Harmonic Oscillator Energy Research Articles

On the thermodynamics of quantum-electrodynamic frequency shifts

The spectra of atoms exposed to black-body radiation suffer temperature-dependent frequency shifts predicted earlier and now becoming measurable, whose thermodynamic status is clarified. To a good approximation, they are calculable as differences between the coupling-induced shifts in the Helmholtz free energies of the coupled system atom plus quantised electromagnetic field, in thermal equilibrium subject to the constraint that the atom is in a given state li). Effectively, the state label i plays the same role as do macroscopically controlled variables in ordinary thermodynamics. With kT well below the electron rest-mass energy, such calculations require only the forward scattering amplitude of light from the atom; some previously overlooked but theoretically interesting relativistic features of the exact expressions are pointed out. For a harmonic oscillator, uniquely, the constrained free-energy shift is the same for all levels (and therefore also the same as the shift in unconstrained equilibrium); consequently there is no thermal frequency shift at all. Recent comments on black-body-induced Stark shifts by Ford et al. (1985-6) are based on their calculation of the free energy of a harmonic oscillator in unconstrained equilibrium with the field. It is shown that as regards frequency shifts their comments are incompatible with, and misrepresent, previous correct work on such shifts.

The effects of vibration-rotation interaction on the quadrupole hyperfine structure of molecular rotational levels

By using a contact transformation method similar to that commonly employed when determining higher-order corrections to the harmonic oscillator and rigid rotor energy levels of molecules, analogous centrifugal distortion and anharmonic corrections to the nuclear quadrupole coupling energies have been obtained for molecules containing one quadrupolar nucleus. The J, K dependence and v, l dependence of these higher-order corrections to the quadrupole hyperfine energies can be cast in a form which is remarkably similar to the form taken for ordinary vibrational and rotational energy corrections, a result which was not evident from earlier partial treatments of this general problem. Results are obtained here for asymmetric top, symmetric top, spherical top, and linear molecules.

The dependence of the harmonic oscillator energy level spacing on the mass number of nuclei

Approximate expressions of h ̵ ω as functions of A of nuclei are derived, showing “discontinuities in the slope” of h ̵ ω at the closed shells. A simplified expression is also given. The dependence on A is established through a Fermi distribution, its parameters ϱ 0 and a being determined by fitting either to values of h ̵ ω reproducing the experimental values of charge rms radii of nuclei: 〈r 2〉 exp 1 2 or directly to 〈r 2 ex p 1 2 .

PROBABILITY DISTRIBUTIONS OF ENERGY OF ONE-DIMENSIONAL PARTICLES OF TWO KINDS IN THE ENERGY EIGENSTATES

Starting from Voe Neumann's axiomatic system of quantum mechanics, this paper calculates the probability distributions of the kinetic and potential energy of the one-dimensional harmonic oscillator and the particle in one-dimensional infinitely deep square potential well. The results are compared with the classical cases, and it is shown that their relation conforms to Bohr's correspondence principle.Then we put forward a new viewpoint. it seems that the concept of the total energy of the particle in quantum mechanics should be deliberated again; the two basic hypotheses in quantum mechanics–“Bohr's probabilistic interpretation” and “the postulate that the particle's energy eguals the eigenvalue in the energy eigenstate” ars not logically harmonious. At the same time, the limitation of von Neumann's axiomatic system is pointed out.

The ground states of the nucleide44Ti

Ground state binding energies and equilibrium harmonic oscillator energies are computed for the various ground state configurations of the doubly even nucleide44Ti. Employed for the study is a variational calculation carried out in a Cartesian harmonic oscillator basis of states employing a tripartite interaction potential composed of Wigner central forces coupled to Majorana exchange forces each having Gaussian radial dependence.

Semiempirical formulas for single-particle energies of neutrons and protons

The stepwise multiple linear regression technique has been used to analyze the single-particle energies of neutrons and protons in nuclei along the line of beta stability. Their regular and systematic trends lead to semiempirical model-independent formulas for single-particle energies of neutrons and protons in the bound nuclei as functions of nuclear parameters A and Z for given states specified by nl j . These formulas are almost as convenient as the harmonic oscillator energy formulas to use. The single-particle energies computed from these formulas have been compared with the experimental data and are found in reasonable agreement.

