Ladder of states

Fig. 1.12. The three-level system showing the creation of a Floquet ladder of states. Also shown is the weak one-photon coupling needed to make a transition to the Floquet ladder from the ground state. f i2 and Q-n are the Rabi frequencies between the states...

There are several other interesting topics in quantum optics which we would like to be able to study. For example, we would like model problems in double resonance spectroscopy, where there are two electromagnetic fields with possibly different polarizations simultaneously interacting with a molecule. This problem resembles the multiple photon excitation problem in that there is population migration along ladders of states, but in this case there can be a vastly larger number of quantum levels to treat — on the order of 2(2J+1). At room temperature, the most probable value of J for SF is about 60, which implies a 250 state calculation. 

In addition, Dirac noted that Eq. (3.48) is identical to the classical expression for the energy of a harmonic oscillator with unit mass (Eq. 2.28). The first term in the braces corresponds formally to the kinetic energy of the oscillator the second, to the potential energy. It follows that if we replace Pj and Qj by momentum and position operators Pj and Q, respectively, the eigenstates of the Schrodinger equation for electromagnetic radiation will be the same as those for harmonic oscillators. In particular, each oscillation mode will have a ladder of states with wavefunctions r y and energies... 

Each mode has a set of energy levels that form a regular ladder of states, with energies En given by... 

Nitmerotts examples of chmbing the ladder can be fotmd in textbooks for secondary edncation. For example, textbooks start the stndy of the snbject of salts with the (strb-) microscopic particles of atoms and molectrles, followed by how atoms theoretically ate converted into iotts, and how ionic srrbstances ate brrilt from charged ions. Textbooks continne with the macroscopic properly of the soln-bility of ionic snbstances in water. Snbseqnently mote complex ions, snch as strl-phates and nitrates, ate addressed to become part of the stndents repertoire ns-ing the sub-microscopic world of chemistry and the symbolic representations. For other subjects, such as organic chemistiy, the pathway for stndy from the basic sub-microscopic particles and related chemical principles to making sense of a relevant macro-world of applications (e.g. production of medicines) is very long. Moreover, the sub-microscopic world of state-of-the-art chemistry has become very complex. 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

The major structural types found for lithium amide complexes in the solid state are illustrated in Fig. 34. These comprise ladders of limited extent when the L Li ratio is less than 1 1 (Fig. 34a), dimeric (NLi)2 rings, when this ratio is 1 1 and, usually, when the complexants are monodentate (Fig. 34b), and monomers, both contact-ion pairs (CIPs) and solvent-separated ion pairs (SSIPs) (Fig. 34c). Monomers occur always when there are two or more monodentate complexants per Li. This also is usual with bidentate ligands, and is always found when the ligands have higher denticity. 

With this classification of states we may write the following generalized Dimer-RVB state for a two-leg spin ladder,... 

The ground-state energy for a ladder of length N is estimated. In the thermodynamic limit TV - oc one can find a closed expression for the density energy per site [22],... 

Also,if the solutions had an initial Si02/Al20 ratio of 73,S formed but no Y, while if this initial ratio was 102,P formed but no S. This behaviour suggests caution in interpreting all crystallisation sequences as examples of Ostwald s rule of successive transformations. The rule states that in a crystallisation sequence the new phases replace each other in the order of a step by step descent of a ladder of increasing thermodynamic stability. An example in a hydrothermal system is ( ) ... 

A typical problem of interest at Los Alamos is the solution of the infrared multiple photon excitation dynamics of sulfur hexafluoride. This very problem has been quite popular in the literature in the past few years. (7) The solution of this problem is modeled by a molecular Hamiltonian which explicitly treats the asymmetric stretch ladder of the molecule coupled implicitly to the other molecular degrees of freedom. (See Fig. 12.) We consider the the first seven vibrational states of the mode of SF (6v ) the octahedral symmetry of the SF molecule makes these vibrational levels degenerate, and coupling between vibrational and rotational motion splits these degeneracies slightly. Furthermore, there is a rotational manifold of states associated with each vibrational level. Even to describe the zeroth-order level states of this molecule is itself a fairly complicated problem. Now if we were to include collisions in our model of multiple photon excitation of SF, e wou d have to solve a matrix Bloch equation with a minimum of 84 x 84 elements. Clearly such a problem is beyond our current abilities, so in fact we neglect collisional effects in order to stay with a Schrodinger picture of the excitation dynamics. 

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