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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.

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# My bra-ket calcs seem to be going wrong - help!

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