Hamiltonian deep well

The one-dimensional potential depicted in Fig. 7(a) provides an illustration of this effect. The Schrodinger equation can be solved with the method used for the square-well case above. Each well gives rise to a nearly independent progression of states. For fi = 2p = 1 and other potential parameters indicated in Fig. 7 one finds that the system has two bound states and a resonance state at 9.46 — i 0.11 localized above the deep outer well. There is also another resonance in the system, E = 9.8 - i 0.002. Its width is very small because this state belongs to the shallow inner well, which is separated from the continuum by a potential barrier. Suppose that we force — by varying a parameter in the Hamiltonian — the narrow state (denoted n) in the shallow well to move across the broader resonance (b) belonging to the deep minimum. The relative positions of the two states can be, for example, controlled by shifting the infinite wall at the... 

Applying an optical lattice provides a periodic structure for the polar molecules described by the Hamiltonian of Equation 12.1, with yjj) given by Equation 12.32. In the limit of a deep lattice, a standard expansion of the field operators i] (r) = w(r - Ri)b] in the second-quantized expression of Equation 12.1 in terms of lowest-band Wannier functions w(r) and particle creation operators b] [107] leads to the realization of the Hubbard model of Equation 12.9, characterized by strong nearest-neighbor interactions [85]. We notice that the particles are treated as hardcore because of the constraint Rq. The interaction parameters Uy and Vyk in Equation 12.9derive from theeffective interaction V ( ri ), and in the limit of well-localized Wannier functions reduce to... 

The Born-Oppenheimer approximation, whose validity depends on there being a deep enough localized potential well in the electronic energy, has, however, been extensively treated. The mathematical approaches depend upon the theory of fiber bundles and the electronic Hamiltonian in these approaches is defined in terms of a fiber bundle. It is central to these approaches, however, that the fiber bundle should be trivial, that is that the base manifold and the basis for the fibers be describable as a direct product of Cartesian spaces. This is obviously possible with the decomposition choice made for O Eq. 2.42 but not obviously so in the choice made for O Eq. 2.43. 

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