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Starting with,
[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
[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