Landau Levels as a Limiting Case of a Model with the Morse-Like Magnetic Field

Abstract

We consider the quantum mechanics of an electron trapped on an infinite band along the $x$-axis in the presence of the Morse-like perpendicular magnetic field $\vec{B}=-B_0{e}^{-\frac{2\pi }{a_0}x}\hat{k}$ with $B_0>0$ as a constant strength and $a_0$ as the width of the band. It is shown that the square integrable pure states realize representations of $su(1,1)$ algebra via the quantum number corresponding to the linear momentum in the $y$-direction. The energy of the states increases by decreasing the width $a_0$ while it is not changed by $B_0$. It is quadratic in terms of two quantum numbers, and the linear spectrum of the Landau levels is obtained as a limiting case of $a_0\to \infty$. All of the lowest states of the $su(1,1)$ representations minimize uncertainty relation and the minimizing of their second and third states is transformed to that of the Landau levels in the limit $a_0\to \infty$. The compact forms of the Barut-Girardello coherent states corresponding to $l$-representation of $su(1,1)$ algebra and their positive definite measures on the complex plane are also calculated.

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Cited by in corpus (2)

• Generalized $su(2)$ coherent states for the Landau levels and their nonclassical properties
• Approach of the Generating Functions to the Coherent States for Some Quantum Solvable Models