Registration is now open for this year's LibreFest! Join us virtually the week of July 13.

8.E: Central Potentials (Exercises)

Last updated
Save as PDF

Page ID: 15775

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

A particle of mass $m$ is placed in a finite spherical well:
\[V(r) = \left\{ \begin{array}{lcl} -V_0&\mbox{\hspace{1cm}}&\mbox{for $r\leq a$}\\ 0&&\mbox{for $r>a$} \end{array} \right. , \nonumber \]

with $V_0>0$ and $a>0$. Find the ground-state by solving the radial equation with $l=0$. Show that there is no ground-state if $V_0\,a^2< \pi^2\,\hbar^2/(8\,m)$.
Consider a particle of mass $m$ in the three-dimensional harmonic oscillator potential $V(r)=(1/2)\,m\,\omega^2\,r^2$. Solve the problem by separation of variables in spherical coordinates, and, hence, determine the energy eigenvalues of the system.
The normalized wavefunction for the ground-state of a hydrogen-like atom (neutral hydrogen, ${\rm He}^+$, ${\rm Li}^{++}$, et cetera.) with nuclear charge $Z\,e$ has the form
\[\psi = A\,\exp(-\beta\,r), \nonumber \]

where $A$ and $\beta$ are constants, and $r$ is the distance between the nucleus and the electron. Show the following:
1. $A^2=\beta^{\,3}/\pi$.
2. $\beta = Z/a_0$, where $a_0=(\hbar^2/m_e)\,(4\pi\,\epsilon_0/e^2)$.
3. The energy is $E=-Z^2\,E_0$ where $E_0 = (m_e/2\,\hbar^2)\,(e^2/4\pi\,\epsilon_0)^2$.
4. The expectation values of the potential and kinetic energies are $2\,E$ and $-E$, respectively.
5. The expectation value of $r$ is $(3/2)\,(a_0/Z)$.
6. The most probable value of $r$ is $a_0/Z$.
An atom of tritium is in its ground-state. Suddenly the nucleus decays into a helium nucleus, via the emission of a fast electron that leaves the atom without perturbing the extranuclear electron, Find the probability that the resulting ${\rm He}^+$ ion will be left in an $n=1$, $l=0$ state. Find the probability that it will be left in a $n=2$, $l=0$ state. What is the probability that the ion will be left in an $l>0$ state?
Calculate the wavelengths of the photons emitted from the $n=2$, $l=1$ to $n=1$, $l=0$ transition in hydrogen, deuterium, and positronium.
To conserve linear momentum, an atom emitting a photon must recoil, which means that not all of the energy made available in the downward jump goes to the photon. Find a hydrogen atom’s recoil energy when it emits a photon in an $n=2$ to $n=1$ transition. What fraction of the transition energy is the recoil energy?
Show that the most probable value of $r$ in the hydrogen ground state is $a_0$.
Let $R_{nl}(r)= v_{nl}(r/a_0)/(r/a_0)$, where $R_{nl}(r)$ is a properly normalized radial hydrogen wavefunction corresponding to the conventional quantum numbers $n$ and $l$, and $a_0$ is the Bohr radius. [ex4 .fgh]
1. Demonstrate that
  \[\frac{d^2v_{nl}}{dy^2} = \left[\frac{l\,(l+1)}{y^2} -\frac{2}{y} + \frac{1}{n^2}\right]v_{nl}. \nonumber \]
2. Show that $v_{nl}\sim y^{\,1+l}$ in the limit $y\rightarrow 0$.
3. Demonstrate that
  \[\left(\frac{1}{n^2}-\frac{1}{m^2}\right)\int_0^\infty v_{nl}(y)\,v_{ml}(y)\,dy = 0. \nonumber \]
4. Hence, deduce that
  \[\int_0^\infty r^2\,R_{nl}(r)\,R_{ml}(r)\,dr=0 \nonumber \]
  
  for $n\neq m$.

Search

Text Color

Text Size

Margin Size

Font Type