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Let's say I have a wavefunction $\Psi(x,t) = A e^{i(kx−ωt)}$. Now I complex conjugate it which gives me $\Psi^∗(x,t)$.

My first question is: Does $\Psi^∗(x,t)$ live in the dual of a Hilbert space?

My second question is: Can I see $\Psi^∗(x,t)$ as a linear functional which can be applied on wavefunction $\Psi(x,t)$ in order to get a number? Thanks

BioPhysicist
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Tomáš Kubalík
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    "Does the Ψ∗(x,t) live in the dual of a Hilbert space?" Almost. It's linear w.r.t. real scalars, but not w.r.t. complex ones. "Can I see Ψ∗(x,t) as a linear functional which can be applied on wave function Ψ(x,t) and I get a number?" Isn't that the definition of the dual space? – Sean E. Lake Oct 02 '22 at 20:07
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    You can define an anti-linear map $C:L^2\longrightarrow L^2$ by $C:\psi \mapsto \bar\psi$, cf. here. But note that on a general Hilbert space complex conjugation is not canonical, as discussed here. – Tobias Fünke Oct 02 '22 at 20:17
  • Reminder that answers should be posted as answers – BioPhysicist Oct 02 '22 at 20:19

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