Oscillators, 3-dimensional harmonic Hamiltonian

Now consider a system of N one-dimensional harmonic oscillators with the Hamiltonian... 

The quantum mechanical hamiltonian for a one-dimensional harmonic oscillator is given by... 

In Eq. (3.1), //<, is a three- or four- dimensional harmonic oscillator Hamiltonian... 

This is the classical Hamiltonian of the three-dimensional harmonic oscillator. By letting p — 0, one has... 

The 3/V —6 one-dimensional Schrodinger equations (6.50) are easily solved. The one-dimensional harmonic-oscillator Hamiltonian is... 

For a one-dimensional harmonic oscillator, with the potential in Eq. (1.7), the Hamiltonian is... 

The Hamiltonian for the three-dimensional harmonic oscillator can be written in the form... 

The problem considered is the one-dimensional harmonic oscillator perturbed by cubic and quartic potential terms. Thus, the unperturbed Hamiltonian operator is... 

It is clear from its form that this partition function wdll generate a correct canonical distribution for the free one-dimensional particle. The NosAHoover chains have successfully solved the pathology that had existed related to the condition Fj = 0. Let s investigate the application of the NosAHoover chains to a slightly more complex problem a one-dimensional harmonic oscillator with Hamiltonian,... 

The zeroth order Hamiltonian is a sum of one-dimensional harmonic oscillator operators. Eigenvalues and eigenfunctions of H0 are designated according to the equation,... 

Until now, our treatment has been built in exactly the same terms that might have been used in work on normal modes of vibration in the latter part of the nineteenth century. However, it is incumbent upon us to revisit these same ideas within the quantum mechanical setting. The starting point of our analysis is the observation embodied in eqn (5.19), namely, that our harmonic Hamiltonian admits of a decomposition into a series of independent one-dimensional harmonic oscillators. We may build upon this observation by treating each and every such oscillator on the basis of the quantum mechanics discussed in chap. 3. In light of this observation, for example, we may write the total energy of the harmonic solid as... 

The matrix equation (18) is block-diagonal (as shown) only if the coupling modes, a = 2,5e, and 5a, are mutually orthogonal. The two-dimensional harmonic oscillator Hamiltonians for the a = 2 and 5e modes are given by... 

However, we can make a correlation of this crystal lattice with a harmonic oscillator. The quantum theory of a single 1-dimensional oscillator predicts a Hamiltonian function of the form ... 

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 Hamiltonian of the g-deformed, 3-dimensional harmonic oscillator must be defined in such a way as to satisfy the following requirements ... 

The generalization of a force constant, k, and reduced mass, fi, from a one dimensional harmonic oscillator to the normal mode oscillators of a polyatomic molecule is accomplished by the F, G matrix methods of Wilson, et al., (1955). For the present discussion it is sufficient to know that a force constant and a reduced mass may be uniquely defined for each of 3N — 6 linearly independent sets of internal coordinate displacements in a polyatomic molecule (see also Section 9.4.12).] The harmonic oscillator Hamiltonian... 

The construction of a Hamiltonian is normally an easy problem. The solution of the Schrodinger equation, on the contrary, represents a serious problem. It can be solved exactly for several model cases a particle in a box (one-, two- or three-dimensional), harmonic oscillator, rigid rotor, a particle passing through a potential barrier, hydrogen atom, etc. In most applications only an approximate solution of the Schrodinger equation is attainable. 

To summarize, we have seen that it is possible to explain the degeneracy pattern characteristic of a three-dimensional harmonic oscillator by introducing, as a proper symmetry group of the Hamiltonian operator (here referred to as the degeneracy group), a group of unitary transformations in a three-dimensional complex space. 

This expression includes terms up to two-body interactions. The algebraic Hamiltonian (2.27) of the ( -dimensional) harmonic oscillator is, of course, a special case of Eq. (2.36). One then observes that it is possible to arrange the Hamiltonian (2.36), in the framework of a dynamical algebra, by explicitly introducing the bilinear products... 

The operators fk, defined by Eq. (25) and corresponding to the action integrals of the bath modes, are the Hamiltonian operators of one-dimensional harmonic oscillators with unit frequency. Therefore, the wave function representing the eigenstates in Eq. (31b) are given by... 

Clemenger [42] has studied the effect of ellipsoidal deformations in alkali clusters with N less than 100, using a modified three-dimensional harmonic oscillator model. The model considers different oscillator frequencies along the z axis (taken as symmetry axis) and perpendicular to the z axis. The model Hamiltonian used by Clemenger also contains an anharmonic term. Its purpose is to flatten the bottom of the potential well and to make it to resemble a rounded square-well potential. 

Recall that the one-dimensional harmonic oscillator with spring constant 2 has potential energy function U(q) = with Hamiltonian... 

Putting the electronic energy at the origin Q = Qy =0 to be zero, the term Ho reduces to the Hamiltonian of the isotropic two-dimensional harmonic oscillator (with frequency w). 1 denotes the 2x2 unit matrix. 

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