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Chinese Physics B

Time evolution of angular momentum coherent state derived by virtue of entangled state representation and a new binomial theorem*

Ji‐Suo Wang, Xiang‐Guo Meng, Hong-Yi Fan

Published Aug 13, 2019• 8 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/ab3a90

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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.
A new complex function space whose basis is the single-variable Hermite polynomial H2jξ*+τξ2τ is constructed, which is related to both entangled state representation and spin coherent state in Schwinger bosonic realization. New binomial theorem involving two-variable Hermite polynomial is derived, which helps to constitute the new complex function space. We also present a new integration transformation of the basis H2jξ*+τξ2τ with its reciprocal transformation which is useful to deriving some operator identities.
In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.
After recognizing that Majorana first formulated what later became known as the Bloch equation fourteen years before Felix Bloch, we propose the name Majorana–Bloch equation. Building on our previous work for spin-½, we have extended the derivation of time evolution from the Majorana–Bloch equation through the von Neumann equation to systems of arbitrary angular momentum in pure coherent states. By exploiting the polynomial form of the coherent-state projector, we establish a one-to-one correspondence between classical and quantum dynamics on the coherent-state manifold. Taking expectation values in coherent states also recovers the classical equation through the Ehrenfest theorem.
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.