8.E: Central Potentials (Exercises) — Read

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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 $n\neq m$.

This page titled 8.E: Central Potentials (Exercises) was last modified on Fri, 13 Mar 2026 03:53:14 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.

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