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#### Impact of the quantized transverse motion on radiation emission in a Dirac harmonic oscillator

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1805.05167.pdf

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##### Citation

Wistisen, T. N., & Di Piazza, A. (2018). Impact of the quantized transverse motion
on radiation emission in a Dirac harmonic oscillator.* Physical Review A,* *98*(2):
022131. doi:10.1103/PhysRevA.98.022131.

Cite as: https://hdl.handle.net/21.11116/0000-0002-4EDB-1

##### Abstract

We investigate the radiation emitted by an ultrarelativistic electron

traveling in a 1-dimensional parabolic potential. Having in mind a simplified

model for beamstrahlung, we consider the realistic case of the electron motion

being highly directional, with the transverse momentum being much smaller than

the longitudinal one. In this case we can find solutions of the Dirac equation

and we calculate exactly the radiation emission using first-order perturbation

theory. We compare the results obtained to that obtained via the semi-classical

method of Baier and Katkov which includes quantum effects due to photon recoil

in the radiation emission but ignores the quantum nature of the electron

motion. On the one hand, we confirm a prediction of the semi-classical method

that the emission spectrum should coincide with that in the case of a linearly

polarized monochromatic wave. On the other hand, however, we find that the

semi-classical method does not yield the exact result when the quantum number

describing the transverse motion becomes small. In this way, we address

quantitatively the problem of the limits of validity of the semi-classical

method, which is known, generally speaking, to be applicable for large quantum

numbers. Finally, we also discuss which beam conditions would be necessary to

enter the studied regime where also the motion of the particles must be

considered quantum mechanically to yield the correct spectrum.

traveling in a 1-dimensional parabolic potential. Having in mind a simplified

model for beamstrahlung, we consider the realistic case of the electron motion

being highly directional, with the transverse momentum being much smaller than

the longitudinal one. In this case we can find solutions of the Dirac equation

and we calculate exactly the radiation emission using first-order perturbation

theory. We compare the results obtained to that obtained via the semi-classical

method of Baier and Katkov which includes quantum effects due to photon recoil

in the radiation emission but ignores the quantum nature of the electron

motion. On the one hand, we confirm a prediction of the semi-classical method

that the emission spectrum should coincide with that in the case of a linearly

polarized monochromatic wave. On the other hand, however, we find that the

semi-classical method does not yield the exact result when the quantum number

describing the transverse motion becomes small. In this way, we address

quantitatively the problem of the limits of validity of the semi-classical

method, which is known, generally speaking, to be applicable for large quantum

numbers. Finally, we also discuss which beam conditions would be necessary to

enter the studied regime where also the motion of the particles must be

considered quantum mechanically to yield the correct spectrum.