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# Quantum Mechanics I ## Problem Set 6 ### Problem 1 Consider a particle of mass $\mu$ in a central potential $V(r) = - \frac{k}{r} + \frac{g}{r^3}$ where $k, g > 0$. * (a) Find the energy eigenvalues $E_{n,l}$ for the bound states of this potential * \(b) Accurately sketch the effective...

# Quantum Mechanics I ## Problem Set 6 ### Problem 1 Consider a particle of mass $\mu$ in a central potential $V(r) = - \frac{k}{r} + \frac{g}{r^3}$ where $k, g > 0$. * (a) Find the energy eigenvalues $E_{n,l}$ for the bound states of this potential * \(b) Accurately sketch the effective potential for a few values of $l\). * \(c\) Find the radius $r_*$ of the circular orbit. ### Problem 2 Consider the scattering of a particle of mass $m$ off an impenetrable sphere of radius $a$, $V(r) = \begin{cases} \infty & r \leq a \\ 0 & r > a \end{cases}$ * \(a\) Calculate the $s$-wave phase shift $\delta_0(k)$. * \(b\) Calculate the $s$-wave scattering cross section $\frac{d \sigma}{d \Omega}$ and the total cross section $\sigma$. * \(c\) At what energies does the total cross section $\sigma$ reach its maximum value? ### Problem 3 Consider the scattering of a particle of mass $m$ off the potential $V(r) = V_0 \theta(a-r) = \begin{cases} V_0 & r \leq a \\ 0 & r > a \end{cases}$ * \(a\) Calculate the $s$-wave phase shift $\delta_0(k)$. * \(b\) Determine the condition for $V_0$ such that there is no scattering at zero energy. ### Problem 4 Using the method of partial waves, calculate the differential cross section $\frac{d \sigma}{d \Omega}$ for scattering from the potential $V(r) = \frac{\alpha}{r^2}$ where $\alpha$ is a constant. **Hint:** *You may find it useful to express the radial equation in terms of dimensionless variables.*