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The Quantum Mechanics Solver 28 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 27 Bloch Oscillations The possibility to study accurately the quantum motion of atoms in standing light fields has been used recently in order to test several predictions relating to wave propagation in a periodic potential. We present in this chapter some of these observations related to the phenomenon of Bloch oscillations. 27.1 Unitary Transformation on a Quantum System Consider a system in the state t which evolves under the effect of a Hamiltonian H t . Consider a unitary operator D t . Show that the evolution of the transformed vector l t D t t is given by a Schrodinger equation with Hamiltonian unr H t D t H t Dt t ihdDt Dt t . 27.2 Band Structure in a Periodic Potential The mechanical action of a standing light wave onto an atom can be described by a potential see e.g. Chap. 26 . If the detuning between the light frequency and the atom resonance frequency a A is large compared to the electric dipole coupling of the atom with the wave this potential is proportional to the light intensity. Consequently the one-dimensional motion of an atom of mass m moving in a standing laser wave can be written H 2 Uo sin2 koX 2m 278 27 Bloch Oscillations where X and P are the atomic position and momentum operators and where we neglect any spontaneous emission process. We shall assume that k0 a c and we introduce the recoil energy ER hQjk2 2m . 27.2.1. a Given the periodicity of the Hamiltonian H recall briefly why the eigenstates of this Hamiltonian can be cast in the form Bloch theorem W eqX uq where the real number q Bloch index is in the interval k0 k0 and where uq is periodic in space with period Aq 2. b Write the eigenvalue equation to be satisfied by uq . Discuss the corresponding spectrum i for a given value of q ii when q varies between k0 and k0. In the following the eigenstates of H are denoted n q with energies En q . They are normalized on a spatial period of extension Aq 2 n k0. 27.2.2. Give the energy levels in terms of the indices n and q in the case Uq 0. .
