Probabilities of Bremsstrahlung Emission of Photons at Low-Energy Electron-Nuclear Collisions in a Magnetic Field

Abstract

We consider quantum-mechanical probabilities of bremsstrahlung of photons in the case of low-energy Coulomb collisions in a magnetic field, where the scattering center so perturbs the state of the incident electron that the motion of the latter becomes quasi-bound. Quantum formulas for the spectral power of bremsstrahlung radiation are obtained from the classical formulas by replacing the Fourier amplitudes of the particle velocity with matrix elements of the velocity operator for wave functions, which are normalized by the condition of a unit flux being incident on the nucleus (or an equivalent outgoing flux), with summation over finite Landau levels and quantized values of the impact parameter. Equivalent forms of the specified matrix elements, which are expressed in terms of the Coulomb field and annihilation/creation operators for the eigenfunctions of the operator of the squared impact parameter, are presented. The obtained presentations for the spectral power of bremsstrahlung radiation in the case of quasi-bound electron motion allow one to translate the results of calculating this value in the classical limit to the quantum case, which is typical of white dwarfs with the strongest magnetic fields.