Schrodinger equation for a charged particle interacting with the plane wave electromagnetic field is solved exactly. The exact analytic solution and the perturbative solution up to second order are compared.PACS numbers: 03.65. Ge, 42.50.Ct, The exact solution of Dirac's equation for an electron in an external plane wave electromagnetic field was first obtained by Volkov [1] in 1935 and it's Green function was derived by Schwinger [2]. The state of the charged particle, known as Volkov state, has been used extensively to explore numerous quantum phenomena such as Compton scattering, photo-ionization, bremsstrahlung processes and Kapitza-Dirac problem (scattering of an electron by a standing light wave), etc. However, an exact analytic solution of Schrodinger equation for a charged particle interacting with a plane wave electromagnetic field has never been obtained. The subject of the interaction of the charged particle with radiation is investigated using perturbation theory in almost any quantum mechanics textbooks. However, the behavior of a charged particle in the strong radiation fields by high-power laser can not be investigated by perturbation theory. To deal with this problem, new approximation methods have been developed [3,4,5,6,7,8,9]. For example, in dipole approximation, electromagnetic field is assumed to be purely time-dependent for the study of ionization of atoms by intense laser pulses. In this approach, the magnetic field is neglected. As opposes to dipole approximation, the effect of magnetic field can be taken into account by expanding the potential to first order in the space coordinate. In this paper, we will solve exactly the Schrodinger equation for an electron in a linearly polarized monochromatic plane wave propagating in the z-direction. The vector potential is given by A = (A 0 e ik(z−ct) , 0, 0). The laser pulse with the elliptic polarization will be discussed later.where ∇.A = 0. To solve this equation, let us transform z-coordinate as s = z − ct. Then the time derivative operator transforms as ∂/∂t → ∂/∂t − c∂/∂s. The equation (1) becomeswhere α is defined as α = eA 0 hc and ∇ 2 = ∂ 2 x + ∂ 2 y + ∂ 2 s . We introduce the following ansatz; i.e. we look for the solution of (2) in the formwhere ǫ and k ′ x , k ′ y are all constants. If we substitute (3) into the equation (2), we obtainLet us make a complex transformation on the coordinate as u = exp(iks). This is reasonable since the Hamiltonian is not Hermitian. Using the transformation for the derivative operator d ds = iku d du , the equation (4) becomeswhere. To solve (5), the last transformation is introduced as
