Phy5645/HydrogenAtomProblem

(Submitted by Team 6)

This problem taken from Quantum Mechanics: Concepts and Applications by Nouredine Zettili: Exercise 6.3

An electron in a hydrogen atom is in the energy eigenstate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{2,1,1} \left(r, \theta, \phi \right) = Nre^{-\frac{r} {2a_o}}Y_{1,-1}\left(\theta, \phi \right)} . (a) Find the normalization constant, N.

Take the volume integral of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\psi*} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{1,-1}\left(\theta, \phi \right) = \sqrt{\frac{3}{8\pi}}sin(\theta)e^{-i\phi} } and as such the phi dependence in the integral vanishes :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int _{\phi= 0}^{2\pi} \int_{\theta = 0}^{\pi} \int_{r=0}^{\infty} N^{2}r^{2}sin^{2}(\theta) \frac{3}{8\pi}e^{-\frac{r} {a_o}}r^{2}sin(\theta)drd\theta d\phi}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Longrightarrow \frac{3N^2}{8\pi} \int_{0}^{\pi} sin^{3}(\theta)d \theta \int_{0}^{2\pi} e^{-2i\phi}d\phi \int_{ 0}^{\infty}r^{4}e^{- \frac{r}{a_o}}dr = 1 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Longrightarrow \frac{3N^2}{8\pi} \frac{4}{3}(2\pi)(24a^5) = 1}

Therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^{2}\left(24a^{5}\right) = 1 } so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \sqrt\frac{1}{24a^5}}

(b)What is the probability per unit volume of finding the electron at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = a_{o}, \theta = 45^{\circ}, \phi = 60^{\circ}? }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\psi* = N^{2}r^{2}sin^{2}(\theta)\left(\frac{3}{8\pi}\right)e^{- \frac{r}{a_o}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Longrightarrow \left(\frac{1}{24{a_o}^5}\right) {a_o}^2 sin^{2}\left(\frac{\pi}{4}\right)\left(\frac{3}{8\pi}\right)e^{-1} = \left(\frac{\pi e^{-1}}{128{a_o}^3}\right) = \frac{0.009}{{a_o}^3}}

(c) What is the probability per unit radial interval (dr) of finding the electron at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a =a_{o} ? }

Average over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = 2a_{o}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| Y_{1,-1}\right|^2 = \left(\frac{3}{8\pi}\right)sin^{2}(\theta) = \left(\frac{3}{16\pi}\right)} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\psi* = N^{2}r^{2}e^{-\frac{r}{a}}s \left| Y_{1,-1}\right|^{2} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Longrightarrow \left(\frac{1}{24{a}^5}\right)(2a)^{2}e^{-\frac{2a} {a}}\left(\frac{3}{16\pi}\right) = \left(\frac{1}{32\pi a} \right)e^{-2} = \frac{0.0013}{a}}

(d) If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{ \hat L^2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{ \hat L_{z}}} are made, what will the results be?

l=1, m = -1 are the l and m of the eigenstate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{1,-1}(\theta, \phi)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat L^2 = \hbar^{2} l (l +1) = 2\hbar^{2} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat L_z = \hbar m = -\hbar }