“So,” Herr Schrodinger says to us, “I’m looking for a function
of x. It needs to be equal to the negative of its second
derivative, up to a constant factor. When x is zero, the
function itself should be zero. And when x equals L for some
constant L greater than zero, the function also needs to be
zero. Can you find me such a function?” Sure we can. As you
know, the derivative of a function is its rate of change. Its
second derivative is the rate of change of the rate of change.
There are functions which are equal to their derivative (and
second derivative), e^x is one of them. But we’re looking for a
function which is equal to the negative of its own second
derivative. There’s two of them, the sine and the cosine
function. And since the sum of a derivative is the derivative of
the sum, we know the answer to Schrodinger’s problem is:
Differentiate that sucker twice, and you’ll sure enough get the
negative of the very same function back, times an overall
constant which happens to be the square of k. We have to