There has been an example problem, where I asked you to show ``that if
(x, t) and
(x, t) are both solutions of the
time-dependent Schrödinger equation, than
(x, t) + (x, t)
is a solution as well.'' Let me review this problem
- (x, t) + V(x)(x, t)
=
(x, t)
- (x, t) + V(x)(x, t)
=
(x, t)
- [(x, t) + (x, t)] + V(x)[(x, t) + (x, t)]
=
[(x, t) + (x, t)],
(10.4)
where in the last line I have use the sum rule for derivatives. This is
called the superposition of solutions, and holds for any two solutions to the
same Schrödinger equation!
Question: Why doesn't it work for the time-independent Schrödinger equation?