Two-level problem

In calculating Pj[oo), the Rosen-Zener approximation, familiar from the two-level problem in atomic physics, can be used for f/co) and this leads to the compact expression, for any pulse-like F(t),... 

In a general description of all these cases we start with the solution at Q = 0, which is a 2-fold degenerate or pseudodegenerate state and include the Q-dependent interaction V(r,Q) as a perturbation. The two-level problem for the electronic states yields (H is the Hamiltonian) ... 

In the study of mixtures, differentiation between enantiomers is a two level problem which is somewhat independent of whether the LC system is chiral or conventional. The problems common to both systems are the effects of overlapping bands on the performance of the detectorfs). Overlap can be between chiral-achiral species on the one hand and co-eluted chiral-chiral with achiral on the other. On first thought the chiral-achiral distinction should be relatively easy if a chiroptical detector is used because the achiral compounds will not interfere with the detection measurement. In addition the ability of the chiroptical detector to measure both positive and negative signals makes the confirmation of the enantiomeric structure elementary [3,4], As pointed out earlier, enantiomers co-elute from conventional columns and two detectors in sequence will provide the information to measure the enantiomeric ratio provided the mixture is not racemic. Partial or total overlap of the band for a non-chiral species with the chiral eluate band increases significantly the difficulty in measuring an enantiomeric ratio. In this instance the total absorbance that is measured may include a contribution from the non-chiral species which without correction will lead to an overestimation of the amount of chiral material and an erroneous value for the enantiomeric ratio. Under these circumstances there is no other LC option but to develop a separation that is based upon a chiral system. 

At the resonance w(t) = A(x), the adiabatic potentials i.e. the eigenvalues of (5.9) show avoided crossing and the population splits into the two adiabatic Floquet states. In the case of quadratically chirped pulses, the instantaneous frequency meets the resonance condition twice and near-complete excitation can be achieved due to the constructive interference. The nonadi-abatic transition matrix Ujj for the two-level problem of (5.9) is given by the ZN theory [33] as... 

Unfortunately, this maneuver links au(t) to all the other coefficients an(t). For simplicity, then, assume a two-level problem, with only states i (initial lower) and / (final upper) to worry about ... 

Thus, the results shown by Eqs. (15-7a) - (15-7h) can be obtained from the two-levels problem, which often appears in quantum mechanics. Here, the following relations are useful. 

In this appendix we generalise the expressions of the diabatic quantities first introduced in Sec. 2 for the ideal case of an exact two-level problem to a more realistic description. In a normal situation, the Hamiltonian has an infinite number of eigenstates, and there is no finite number of strictly diabatic states [76] that can describe a given pair of adiabatic states [77-80]. Instead, one can define a unitary transformation of the adiabatic states generating two quasidiabatic states characterised by a residual non-adiabatic coupling, as small as possible, but never zero (see, e.g., [5,24,32-35]). In practice, the electronic Hilbert space is always truncated to a finite number of configurations. In what follows, we consider the case of MCSCF wavefunctions and make use of generalised crude adiabatic states adapted to this. 

Where the medium is initially in an excited eigenstate, the resonant active process in this two-level problem is one-photon emission, or fluorescence. In this case, the resonant state would lie below the initial state. The field gains energy through the interaction while the medium loses energy. 

Despite the increase in number of levels, the list of quantum properties assigned to the two-level problem also applies to this case. How then can we propose to approximate the dynamics of this system using classical mechanics ... 

Perturbation theory is an extremely useful analytic tool. It is almost always possible to treat a narrow range of. /-values in a multistate interaction problem by exactly diagonalizing a two-level problem after correcting, by nondegenerate perturbation theory or a Van Vleck transformation, for the effects of other nearby perturbers. Such a procedure can enable one to test for the sensitivity of the data set to the value of a specific unknown parameter. 

Hzf and Hfield are respectively the zero-field and field-dependent parts of H and Mfii is the tuning rate for the energy of basis function iM) in the field F. The Hamiltonian matrix for this A J = 0 two-level problem is... 

Hi2- However, analysis of the two-level problem (related to accidental predissociation, discussed in Section 7.13, autoionization, discussed in Section 8.4, and Intramolecular Vibrational Redistribution, discussed in Section 9.4.14) provides insights into the unique effects that derive from widths and decay rates of the basis states. 

These results are very similar to the normal two-level problem, except (i) the real parts of the energies exhibit level repulsion, but this repulsion is reduced (but never reversed) by the <5r2/2<5e term in the denominator of Eqs. (9.3.12a) and (9.3.12b) (h) the imaginary parts of the energies exhibit level attraction (the interaction causes the widths of the mixed states to become more similar), and this attraction is reduced (but never reversed) by the 48e2/8Y term in the denominator of Eqs. (9.3.12a) and (9.3.12b) (Hi) when <5T — 0, the level widths are unaffected by the interaction (iv) when Se = 0 but <5Tj >> V, the level positions are unaffected by the interaction (because the narrower level is symmetrically surrounded by the much broader level and the usual level shift by V is suppressed). This amounts to an intuitively-sensible extension of the normal (real E, Hermitian H) two-level problem. 

Despite these difficulties, Lasaga and Karplus have discussed the calculation of excitation energies based on an operator equation related to (3). Simons and Dalgaard have proposed a perturbation approach to a similar operator problem. To date, however, numerical applications have been limited to the analysis of the singlet excitation of ethylene in Pariser-Parr-Pople (PPP) model, a two-level problem. ... 

Since the electron density is strongly peaked alE, it is possible to treat the drift mobility problem as a two-level problem a conducting level near the mobility edge with an effective density of states and a single trap level at E. This trap level has the unusual feature that its energy position E and number density are both time dependent. 

The equivalence of any two-level problem to the problem of a particle of spin 1/2 in magnetic field is well known [10] and has been used to treat the one-electron part of a two-orbital problem in terms of a fictitious spin of 1/2 in a fictitious magnetic field. The equivalence is obvious from the fact that any 2x2 matrix is fully characterized by its four complex elements and therefore can be written as a linear combination of the unit matrix (a /3) and the three Pauli matrices Oy, a ). The one-electron part Hi of a... 

Note that the transformation U is its own inverse U = U. In the x-basis, only the states (0) and (c) are coupled by the rf field. With this reduction of the problem to a two-level problem, we can write down how the amplitudes evolve in the x-basis under the influence of the rf magnetic field applied from time h to time h + x. 

---