Oscillators, 3-dimensional harmonic angular momentum

The two first equations follow from the commutation relations of the angular momentum components, Jf F = X, Y, Z, along space fixed axes. The latter two are not as obvious as they may appear at first glance. Since Jx and Jy do not commute with h we cannot factorize the vector space to treat // and Jz in a separated basis of vectors, (In, />, such as it is usually assumed when discussing the two-dimensional harmonic oscillator. Jy + iJx cannot be used as ladder operators of Jz, and similarly it may be shown that the usual ladder operators85 for H are inapplicable as well, since they do not commute with/2. 

As is well known, the standard 3-dimensional harmonic oscillator is a manifestation of the standard u 3) D so(3) symmetry. It is instructive to see how a g-deformed version of the 3-dimensional harmonic oscillator is related to the u,(3) algebra and its so,(3) subalgebra. The construction of the Hamiltonian of the g-deformed, 3-dimensional harmonic oscillator is a non-trivial problem, because one has first to construct the square of the g-deformed angular momentum operator. 

The bending vibration of a linear ABC molecule may be treated as a two-dimensional isotropic harmonic oscillator. The v,l) basis set is particularly convenient, where v and l are respectively the vibrational and vibrational angular momentum quantum numbers. The allowed values of l are —v, —v + 2,. .. v. Since there are two quantum numbers, two pairs of creation, annihilation operators are needed to generate all basis states from the v = 0, 1=0) zero-point state. These are, following the notation of Cohen-Tannoudji, et al., (1977), a, ay and at,a9, where... 

Q and A are the projections of vectors j and 1 on the symmetry axis (z), respectively. The vector j is the total, 1 is the orbital angular momentum of the particle. N and Uz are quantum numbers of the three-dimensional harmonic oscillator potential of an axially symmetric nucleus, N - Uy- - n. ... 

The space of the 3-dimensional q-deformed harmonic oscillator consists of the completely S3nnmetric irreducible representations of the quantum algebra u (3) [12-14]. In this space a deformed angular momentum algebra, sOg(3), can be defined [10]. The Hamiltonian of the 3D -HO is defined so that it satisfies the following requirements ... 

We now need a systematic way to evaluate matrix elements like — Rgf v y. This is provided by the second quantization formulation [5] of the one-dimensional harmonic oscillator problem, which parallels in some ways the ladder operator treatment of angular momentum. The harmonic oscillator Hamiltonian is... 

The two degrees of freedom associated with the ring puckering are, therefore, an ordinary vibration and a type of one-dimensional rotation in which the phase of the puckering moves around the ring the latter is not, however, a true rotation since there in no angular momentum about the axis of rotation, and so is described as a pseudo-rotation . This separation of the wave equation is not exact, but it has been stated that exact separation is possible and, on the assumptions of harmonic oscillations and small amplitudes of vibrations, leads to the same results as those given. 

Here for the non-vanishing terms the nmnerical value of M must be equal to that of M. (28) predicts that only states which have equal total angular momentum and where the nuclear angular momentum differs by 2, perturb each other. The important fact that the perturbation matrix (28) reduces to irreducible parts with respect to the total angular momentum, allows us to treat the [>erturbation calculations separately for each value of M = 0, 1, 2,... To evaluate (28) numerically we only have to compute the integrals dependent on r and determine the selection rules with respect to N, This becomes a calculation of the matrix elements of for a two-dimensional harmonic oscillator, which can be performed by squaring the matrix for r. As selection riiles for N we obtmn N = N 2, The matrix elements different from zero become... 

In three-dimensional formulation, the condition k1 + k += 0 ov A + BY is satisfied, without loss of generality on setting k = 0, ik = to describe a threefold degenerate state by the magnetic quantum number w/ = 0, 1. Equating all constants to zero, by the mathematical separation of the physically entangled x and y coordinates, not only avoids the use of complex functions, but also destroys the ability to describe the angular momentum of the system. The one-dimensional projection appears as harmonic oscillation, e.g,... 

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