The Infinite Square Well Now we are ready to solve the time independent Schrödinger equation (TISE) to get ψ(x) and E. (Recall Ψ(x,t) = ψ(x)e −i2  Et/h ) Since the particle can never be outside the box, this requires ψ(x) = 0 in x ≤ 0 and x ≥ a

Inside the box 0 < x < a, since U(x) = 0, the TISE becomes The 1-D TISE is a second-order ordinary differential equation which has a general solution of where A and B are any complex constants or ψ (x) =Asinkx+Bcoskx.

The constants A and B, and the energy E are determined by the boundary conditions (continuity requirements): ψ(0) = 0, ψ(a) = 0, and normalization Hence, the allowed values of E are

The solutions are As a collection, the functions  n (x) have some interesting and important properties: 1. They are alternately even and odd, with respect to the center of the well.  1 is even,  2 is odd,  3 is even, and so on.

2. As you go up in energy, each successive state has one more node (zero crossing).  1 has none (the end points don't count),  2 has one,  3 has two, and so on. 3. They are mutually orthogonal, in the sense that whenever m≠n; 4. They are complete, in the sense that any other function, f(x), can be expressed as a linear combination of them:

The expansion coefficients (c n ) can be evaluated--for a given f(x)--by a method I call Fourier's trick, which beautifully exploits the orthonormality of {  n }: Multiply both sides of Equation by  m *(x), and integrate. These four properties are extremely powerful, and they are not peculiar to the infinite square well.