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Fortschritte der Physik

Classical Propagator for Quadratic Quantum Systems. Example of a Trapped Ion

Olga V. Man’ko

Published May 1, 2000• 1 citation
Publisher

Abstract

The “classical propagator” for quadratic quantum systems is presented and its relation to the quantum propagator is shown. New formalism of quantum mechanics, based on the probability representation of the quantum state, is applied to the problem of an ion in a Paul trap, for which the classical propagator is written in the explicit form.

DOI
10.1002/(sici)1521-3978(200005)48:5/7<643::aid-prop643>3.0.co;2-k

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The “classical propagator” for quadratic quantum systems is presented and its relation to the quantum propagator is shown. New formalism of quantum mechanics, based on the probability representation of the quantum state, is applied to the problem of an ion in a Paul trap, for which the classical propagator is written in the explicit form.
A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.
In this work we present the simplest generic form of the propagator for the time-dependent quadratic Hamiltonian. We manifest the simplicity of our method by giving explicitly the propagators for a free particle in time-dependent electric field and the Paul trap. Exact transition amplitudes and uncertainties are calculated analytically for the Paul trap and harmonic oscillator (HO). The results show that near the instability regions very large quantum mechanical uncertainties are obtained as demonstrated in a special figure. The method is also applied to calculating the trajectory of a classical forced time-dependent HO.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
Review of probability representation of quantum mechanics where the system states are identified with fair probability distributions is presented. Examples of qubits and qutrits are given in detail. The explicit expressions of the qubit density matrices and qubit state vectors in terms of probabilities of dichotomic random variables are discussed. The wave function of a qubit state, and its phase which is not contained in the density matrix of the state, are expressed in terms of the dichotomic probabilities. The generalisation of the probability representation to the case of qudit state is considered. The Schrödinger–like equation for spectra of Hermitian and non Hermitian Hamiltonians is written in the new form of equation for the dichotomic probabilities.