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What am I doing wrong here?
Let [itex]\psi[/itex] be a ket whose representation in the X basis is given by
[itex]\psi(x)\ =\ \langle x|\psi\rangle\ =\ e^{-x^{2}/2}[/itex]
Then
[itex]\psi(-x)\ =\ \langle -x|\psi\rangle\ =\ e^{-x^{2}/2}\ = \psi(x)[/itex] (1)
But we also have:
[itex]\psi(-x)\ =\ \langle -x|\psi\rangle[/itex] (2)
[itex]\ =\ \langle (-1)\times x)|\psi\rangle[/itex] (3), by the linearity of the inner product
[itex]\ =\ (-1)^*\times\langle x|\psi\rangle[/itex] (4)
[itex]\ =\ -\langle x|\psi\rangle[/itex] (5)
[itex]\ = -\psi(x)[/itex] (6)
and this contradicts (1).
I must have gone wrong here somewhere. I think it might be in (2) or (3). But I can't see the problem.
Thank you very much for any help.
Let [itex]\psi[/itex] be a ket whose representation in the X basis is given by
[itex]\psi(x)\ =\ \langle x|\psi\rangle\ =\ e^{-x^{2}/2}[/itex]
Then
[itex]\psi(-x)\ =\ \langle -x|\psi\rangle\ =\ e^{-x^{2}/2}\ = \psi(x)[/itex] (1)
But we also have:
[itex]\psi(-x)\ =\ \langle -x|\psi\rangle[/itex] (2)
[itex]\ =\ \langle (-1)\times x)|\psi\rangle[/itex] (3), by the linearity of the inner product
[itex]\ =\ (-1)^*\times\langle x|\psi\rangle[/itex] (4)
[itex]\ =\ -\langle x|\psi\rangle[/itex] (5)
[itex]\ = -\psi(x)[/itex] (6)
and this contradicts (1).
I must have gone wrong here somewhere. I think it might be in (2) or (3). But I can't see the problem.
Thank you very much for any help.