Physics Letters A 355 (2006) 348–351 www.elsevier.com/locate/pla An exact solution for electrons in a time-dependent magnetic field D. Laroze, R. Rivera ∗ Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile Received 19 October 2005; received in revised form 14 February 2006; accepted 3 March 2006 Available online 10 March 2006 Communicated by J. Flouquet Abstract In this work we study the dynamics of electrons in presence of a uniform time-dependent magnetic field. An exact solution for the wave function time evolution is obtained when the initial state is a superposition of Landau levels. For a given time dependence of the magnetic field, the time-evolved wave function differs from the initial wave function by a dynamic phase factor.  2006 Elsevier B.V. All rights reserved. PACS: 71.10.Ca; 71.70.Di; 03.65.-w Keywords: Time-dependent problems; Quantum mechanics; Exact solution; Caustics 1. Introduction The dynamics of electrons in magnetic fields has played a fundamental role in physics and its application to technology in a wide spectrum of topics, such as the Aharonov–Bohm effect [1–4], the bidimensional electron gas at the interface of semi- conductor heterostructures [5], cyclotron resonance [6], mag- netoplasmon resonance [7], magnetoresistance [8], and electro- magnetic lenses with time-dependent magnetic fields [9–12]. In the case of time-dependent quantum systems it appears in ad- dition the phenomenon of dynamic phases [13,14], illustrated for example in Refs. [15,16] in the case of the time-dependent harmonic oscillator. All these situations are interesting in them- selves, and present rather complex mathematical structures; therefore exact analytic solutions have been found only in a few special cases. In the present work, we study the dynamics of electrons in the presence of a uniform time-dependent magnetic field, and we find an exact solution for the corresponding propagator of the Schrödinger equation. An analytical expression for the time-evolved wave function is found when the initial state is a * Corresponding author. Tel.: +56 32 273136; fax: +56 32 273529. E-mail addresses: david.laroze@usm.cl (D. Laroze), rrivera1@vtr.net (R. Rivera). superposition of Landau levels; in particular it is shown that a dynamical phase appears in the time evolution of the wave func- tion as a consequence of the time-dependence of the magnetic field. 2. Theoretical model In this section we will develop the quantum mechanical for- malism to describe the dynamics of electrons in a uniform time-dependent magnetic field, B(t). Since the corresponding Hamiltonian is time-dependent, energy will not be conserved and there will not be an energy spectrum. Therefore, the prob- lem must be approached by directly solving the time-dependent Schrödinger equation: (1) 1 2m  P + e c A  2 Φ(r,t) = i ¯ h ∂ ∂t Φ(r,t), where m is the electron mass, q =−e is the electron charge, c is the speed of light and the vector potential is chosen to be A =−r × B/2. The temporal evolution of the system from an initial instant t 0 to a final instant t f is given by (2) Φ(r f ,t f ) =  d 3 r i G(r f ,t f , r i ,t 0 )Φ 0 (r i ,t 0 ), where G(r f ,t f , r i ,t 0 ) is the propagator and Φ 0 (r i ,t 0 ) is the initial wave function. If the magnetic field has the form B(t) = 0375-9601/$ – see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.03.002