Comparing the Schrödinger and Dirac Descriptions of an Electron in a Uniform Magnetic Field

Villamizar, David Velasco; Russell, Benjamin

doi:10.1590/1806-9126-RBEF-2018-0035

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Produtos e Materiais Didáticos • Rev. Bras. Ensino Fís. 40 (4) • 2018 • https://doi.org/10.1590/1806-9126-RBEF-2018-0035 copy

Comparing the Schrödinger and Dirac Descriptions of an Electron in a Uniform Magnetic Field

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

In this article we present a detailed description, using ladder operators, of an electron in a uniform magnetic field evolving under the Schrödinger equation. We go on to describe the same physical system in terms of relativistic quantum mechanics using the Dirac equation and to compare the two models in detail. The main differences between these two quantum mechanical approaches are discussed and we observe specifically how the relativistic phenomena modify the description of this particular quantum system by isolating effects which only exist in the relativistic model.

Keywords:
Quantum Mechanics; Schrödinger Equation; Dirac Equation; Uniform Magnetic Field

$n$	$m_{l}$	$F_{n, m_{l}} (ρ, φ)$
0	0	$F_{0, 0} = \frac{β}{\sqrt{π}} e^{- β^{2} ρ^{2} / 2}$
1	1	$F_{1, 1} = \frac{β}{\sqrt{π}} (β ρ) e^{- β^{2} ρ^{2} / 2} e^{i φ}$
2	2	$F_{2, 2} = \frac{β}{\sqrt{2 π}} {(β ρ)}^{2} e^{- β^{2} ρ^{2} / 2} e^{i 2 φ}$
	0	$F_{2, 0} = \frac{β}{\sqrt{π}} [{(β ρ)}^{2} - 1] e^{- β^{2} ρ^{2} / 2}$
3	3	$F_{3, 3} = \frac{β}{\sqrt{6 π}} {(β ρ)}^{3} e^{- β^{2} ρ^{2} / 2} e^{i 3 φ}$
	1	$F_{3, 1} = \frac{β}{\sqrt{2 π}} [{(β ρ)}^{3} - 2 (β ρ)] e^{- β^{2} ρ^{2} / 2} e^{i φ}$
4	4	$F_{4, 4} = \frac{β}{2 \sqrt{6 π}} {(β ρ)}^{4} e^{- β^{2} ρ^{2} / 2} e^{i 4 φ}$
	2	$F_{4, 2} = \frac{β}{\sqrt{6 π}} [{(β ρ)}^{4} - 3 {(β ρ)}^{2}] e^{- β^{2} ρ^{2} / 2} e^{i 2 φ}$
	0	$F_{4, 0} = \frac{β}{\sqrt{4 π}} [{(β ρ)}^{4} - 4 {(β ρ)}^{2} + 2] e^{- β^{2} ρ^{2} / 2}$
5	5	$F_{5, 5} = \frac{β}{2 \sqrt{30 π}} {(β ρ)}^{5} e^{- β^{2} ρ^{2} / 2} e^{i 5 φ}$
	3	$F_{5, 3} = \frac{β}{2 \sqrt{6 π}} [{(β ρ)}^{5} - 4 {(β ρ)}^{3}] e^{- β^{2} ρ^{2} / 2} e^{i 3 φ}$
	1	$F_{5, 1} = \frac{β}{\sqrt{12 π}} [{(β ρ)}^{5} - 6 {(β ρ)}^{3} + 6 (β ρ)] e^{- β^{2} ρ^{2} / 2} e^{i φ}$

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E-mail: marcio@sbfisica.org.br

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[1] *Endereço de correspondência: david.velasco.v@gmail.com.