Abstract
In this paper we introduce a new method for constructing coherent states for 2D harmonic oscillators. In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2D harmonic oscillator. We define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the S U ( 2 ) coherent states, where these are then used as the basis of expansion for Schrödinger-type coherent states of the 2D oscillators. We discuss the uncertainty relations for the new states and study the behaviour of their probability density functions in configuration space.
Highlights
Degeneracy in the spectrum of the Hamiltonian is one of the first problems we encounter when trying to define a new type of coherent state for the 2D oscillator
We aim to develop an approach for constructing coherent states for 2D oscillators in both isotropic and commensurate anisotropic settings
In order to generalise coherent states to the commensurate anisotropic oscillator, we introduce two integers, p, q, in the Hamiltonian as where the frequencies are related by ωx pω and ωy qω, and the ratio, p q
Summary
Degeneracy in the spectrum of the Hamiltonian is one of the first problems we encounter when trying to define a new type of coherent state for the 2D oscillator. These definitions were formalised by Glauber and Sudarshan [8,9], but these minimal uncertainty wave-packets were first studied by Schrödinger [10], and so we will refer to them as Schrödinger-type coherent states throughout These properties can be used to show that the states |z form an over-complete basis, and they resolve the identity in the following way: d2z |z z| = ∑ |n n| = IH. We will construct two new ladder operators as linear combinations of the operators in (7) and proceed to define a single indexed Fock state for the 2D system which yields the SU(2) coherent states, as well as extend the definitions in Section 2 to obtain Schrödinger-type coherent states for the 2D system. The results are essentially the same as those in (9) and (10), but they are tuned by the continuous parameters α, β introduced in (12)
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