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 time-dependent Hamiltonian. The invariant operator method is very useful in solving the quantum solutions of time-dependent Hamiltonian systems like this. The invariant operator of the system is represented in terms of the lowering operatorâ(t) and the raising operatorâ † (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.