- #1

JK423

Gold Member

- 394

- 7

**any**state can be written as a linear sum of the eigenstates.

But is the word 'any' correct? I haven't seen a proof of the above completeness theorem, so i don't know what the mathematical requirements are.

For example, consider the eigenstates of the Hamiltonian H

_{0}that correspond to an infinite square well [0,L]. These eigenstates vanish for x<0 and x>L. But that's a little restrictive! Because i cannot expand ANY state in this basis, for example those states that are also defined in the x<0, x>L region. Am i wrong?

One last question. If your Hamiltonian of your system is H=H

_{0}+V, where H

_{0}is a random Hamiltonian and V the potential. Since the eigenstates of H

_{0}form a complete set, then i can expand any state of the system in this basis, even though the system is described by H and not H

_{0}. Is this correct?