Isotropic harmonic oscillator

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrddinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian assuming the fonn [95]... 

Derive the result that the degeneracy of the energy level E for an isotropic three-dimensional harmonic oscillator is (n + l)(n + 2)/2. 

The energy of the isotropic harmonic oscillator in three dimensions can be written as... 

Explicit forms of the coefficients 7j and A depend on the coordinate system employed, the level of approximation applied, and so on. They can be chosen, for example, such that a part of the coupling with other degrees of freedom (typically stretching vibrations) is accounted for. In the space-fixed coordinate system at the infinitesimal bending vibrations, Ty, + T f. reduces to the kinetic energy operator of a two-dimensional (2D) isotropic harmonic oscillator,... 

Figure 2.6 The potential V(r) that corresponds to the dynamical symmetry (I). The potential is nonrigid because [cf. Eq. (2.113)] the rotational spacings are comparable to the vibrational ones. Tn the harmonic limit V(r) is the potential of an isotropic harmonic oscillator.

What are the electric-dipole selection rules for a three-dimensional harmonic oscillator exposed to isotropic radiation ... 

Solve the Schrodinger equation for the two-dimensional isotropic harmonic oscillator using plane polar coordinates. First show that... 

We have considered only one particular degenerate vibrational level. The general case is dealt with by solving the isotropic two-dimensional harmonic oscillator in plane polar coordinates (Problem 6.19). The result is... 

The first part of the Hamiltonian (16), Hc.o.m, describes the center-of-mass contribution, as in the quasi-one-dimensional cases, and contributes the eigenenergy of a two-dimensional isotropic harmonic oscillator to the total energy. The second part of the Hamiltonian, Hint, depends on the antisymmetric coordinates xa and ya, and represents the contribution to the total energy due to the internal degrees of freedom. 

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

In this section we shall apply the realizations of so(2, 1) to physical systems, such as the nonrelativistic Coulomb problem, the three-dimensional isotropic harmonic oscillator, Schrodinger s relativistic equation (Klein-Gordon... 

The practical form for the partition function, gpk, [needed in Eq. (15)] for any isomer of a given species is computed as a product of the translational partition function for the center of mass times the partition function of (3p—3) independent isotropic harmonic oscillators (we regard all internal degrees of freedoms as vibrational modes), each truncated at the energy e e is a positive number) ... 

Let us take two spherically symmetrical systems, each with a polaris-ability a, say two three-dimensional isotropic harmonic oscillators with no permanent moment in their rest position. If the charges e of these oscillators are artificially displaced from their rest positions by the displacements... 

Consider now one of these variable and its contribution to the potential energy, z(r) = 27rg 2(7Xz(r)2. This is the potential energy of a three-dimensional isotropic harmonic oscillator. The total potential energy, Eq. (16.82) is essentially a sum over such contributions. This additive form indicates that these oscillators are independent of each other. Furthermore, all oscillators are characterized by the same force constant. We now also assume that all masses associated with these oscillators are the same, namely we postulate the existence of a single frequency Ms., sometimes referred to as the Einstein frequency of the solvent polarization fluctuations, and Ws are related as usual by the force constant... 

It is strange that the analogous regular effects for other types of boundary conditions for the hydrogen atom have not been attempted to date. Similar trends for the confined isotropic harmonic oscillator will be discussed in Section 8. 

There is a general tendency for degenerate states of the free problem confinement results in a greater change in states with smaller l than in states with large l. For example, one can demonstrate a similar shift pattern for the isotropic harmonic oscillator which will be taken up in Section 7. 

In this section we shall state some exact results for the spherically confined isotropic harmonic oscillator inside impenetrable walls. The eigenspectral regularities and the characterization of energy states in terms of the electron density and its derivatives at the equilibrium position will be considered. 

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