$$\require{cancel}$$

# 8.E: Central Potentials (Exercises)


1. A particle of mass $$m$$ is placed in a finite spherical well: V(r)=\left\{\begin{array}{ll}
-V_{0} & \text { for } r \leq a \\
0 & \text { for } r>a
\end{array}\right. 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 $$V_0\,a^{\,2}< \pi^{\,2}\,\hbar^{\,2}/(8\,m)$$.
2. 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.
3. 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 $$a_0/Z$$.
4. 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?
5. Calculate the wavelengths of the photons emitted from the $$n=2$$, $$l=1$$ to $$n=1$$, $$l=0$$ transition in hydrogen, deuterium, and positronium.
6. 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?
