1. The atom vortices are states of atoms which, are characterized by the fact that the angular momentum of the atom along its propagation axis is quantized. The wave-function of an atom vortex is given in cylindrical coordinates as: (p, p, z) = AJe(ap)eitdelkz S Where A, a are constants and f = +1, +2, 13, ... is the helicity of the vortex which is associated with its angular momentum along z-axis given by L = he. The vortex carries also a momentum along z-axis given by P = hk. The function Je(ap) is a Bessel function of the first kind. Express the wave-function in spherical coordinates. Show if this function has a definite parity or not. iii) Under what conditions the following wave-function has a definite parity and what is its value? (p, p, z) = Ale(ap)elldelkz + BJm(ap)eimde-ikz Remember that the parity operator acts on a function as PF r,0, Q = Fr,x -0,4+5)