Spectrum of Quantized Energy for a Lengthening Pendulum
Jeong Ryeol Choi, Ji Nny Song, Seong Ju Hong +3 more
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 operator ǎ(t) and the raising operator ǎ†(t). The Schrödinger solutions ψn(θ, t) whose spectrum is discrete are obtained by means of the invariant operator. The expectation value of the Hamiltonian in the ψn(θ, t) state is the same as the quantum energy. At first, we considered only θ2 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 θ4 term but also up to θ6 term in the Hamiltonian, using the perturbation theory.
AIP conference proceedings•Jan 1, 2010•0 citations