In the case of harmonic quantum oscillators, the continued fractions F define not only fractal sets of natural angular frequencies [[omega].sub.jk], angular accelerations [a.sub.jk] = c x [[omega].sub.jk], oscillation periods [[tau].sub.jk] = 1/[[omega].sub.jk] and wavelengths [[lambda].sub.jk] = c/[[omega].sub.jk] of the chain system, but also fractal sets of energies [E.sub.jk] = [??] x [[omega].sub.jk] and masses [m.sub.jk] = [E.sub.jk]/[c.sup.2] which correspond with the eigenstates
of the system.
For the most part, the lowest eigenstates
in simulations A and B do fall into the categories above.
Similarly, the vectors [[psi].sup.(od).sub.m] (j), j = 1, ..., M, are the eigenstates
of the tridiagonal M x M matrix [H.sup.(od)]:
And if they had dealt with the case of the non-relativistic Schrodinger equation, they could take the eigen states of [[sigma].sub.z] with the eigenvalues [+ or -] 1 as its solution for arbitrary momentum, i.e., setting the eigenstate
[[sigma].sub.z] = 1 as the initial state and the eigenstates
[[sigma].sub.z] = [+ or -] 1 as the two final states could be considered to be valid.
The molecular projected self-consistent Hamiltonian (MPSH) eigenstates
near Fermi level for both devices are shown in Figure 5, which are the perturbed molecular orbitals due to coupling between molecule and the electrodes.
(Here, [X.sub.n] and [x.sub.e] represent the sets of nuclear and electronic coordinates, resp., and the [[psi].sup.e.sub.i] are eigenstates
of the electronic Hamiltonian at fixed [X.sub.n].) The strengths of the present approach include the retention of all of the 3N nuclear degrees of freedom and incorporation of both the excitation due to a laser pulse and the subsequent deexcitation at an avoided crossing near a CI.
de Bievre, "Scarred eigenstates
for quantum cat maps of minimal periods," Communications in Mathematical Physics, vol.
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where they give rise to the eigenstates
of the quantum harmonic oscillator.
Sun, "Symmetries of the eigenstates
in an anisotropic photonic crystal," Phys.
where [[sigma].sup.0.sub.j] are the canonical populations of the diatomic rotor, |jm> are the usual eigenstates
for the unperturbed rotor, [C.sup.jm.sub.j'm'] are the coefficients of the state |jm(t)> in the |j'm'> basis, and <***> is the average over the translational degrees of freedom for both the solvent and the solute.
Among their topics are Isaac Newton and prehistory, Max Planck and the birth of the quantum adventure, Neils Bohr introduces the quantum into atomic physics, Erwin Schrodinger and wave mechanics, eigenstates
in the subnuclear world, and Richard Feynman and the strange theory of light and matter.