The following sums are guaranteed to be equal:

Eigenfunction sum:

\frac{1}{2π} ∑_{k=−∞}^{∞} e^{ikx}e^{−ik^{2}t}

Image sum:

\frac{1}{\sqrt{4πit}} ∑_{n=−∞}^{∞} e^{i(x−2πn)^{2}/4t}

(What these functions have to do with the Schrödinger equation will be explained early in our course.)

To evaluate the sums numerically, we have to replace ±∞ with some large integers ±N. We plot both sums and increase N until the two curves almost agree, to be sure that we have taken enough terms.

Also, to make sure that the sums converge, we give t a small negative imaginary piece, −ip. This makes the terms in the sum decrease exponentially at large k and n. We are interested in the limit when p → 0.

We will plot just the real part of the function, which reduces to the cosine sums on the Web page. The first part of the Maple session holds t fixed at various values and plots the resulting function of x. The second part of the Maple session (yielding the graphs on the Web page) sets x = 0 and plots the resulting function of the real part of t, with the imaginary part (the cutoff parameter) fixed at some small negative value.