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 B = − B 0 e − 2 π a 0 x k with B0 > 0 as a constant strength and a0 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 a0 while it is not changed by B0. It is quadratic in terms of two quantum numbers, and the linear spectrum of the Landau levels is obtained as a limiting case of a0 → ∞. 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 a0 → ∞. 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.