appealing approach, as it approximates the propagator by a closed-form expression (a sum over a finite number of classical paths). In this paper we solve the classical path enumeration problem. We obtain closed-form expressions for the initial velocity, bounce times, focal times, action, van Vleck determinant, and Morse index for each classical path. We calculate the propagator within the semiclassical approximation. The numerical results agree with eigenfunction expansion results away from caustics. We derive mappings between the 1SB and SB, which explains why each bounce of the 1SB increases the Morse index by 2 and results in a phase change of π . We interpret the semiclassical Feynman PI to obtain visualizations of matter wave propagation based on interference between classical paths, in analogy with the traditional visualization of light wave propagation as interference between classical ray paths.