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Journal of Contemporary Physics (Armenian Academy of Sciences)

Approximate solution of the Schrödinger equation for an atom in the field of a standing wave

Lilit A. Hovhannisyan, G. A. Muradyan, A. Zh. Muradyan

Published Dec 26, 2013• 0 citations
Publisher

Abstract

We present an approximate method of calculation of energy spectrum and wave functions of translational motion of a two-level atom in the field of a standing wave at zero value of the quasimomentum. Wave functions in the momentum representation are expressed in terms of confluent hypergeometric functions in case of continuous spectrum and in terms of Hermitian polynomials in case of discrete spectrum. The energy spectrum exhibits infinite compaction at transition from discrete part to continuous, as it is the case for, e.g., Coulombian potential.

DOI
10.3103/s1068337214010022

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We present an approximate method of calculation of energy spectrum and wave functions of translational motion of a two-level atom in the field of a standing wave at zero value of the quasimomentum. Wave functions in the momentum representation are expressed in terms of confluent hypergeometric functions in case of continuous spectrum and in terms of Hermitian polynomials in case of discrete spectrum. The energy spectrum exhibits infinite compaction at transition from discrete part to continuous, as it is the case for, e.g., Coulombian potential.
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Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB0_00​ method. Solution in the whole region is built with the help of simple connection formulas we derive from basic requirements of continuity and finiteness for the wave function in quantum mechanics. We show that, for conservative quantum systems, not only total energy, but also momentum is the constant of motion. We derive the quantization conditions for two and more turning point problems. Exact energy eigenvalues for solvable and some ``insoluble'' potentials are obtained. The eigenfunctions have the form of a standing wave, Ancos⁡(knx+δn)A_n\cos(k_nx+δ_n)An​cos(kn​x+δn​), and are the asymptote of the exact solution.
The eigenstate problem of the Jaynes–Cummings model on the basis of a complete Hamiltonian, including the centre-of-mass kinetic energy operator, is considered. The energy spectrum and wavefunctions in the standing-wave (SW) and counterpropagating-wave (CPW) cases are calculated and compared with each other. It is shown that in the CPW case: (i) the atomic momentum distribution is asymmetric; (ii) the concept of quasimomentum is not applicable and instead the problem concerns the ordinary momentum; (iii) atomic and photonic state distributions are self-consistent; and as a consequence (iv) the mean number of photons in counterpropagating travelling waves and mean atomic momentum are matched. Explicit analytic expressions are proposed to describe the transition from counterpropagating quantized waves to a standing wave. It is also shown that, if the recoil energy is taken into account, the Doppleron resonance is split into two branches, one of which diverges to Bragg-like resonance in high-order range.
The eigenstate problem of the Jaynes-Cummings model on the basis of complete Hamiltonian, including the center-of -mass kinetic energy operator, is treated. The energy spectrum and wave functions in standing-wave (SW)- and counterpropagating waves (CPW)- cases are calculated and compared with each other. It is shown that in CPW-case i) the atomic momentum distribution is asymmetric and somewhat narrower in general; ii) the concept of quasimomentum is not applicable and instead the ordinary momentum concerns the problem; iii) atomic and photonic state distributions are self-consistent, and, in consequence iiii) mean number of photons in the counterpropagating traveling waves and mean atomic momentum match. Explicit analytic expressions for energy eigenvalues and eigenfunctions are found in Tavis -Cummings-type approximation [Phys. Rev. 170, 379(1968)] and is pointed, that it implies only the bounded-like states for atomic center-of-mass motion. It is also shown that if the recoil energy is taken into account, the Doppleron resonance is split into two branches, one of which diverges to Bragg-like resonance in the high-order range.
The spherically symmetric potential is considered whose dependence on the distance r is described by the smooth composition of Coulomb at r < r0 and oscillator at r > r0 potentials. The boundary distance r0 is determined by the parameters of these potentials. The exact continuous solution of the radial Schrödinger equation is expressed in terms of the confluent hypergeometric functions. The discrete energy levels are obtained. The graphic illustrations for the energy spectrum and the radial wave functions are presented.
The motion of a two-level atom in a standing-wave laser field is studied using an approximation to the resolvent operator in the form of a truncated eigenfunction expansion, with the field treated in the occupation-number representation. The formalism provides expressions for the time-dependent S -matrix for transitions into both ground and excited atomic states. The effective Hamiltonian for purely elastic scattering is constructed, allowing for the direct study of scattering in the Bragg regime. Contact with standard treatments of Bragg scattering in terms of Mathieu functions is obtained by imposition of additional approximations. Comparison of numerical results obtained with and without imposition of these approximations provides an indication of their range of validity. The dependence of excitation probabilities on field strength and detuning is studied. These numerical illustrations provide evidence that in cases of physical interest excitation probabilities can be significant and can be reliably estimated for a broad range of field parameters using the present formalism.