8.E: Central Potentials (Exercises)

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A particle of mass $m$ is placed in a finite spherical well: \begin{equation}V(r)=\left\{\begin{array}{ll}
-V_{0} & \text { for } r \leq a \\
0 & \text { for } r>a
\end{array}\right.\end{equation} 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 \[\psi = A\,\exp(-\beta\,r),\] 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?

Contributors and Attributions

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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