The 2D Infinite Well: In two dimensions, the Schrödinger equation is(a) Given that U is a...

The 2D Infinite Well: In two dimensions, the Schrödinger equation is

(a) Given that U is a constant, separate variables by trying a solution of the form ψ(x, y) = f(x)g(y), then dividing by f(x)g(y). Call the separation constants Cx and Cy.

(b) For an infinite well,

What should f(x) and g(y) be outside this well? What functions would be acceptable standing-wave solutions for f(x) and g(y) inside the well? Are Cx and Cy positive, negative, or zero? Imposing appropriate conditions, find the allowed values of Cx and Cy.

(c) How many independent quantum numbers arc there?

(d) Find the allowed energies E.

(e) Are there energies for which there is not a unique corresponding wave function?