Residents of Flatworld (a two-dimensional world far, far away) have it easy. Although quantum mechanics applies in their world, the equations they must solve to understand atomic energy levels involve only two dimensions. In particular, the Schrodinger equation for the one-electron "flatrogen" atom is, −~22m1r∂∂r(r∂∂r)ψ(r, θ) −~22m1r2∂2 ∂θ2 ψ(r, θ) + U(r)ψ(r, θ) = Eψ(r, θ). (1)
(a) Separate variables by trying a solution of the form ψ(r, θ) = R(r)Θ(θ), then dividing by R(r)Θ(θ). Show that the "θ-equation" can be written as, d2 dθ2 Θ(θ) = CΘ(θ), (2) where C is a separation constant.
(b) To be physically acceptable, Θ(θ) must be continuous, which, since it involves rotation about an axis, means that it must be periodic. What must be the sign of C.
(c) Show that a complex exponential is an acceptable solution for Θ(θ).
(d) Imposing the periodicity condition, find the allowed values of C.
(e) What property is quantized according to the value of C?
(f) Obtain the "r-equation".
(g) Given that U(r) = −b/r, show that a function of the form R(r) = e−r/a is a solution, but only if C ia a certain one of its allowed values.
(h) Determine the value of a, and thus find the ground state energy and wave function of the flatrogen atom.