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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

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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.

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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

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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.

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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:

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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.

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2. Time Independent Schrodinger Equation

2. Time Independent Schrodinger Equation

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