| Fock[1]and Darwin[2]first solved The quantum mechanical problem of an electron in thecombined presence of a uniform magnetic field and a parabolic potential（this model is called Fock-Darwin （FD） model nowonwards.）. The spectrum of this model interpolates between the boundstates of a two dimensional （2D） harmonic oscillator in a homogeneous magnetic field to the caseof the Landau levels of a free particle in a magnetic field[3]. Hence, the FD model has wideapplications in the calculation of dissipative Landau diamagnetism[7], quantum dots[5], orbitalmagnetism of noninteracting fermions in a2D harmonic potential[6],quantum Hall effect[4], and thethird law of thermodynamics[8]can be verified by the FD model.The evolution of the states in homogeneous electric and magnetic fields is One of theimportant research areas related to the FD model. The problem of the evolution of a Gaussian wavepacket in homogeneous electric and magnetic field was first considered by Darwin[9]. Several otherauthors[11-12]investigated the dynamics of a particle in a homogeneous electromagnetic field interms of coherent states[10],whereby they obtained an explicit representation of the Green’sfunction and studied the invariants of the system. The coherent states have been used widely inphase-space path integral in quantum field theory[13]and quantum optics[14]. Kennard[15]firstinvestigated the evolution of a more general wave packet of the harmonic oscillator than that ofcoherent state, now known as a squeezed state. Squeezed states also play an important role inquantum optics[13], and recently quantum information processing[16-21]. Plebaski[22]and Infeld[23]established a relation between the evolution of initial states whose time evolution is known andstates which are derived from such initial states by the application of the squeezing operator（squeezed states）. Recently, J. E. Santos et al[24]achieved the generalization of this relation to theFD Hamiltonian in an homogeneous time-dependent electric.In this paper, we consider a more general problem in light of work of Ref.[24]: how anarbitrary initial state evolves under the FD Hamiltonian in the presence of a homogeneous time-dependent electric field? To answer this question, one only need to find out how the eigenstates of FD Hamiltonian evolve under the additional electric field since eigenstates of FD Hamiltonianform a complete set and any states can be expressed in terms of a linear combination of them. Weuse path integral method to study the time evolution of the eigenstates of the FD Hamiltonianunder an additional time-dependent electric field, and an exact analytical relation between theinitial state and the evolved state will be established. |
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