- #1

JALAJ CHATURVEDI

- 2

- 0

<ψ|Qψ>=<Qψ|ψ>

But what about dr, dphi and d theta

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- Thread starter JALAJ CHATURVEDI
- Start date

- #1

JALAJ CHATURVEDI

- 2

- 0

<ψ|Qψ>=<Qψ|ψ>

But what about dr, dphi and d theta

- #2

Cryo

Gold Member

- 173

- 74

Normally, for full 3d spherical polars ##\langle \psi_1 | \psi_2 \rangle = \int d^3 r \psi_2^{*} \left(\boldsymbol{r}\right)\psi_1 \left(\boldsymbol{r}\right)=\int_0^\infty r^2 dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} \psi_2^{*} \left(r,\,\theta,\,\phi\right)\psi_1 \left(r,\,\theta,\,\phi\right)##

It therefore makes sense to define the inner product for functions of ##\phi## as ##\langle \Phi_1 | \Phi_2\rangle=\int_0^{2\pi} d\phi \Phi_1^{*}\left(\phi\right)\Phi_2\left(\phi\right)##. Now an operator ##\hat{O}## on this Hilbert space (function of ##\phi## in the domain ##0\dots2\pi##) is self-adjoint if:

##\int_0^{2\pi} d\phi \Phi_1^{*}\left(\phi\right)\hat{O}\Phi_2\left(\phi\right)=\int_0^{2\pi} d\phi \Phi_2\left(\phi\right) \hat{O} \Phi_1^{*}\left(\phi\right)##

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