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Communications in Theoretical Physics

Angular Momentum Phase State Representation for Quantum Pendulum

Fan Hong‐Yi, Ji‐Suo Wang

Published Apr 1, 2005• 0 citations
Publisher

Abstract

To consummate the quantum pendulum theory whose Hamiltonian takes bosonic operator formalism and manifestly exhibits its dynamic behaviour in the entangled state representation, we introduce angular momentum state representation and phase state representation. It turns out that the angular momentum state is the partial wave expansion of the entangled state.

DOI
10.1088/0253-6102/43/4/009

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To consummate the quantum pendulum theory whose Hamiltonian takes bosonic operator formalism and manifestly exhibits its dynamic behaviour in the entangled state representation, we introduce angular momentum state representation and phase state representation. It turns out that the angular momentum state is the partial wave expansion of the entangled state.
By virtue of the Einstein–Podolsky–Rosen entangled state, which is the common eigenvector of two particles' relative coordinate and total momentum, we establish the bosonic operator version of the Hamiltonian for a quantum point-mass pendulum. The Hamiltonian displays the correct Schrodinger equation in the entangled state representation. The corresponding Heisenberg operator equations which predict the angular momentum–angle uncertainty relation are derived. The quantum operator description of two quantum pendulums coupled by a spring is also derived.
By virtue of the Einstein-Podolsky-Rosen entangled state, which is the common eigenvector of two panicles' relative coordinate and total momentum, we establish the bosonic operator version of the Hamiltonian for a quantum point-mass pendulum. The Hamiltonian displays the correct Schroedlnger equation in the entangled state representation.The corresponding Heisenberg operator equations which predict the angular momentum-angle uncertainty relation are derived. The quantum operator description of two quantum pendulums coupled by a spring is also derived.
We study how can an angular momentum coherent state ∣ τ 〉 keeps its form-invariant during time evolution governed by the Hamiltonian H = f ( t ) J + + f * ( t ) J − + g ( t ) J z . We discuss this topic in the context of boson realization of ∣ τ 〉 . By employing the entangled state representation ∣ ζ 〉 and deriving a new binomial theorem involving two-subscript Hermite polynomials, we derive the wave function 〈 ζ ∣ τ 〉 , which turns out to be a single-subscript Hermite polynomial. Based on this result the maintenance of angular momentum coherent state during time evolution is examined, and the value of τ ( t ) is totally determined by the parameters involved in the Hamiltonian.