We like to think of the (E ightarrow infty) limit as the limit in which quantum

We like to think of the \(E \rightarrow \infty\) limit as the limit in which quantum mechanics "turns into" classical mechanics, but this clearly has limitations. The limitation that we will consider here is the fact that if the energy density of a particle's wavefunction is too high, then it will create a black hole. For a total energy \(E\), a black hole is created if this is packed into a region smaller than its Schwarzschild radius, \(R_{s} \cdot{ }^{3}\) For energy \(E\), the Schwarzschild radius is

\[\begin{equation*}R_{S}=\frac{2 G_{N} E}{c^{4}} \tag{5.77}\end{equation*}\]

where \(G_{N}\) is Newton's constant and \(c\) is the speed of light.

(a) For a general energy eigenstate \(\left|\psi_{n}\rightangle\) of the infinite square well, determine its Schwarzschild radius. For what value of energy level \(n\) does the Schwarzschild radius equal the size \(a\) of the infinite square well? In this part, you can leave the answer in terms of the constants provided in the problem.

(b) What is the energy \(E_{n}\) for which the size of the well is the Schwarzschild radius? Evaluate this for a well that's the size of an atomic nucleus, \(a=10^{-15}\) m. Compare this energy to some "everyday" object's energy (something like the kinetic energy of a thrown ball, the energy of photons from the sun, etc.).

(c) Using part (a), what energy eigenstate level \(n\) does this correspond to? Take the mass \(m\) of the object to be that of the pion, a sub-atomic particle responsible for binding atomic nuclei. The mass of the pion \(m_{\pi}\) is
\[\begin{equation*}m_{\pi}=2.4 \times 10^{-28} \mathrm{~kg} . \tag{5.78}\end{equation*}\]
How does this compare to the energy level \(n\) where you would predict that the pion would be traveling at the speed of light, \(c\) ? You can use the approximation that \(\hbar=10^{-34} \mathrm{~J} \cdot \mathrm{s}\).