The Shooting Method and the Infinite Square Well

Please wait for the animation to completely load.

In the infinite square well potential, a particle is confined to a box of length L by two infinitely high potential energy barriers:

V = ∞        x ≤ 0 ,             V = 0    0 < x < L ,             V = ∞        xL .

Because of the infinitely high walls we must have that ψ(x) = 0 for x ≤ 0 and ψ(x) = 0 for xL.  From the time-independent Schrödinger equation in one dimension inside the well, 0 < x < L, we have that

−(ħ2/2m)(d2/dx2) ψ(x) = E ψ(x) .

The general solution is (we choose sines and cosines because of the boundary conditions, but complex exponentials are also solutions)

ψ(x) = Asin(kx) + Bcos(kx) ,

where k2 = 2mE/ħ2, but we still need to satisfy the boundary conditions, and determine A, B, and k (and if we know k we also have determined E).  For the wall on the left we need ψ(0) = 0, which means that B = 0, and hence that ψ(x) = Asin(kx). This solution is plotted in the animation.  Change the energy of the solution and note how the wave function shape changes.  For the wall on the right we need ψ(L) = 0 which sets k = nπ/L.

The algorithm used to calculate energy eigenfunctions is called the shooting method.  The shooting method calculates the wave function by numerically solving the time-independent Schrödinger equation.  The solution for the wave function starts with ψ(xleft) = 0 to approximate ψ(x = −∞) = 0, and then numerically solves the time-independent Schrödinger equation, calculating the energy eigenfunction from left to right. Note that all energy values solve the time-independent Schrödinger  equation, but only some of these are referred to as energy eigenvalues (eigen is German for proper or characteristic) which yield valid (proper) bound-state energy eigenfunctions. 

In the case of the present animation, only the energy values that correspond to ψ(L) = 0 are called energy eigenvalues.