Three-dimensional harmonic

Einstein9 was the first to propose a theory for describing the heat capacity curve. He assumed that the atoms in the crystal were three-dimensional harmonic oscillators. That is, the motion of the atom at the lattice site could be resolved into harmonic oscillations, with the atom vibrating with a frequency in each of the three perpendicular directions. If this is so, then the energy in each direction is given by the harmonic oscillator term in Table 10.4... 

The Schrodinger equation for this three-dimensional harmonic oscillator is... 

The energy levels for the three-dimensional harmonic oscillator are, then, given by the sum (equation (4.53))... 

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

Tacoma Narrows bridge % tangent 16 Taylor s series 32-34 tests of series convergence 35-36 thermodynamics applications 56-57, 81 first law 38-39 Jacobian notation 160-161 systems of constant composition 38 three-dimensional harmonic oscillator 125-128... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

Fig. 3 Bneigy levels of the three-dimensional harmonic oscillator. The degree of degeneracy of each level is shown in parenthesis.

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

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

For the three-dimensional harmonic oscillator, V = kxx2 + h.kyy2+ kzz2. Separation of variables gives the wave function as the product of three one-dimensional harmonic oscillator wave functions, and gives the energy as... 

In variance with the hydrogen-like and Slater functions the potential employed to formally construct the gaussian basis states has nothing to do with the real potential acting upon an electron in an atom. On the other hand the solutions of this (actually three-dimensional harmonic oscillator problem) form a complete discrete basis in the space of orbitals in contrast to the hydrogen-like orbitals. 

Important examples of chemical interest include particles that move in the central held on a circular orbit (V constant) particles in a hollow sphere V = 0) spherically oscillating particles (V = kr2), and an electron on a hydrogen atom (V = 1 /47re0r). The circular orbit is used to model molecular rotation, the hollow sphere to study electrons in an atomic valence state and the three-dimensional harmonic oscillator in the analysis of vibrational spectra. Constant potential in a non-central held dehnes the motion of a free particle in a rectangular potential box, used to simulate electronic motion in solids. 

The superfreeon orbitals are taken to be the three-dimensional harmonic oscillator orbitals which lie the following energy sequence... 

The potential energy V of a three-dimensional harmonic oscillator depends on the position of a particle in space, as described by the three coordinates x, y and z. It can thus be considered to be a function of these variables written as V(x,y, z) and defined by the equation... 

Relation (2) is introduced in the theory of de Boer and Hamaker energy constant in London s theory in which the two atoms are represented by three-dimensional harmonic oscillators with a characteristic frequency v and a polarizability a. h is Planck s constant. 

The Schrodinger equation for this three-dimensional harmonic oscillator is (()-lp () lp 1. 7 2 2 2 2... 

---