Let
be a bound state of a given potential with energy
and
be another bound state with energy
such that
. Let
be a point at which both
and
vanish; this is guaranteed to happen at least for
We will now prove that, between any two points
and
at which
vanishes, there must be at least one point at which
vanishes. Let us begin by writing down the Schrödinger equation for each wave function:


Multiplying the first equation by
and the second by
subtracting the second equation from the first, and simplifying, we see that

If we now integrate both sides of this equation from
to any position
and simplify, we see that

The key is to now let
be the first position to the right of
where
vanishes.

Now, if we assume that
does not vanish at or between
and
, then it is easy to see that the left hand side of the previous equation has a different sign from that of the right hand side, and thus it must be true that
must vanish at least once between
and
if