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[itex]\hat{X}\psi = x\psi[/itex]

then,

[itex]x\psi = x\psi[/itex]

[itex]\psi = \psi[/itex]

So the eigenfunctions for this operator can equal anything (as long as they keep [itex]\hat{X}[/itex] linear and Hermitian), right?

Well, McQuarrie says that "the eigenfunctions of a Hermitian operator are orthogonal", which can be checked with:

[itex]\int_{-\infty}^{\infty}\psi^*_m \psi_n\, dx = \langle m | n \rangle = 0[/itex]

But if the eigenfunctions can be anything, then that integral won't always equal zero. What am I missing here?

Thanks