Coherent states for Hamiltonians quadratic in general form

Author(s)

Alberto Silva Pereira

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2016-04-25

Examining board members

Dmitri Maximovitch Guitman; Evaldo Mendonça Fleury Curado; Rodrigo Fresneda; Renato Higa; Adilson Jose da Silva

• Advisor: Dmitri Maximovitch Guitman

Abstract

In this thesis we obtain quantum states that satisfy the Schrödinger equation for quadratic Hamiltonians in the general form and at the same time allow, naturally, to obtain the correspondence with the classical description. For this, we use the method of integrals of motion to construct creation and annihilation operators, which satisfy the algebra of Weyl-Heisenberg. Thus, we obtain the generalized number states (GNS) in the same way that is done for the Fock states. We obtain diferent families of coherent states (CS) that we call generalized CS (GCS), by a superposition of GNS. These states are labeled by a complex constant z which is written in terms of the initial expected values of the coordinate and momentum. We write the GCS in terms of the initial standard deviation of the coordinate, $\\sigma_q$, which provides the minimization of Heisenberg uncertainty relation at the initial instant time. In particular, we obtain for the first time the GCS for the free particle and discuss in detail their properties, such as the completeness relation, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We show that the expected values of coordinated and momentum propagate along the classical trajectory in phas espace. When the Compton wavelength is much smaller than $\\sigma_q$, the CS can be considered a semiclassical state. In addition to the free particle, we obtain for the first time the GCS for the inverted oscillator and discuss in detail their properties. We show that the GCS of diferent systems can be related by imposing conditions on the parameters of the Hamiltonian. Finally, we consider the time-dependent Hamiltonian, especially to obtain the GCS for a harmonic oscillator whose frequency varies explicitly in time. We also show useful models to obtain exact solution for time-dependent systems, by analogy with the spin equation or one-dimensionaltime-independent Schrödinger equation, as well as a method which consists first to find the solution and then determine the shape of the frequency. (AU)

• FAPESP's process: 13/02966-0 - Quantum motion of charge in electromagnetic fields in probability representation and corresponding coherent states
• Grantee: Alberto Silva Pereira
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)