Question #43706

In a theoretical statistical model of beta decay, the probability density of emitting a non-relativistic beta particle of mass $m$ with kinetic energy $E$ is given by $\rho_c(E) = A E \left(1 - \frac{E}{Q}\right)^n$, where $Q$ is the endpoint kinetic energy, $A$ is a normalization constant, and $n$ is a very large integer ($n \gg 1$) representing the phase space degrees of freedom.

To account for the wave nature of the beta particle, a quantum suppression factor $S(E)$ is introduced. $S(E)$ is defined as the ratio of the nuclear volume $V_0$ to the spherical volume $V_\lambda$ associated with the electron's de Broglie wavelength $\lambda_e$ (i.e., $V_\lambda = \frac{4}{3}\pi \lambda_e^3$).

The modified quantum probability density is $\rho_q(E) = B \rho_c(E) S(E)$, where $B$ is another normalization constant. Let $E_c$ and $E_q$ be the most probable kinetic energies of the beta particle in the classical and quantum models, respectively. What is the limiting value of the ratio $\frac{E_q}{E_c}$ for $n \to \infty$?