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I have a notational problem, I know when you define bra and ket you are defining an inner product, but you can see it as an linear operation where the linear operators (bras) act on vectors (kets), but of the same form you can think of the kets as operator over bras but in this case this operators are antilinear, I would think this operator would have to be linear as well.

Therefore I have a doubt there, why do i need to use in this case an antilinear operator? Probably the answer is the inner product is always positive or equal zero given the object (probability) is always positive or equal zero and I use the inner product to calculate this magnitude, which it's really the only thing important here.

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  • $\begingroup$ I can't argue with your last line, but I wonder would this be considered a mathSE post, rather than physics. $\endgroup$
    – user108787
    Commented Aug 7, 2016 at 19:43
  • $\begingroup$ Related: physics.stackexchange.com/q/43069/2451 , physics.stackexchange.com/q/45227/2451 , physics.stackexchange.com/q/216846/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented Aug 7, 2016 at 19:49
  • $\begingroup$ What structure does the inner product of a Hilbert space used in QM have? Proceed from there ... $\endgroup$
    – Sanya
    Commented Aug 7, 2016 at 21:07
  • $\begingroup$ $<a,b>=<b,a>^{*}$ $\endgroup$
    – 7919
    Commented Aug 7, 2016 at 22:28
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    $\begingroup$ You just answered your question yourself ;) The inner product gives you the antilinearity ... $\endgroup$
    – Sanya
    Commented Aug 8, 2016 at 8:06

1 Answer 1

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Bras are linear operators from kets to scalars, since the bra $\left\langle a\right |$ sends the linear combination $\lambda\left|b\right\rangle+\mu\left|c\right\rangle$ to $\lambda\left\langle a,b\right\rangle+\mu\left\langle a,c\right\rangle$.

Kets also act linearly on bras: the ket $\left| a\right\rangle$ sends the linear combination $\lambda\left\langle b\right|+\mu\left\langle c\right|$ to $\lambda\left\langle b,a\right\rangle+\mu\left\langle c,a\right\rangle$.

But the transformation that turns a ket into its corresponding bra ($\left| a\right\rangle\mapsto\left\langle a\right|$) is antilinear: it sends $\lambda\left| a\right\rangle$ to $\lambda^*\left\langle a\right |$.

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