0
$\begingroup$

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

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ "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? $\endgroup$
    – Sean E. Lake
    Commented Oct 2, 2022 at 20:07
  • 1
    $\begingroup$ 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. $\endgroup$
    – Tobias Fünke
    Commented Oct 2, 2022 at 20:17
  • $\begingroup$ Reminder that answers should be posted as answers $\endgroup$
    – BioPhysicist
    Commented Oct 2, 2022 at 20:19

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.