Semiclassical calculation of the Pendulum Period

Abstract. In this paper, we calculate the swing period of the classical pendulum via semiclassical path integration. We point out the significance of the classical periodic orbits and the equivalence of the pendulum’s classical isochronism to the equidistance of the quantum energy levels. We derive the swing period in terms of the semiclassical tunneling time and the fractional revival period. A possible definition of a critical value for the quantum “bounce time” is proposed. This paper intends for graduate students as an illustrating example of applying quantum mechanics to a classical system. It offers valuable insight into some characteristics that the classical and quantum pendulums possess in common. It also intends for a specialist in quantum chemistry where the quantum pendulum dynamics appears in what is known as hindered rotation about some chemical bonds.