A particle in the Bio-Savart-Laplace magnetic field: explicit solutions

D. Yafaev

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A particle in the Bio-Savart-Laplace magnetic field: explicit solutions

D. Yafaev

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Abstract

We consider the Schrödinger operator ${\bf H}=(i\nabla+A)^2 $ in the space $L_2({\mathbb R}^3)$ with a magnetic potential $A $ created by an infinite straight current. We perform a spectral analysis of the operator ${\bf H}$ almost explicitly. In particular, we show that the operator ${\bf H}$ is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions $\exp(-i{\bf H}t)f$ of the time dependent Schrödinger equation. Equations of classical mechanics are also integrated. Our main observation is that both quantum and classical particles have always a preferable (depending on its charge) direction of propagation along the current and both of them are confined in the plane orthogonal to the current.

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D. Yafaev. 2004-03-26. A particle in the Bio-Savart-Laplace magnetic field: explicit solutions. https://arxiv.org/abs/physics/0403131

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