Spectrum of Quantized Energy for a Lengthening Pendulum
Abstract
We considered a quantum system of simple pendulum whose length of string is increasing at a steady rate. Since the string length is represented as a time function, this system is described by a timedependent Hamiltonian. The invariant operator method is very useful in solving the quantum solutions of timedependent Hamiltonian systems like this. The invariant operator of the system is represented in terms of the lowering operator ǎ(t) and the raising operator ǎ^{†}(t). The Schrödinger solutions ψ_{n}(θ, t) whose spectrum is discrete are obtained by means of the invariant operator. The expectation value of the Hamiltonian in the ψ_{n}(θ, t) state is the same as the quantum energy. At first, we considered only θ^{2} term in the Hamiltonian in order to evaluate the quantized energy. The numerical study for quantum energy correction is also made by considering the angle variable not only up to θ^{4} term but also up to θ^{6} term in the Hamiltonian, using the perturbation theory.
 Publication:

Icnaam 2010: International Conference of Numerical Analysis and Applied Mathematics 2010
 Pub Date:
 September 2010
 DOI:
 10.1063/1.3498546
 Bibcode:
 2010AIPC.1281..594R
 Keywords:

 harmonic oscillators;
 mathematical operators;
 functional analysis;
 algebra;
 03.65.Ge;
 02.30.Tb;
 02.30.Sa;
 02.20.Sv;
 Solutions of wave equations: bound states;
 Operator theory;
 Functional analysis;
 Lie algebras of Lie groups