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

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 $\vec{B}=-B_{0}e^{-\frac{2π}{a_{0}}x}\hat{k}$ 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 $a_{0}\rightarrow\infty$. 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 $a_{0}\rightarrow\infty$. 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.