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# 8.E: Central Potentials (Exercises)

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

7. Show that the most probable value of $$r$$ in the hydrogen ground state is $$a_0$$.

8. 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 $\frac{d^{\,2}v_{nl}}{dy^{\,2}} = \left[\frac{l\,(l+1)}{y^{\,2}} -\frac{2}{y} + \frac{1}{n^{\,2}}\right]v_{nl}.$

2. Show that $$v_{nl}\sim y^{\,1+l}$$ in the limit $$y\rightarrow 0$$.

3. Demonstrate that $\left(\frac{1}{n^{\,2}}-\frac{1}{m^{\,2}}\right)\int_0^\infty v_{nl}(y)\,v_{ml}(y)\,dy = 0.$

4. Hence, deduce that $\int_0^\infty r^{\,2}\,R_{nl}(r)\,R_{ml}(r)\,dr=0$ for $$n\neq m$$.

# Contributors

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