Chapter 13.6, Problem 6P

Let $V=0$ in the Schrödinger equation (3.22) and separate variables in 2 -dimensional rectangular coordinates. Solve the problem of a particle in a 2 -dimensional square box, $0<x<l,0<y<l.$ This means to find solutions of the Schrödinger equation which are 0 for $x=0,x=l,y=0,y=l,$ that is, on the boundary of the box, and to find the corresponding energy eigenvalues. Comments: If we extend the idea of a "particle in a box" (see Section 3, Example 3) to two or three dimensions, the box in 2 D might be a square (as in this problem) or a circle (Problem 8 ); in 3 D it might be a cube (Problem 7.17) or a sphere (Problem 7.19). In all cases, the mathematical problem is to find solutions of the Schrödinger equation with $V=0$ inside the box and $\Psi =0$ on the boundary of the box, and to find the corresponding energy eigenvalues. In quantum mechanics, $\Psi$ describes a particle trapped inside the box and the energy eigenvalues are the possible values of the energy of the particle.