In the discussion on formal aspects of quantum mechanics I have shown that the eigenfunctions to the Hamiltonian are complete, i.e., for any ψ(x,t)
|
ψ(x,t) ={ \mathop{∑
}}_{n=1}^{∞}{c}_{
n}(t){ϕ}_{n}(x),
| (10.5) |
where
|
\hat{H}{ϕ}_{n}(x) = {E}_{n}{ϕ}_{n}(x).
| (10.6) |
We know, from the superposition principle, that
|
ψ(x,t) ={ \mathop{∑
}}_{n=1}^{∞}{c}_{
n}(0){e}^{−iEt∕ℏ}{ϕ}_{
n}(x),
| (10.7) |
so that the time dependence is completely fixed by knowing c(0) at time t = 0 only! In other words if we know how the wave function at time t = 0 can be written as a sum over eigenfunctions of the Hamiltonian, we can then determibe the wave function for all times.