Systematics of Nuclear Single-Particle States

Semiempirical model-independent formulas for single-particle energies of neutrons and protons in bound nuclei have been obtained as functions of nuclear parameters $A$ and $Z$ for given states specified by $\mathrm{nlj}$. These formulas are almost as convenient as the harmonic oscillator energy formulas to use. The single-particle energies computed from these formulas have been compared with the experimental data and found in good agreement for occupied states.

Harmonic-oscillator pattern arising from an algebraic approach to chiral symmetry

In a previous work we proposed a saturation scheme for theSU3⊗SU3 chiral algebra within an infinite set ofSU6⊗O3 states. A mixing operator was introduced which transforms the axial charges of theSU6 solution into the physical ones,Qi5. Here we saturate the Weinberg equation for the (mass)2 operator [Q5+, [Q5+,m2]]=0, between meson states, in a perturbative approach. The generatorZ of the mixing operators, for which formerly only the transformation properties underSU6×O3 were given, is now completely established asZ=(W×M)z, whereW is theW-spin operator andM is the co-ordinate of the three-dimensional harmonic oscillator. In a perturbative expansion of the (mass)2 operator, the lowest term consists of two pieces, the harmonic-oscillator energy and a spin-orbit coupling of the form (−1)L+1(L·S+1/2). The resulting (mass)2 consists of families of equispaced linearly rising trajectories.

A study of light 4 n nucleides employing a three-part gaussian interaction

The properties of the light 4 n doubly even nucleides ranging from 4He to 40Ca are studied in a simple restricted Hartree-Fock calculation, employing an effective three-part two-body nucleon-nucleon interaction. This interaction consists of a trio of Wigner plus Majorana exchange forces each having Gaussian radial dependence. A rectangular harmonic oscillator basis of states is utilized. Two classes of parameter sets are studied: one generated from fitting the ground state energy and root-mean-square radius of 4He and ground state energy and density of nuclear matter, and one obtained from fitting the ground state energy and root-mean-square radius of both 4He and 40Ca. The resulting parameters are used to compute the ground state binding energies and harmonic oscillator energies of seven nucleides, 3Be, 12C, 20Ne, 14Mg, 28Si, 32S, and 36Ar, treated as being axially symmetric, with two possible configurations for all but 8Be and 12C considered. Moreover, 28Si is also analysed as a spherically symmetric nucleide. Additionally, the deformations of the axial nucleides are determined.

Quasiparticle-Conserving Hamiltonian for Interacting Electrons and Phonons

The Hamiltonian most often used to describe a system of electrons interacting with lattice vibrations has the asymmetry that the electrons are represented by the creation and annihilation operators which arise from quantum field theory whereas the lattice vibrations are represented by the one-particle ladder operators of the one-dimensional harmonic-oscillator problem. Although the equal spacing of the harmonic-oscillator energy levels and the commutation relations satisfied by the oscillator ladder operators have led to an interpretation of this formalism in terms of the "creation and annihilation" of phonons, the phonons thus defined are not the quasiparticles which are obtained by a field-theoretical formulation of the lattice Hamiltonian in the harmonic approximation. A Hamiltonian is derived in this paper in which both the electrons and the lattice (in the harmonic approximation) are represented in field-theoretical formalism. A lattice creation operator in this formulation actually creates a vibrational mode (in a given state) from the physical vacuum, in contrast to a ladder operator which merely generates transitions between two states of an already existing vibrational mode. As a consequence, the number operator for the lattice quasiparticles commutes with the Hamiltonian, a property which is useful in certain applications. It is shown that even though our Hamiltonian is formally different from the ordinary one, it yields equivalent results for the state vectors of physical interest.

A note on small vibrations

This note exhibits some old results in what is possibly a new form.The simple harmonic oscillator, whose kinetic and potential energies are given in terms of a single coordinate x bywhere b is a positive constant, has the equation of motionThe solution, appropriate to the initial conditions at t = 0 is conveniently written