Question

Write the matrix elements between the $n = 3$ states of the hydrogen atom with magnetic number $m = +1$ as $\langle \phi_{3,l,+1}, z \phi_{3,l',+1} \rangle = C_{ll'} \int_0^\infty dr F_{ll'}(r)$. Compute the angular part of the matrix element $C_{ll'} = \int d^2 \Omega Y_{l,+1}^* z Y_{l',+1}$ whenever is non-vanishing. Here $Y_{l,m}(\theta, \phi)$ are the standard spherical harmonics normalized to $\int d^2 \Omega Y_{l,m}^* Y_{l',m'} = \delta_{ll'} \delta_{mm'}$.

          Write the matrix elements between the $n = 3$ states of the hydrogen atom
with magnetic number $m = +1$ as
$\langle \phi_{3,l,+1}, z \phi_{3,l',+1} \rangle = C_{ll'} \int_0^\infty dr F_{ll'}(r)$.
Compute the angular part of the matrix element $C_{ll'} = \int d^2 \Omega Y_{l,+1}^* z Y_{l',+1}$
whenever is non-vanishing. Here $Y_{l,m}(\theta, \phi)$ are the standard spherical harmonics normalized to $\int d^2 \Omega Y_{l,m}^* Y_{l',m'} = \delta_{ll'} \delta_{mm'}$.

Write the matrix elements between the n = 3 states of the hydrogen atom
with magnetic number m = +1 as
⟨ϕ3,l,+1, z ϕ3,l',+1⟩ = Cll'∫0^∞ dr Fll'(r).
Compute the angular part of the matrix element Cll' = ∫ d^2 Ω Yl,+1^* z Yl',+1
whenever is non-vanishing. Here Yl,m(θ, ϕ) are the standard spherical harmonics normalized to ∫ d^2 Ω Yl,m^* Yl',m' = δll'δmm'.

Added by Roger S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Jacob Fry

11/30/2022

Step 1

Since Y_(1,+1) = sqrt(3/(8*pi)) * sin(theta) * e^(i*phi), we have Y_(1,+1)^{**} = sqrt(3/(8*pi)) * sin(theta) * e^(-i*phi). The integral becomes: C_(11) = \int d^{2}\Omega Y_(1,+1)^{**}\hat{z}Y_(1,+1) = \int sin(theta) d(theta) d(phi) * sqrt(3/(8*pi)) * Show more…

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Write the matrix elements between the n=3 states of the hydrogen atom with magnetic number m=+1 as (phi _(3,1,+1),zphi _(3,1,+1))=C_(11)int_0^{infty} drF_(11)(r). Compute the angular part of the matrix element C_(11)=int d^{2}Omega Y_(1,+1)^{**}hat{z}Y_(1,+1) whenever it is non-vanishing. Here Y_(l,m)( heta ,phi ) are the standard spherical harmonics normalized to int d^{2}Omega Y_(l,m)^{**}Y_(l',m')=delta_{ll'}delta_{mm'}. Write the matrix elements between the n = 3 states of the hydrogen atom with magnetic number m =+1 as (3,1,+1,z3,1,+1) = C1v drFu(r) whenever it is non-vanishing. Here Y_{l,m}( heta,phi) are the standard spherical harmonics normalized to int d^{2}Omega Y_{l,m}Y_{l',m'} = delta_{ll'}delta_{mm'}.