How does the difference in energy levels decrease as you increase $n$ in 1D particle in a box (correspondence principle)? I found from multiple sources that the difference between energy levels must decrease as you increase quantum numbers in order to follow the correspondence principle. The energy of a particle in the box increases proportionally to $n$-squared. This means that the energy spacings increase at high values of $n$. How does the correspondence principle is applicable with respect to energy level differences?
 A: The idea that the difference between energy levels must decrease as you increase the quantum number in order to recover the correspondence principle must be wrong: if it were true, then the quantum harmonic oscillator could not exist in the physical world. What we may require is the fact that the relative energy difference must decrease as a function of $n$. Relative to what? Well, relative to the macroscopic energy of the system $E(n)$. We therefore postulate:
$$
\lim_{n\rightarrow\infty} \Delta_r(n)=\lim_{n\rightarrow\infty} \frac{E(n+1)-E(n)}{E(n)}=0.
$$
If this is true, then for high $n$ (classical limit) increasing the energy quantum number does not produce an appreciable energy difference, at least if compared to the macroscopic energy $E(n)$ we experience. In other words, we lose track of the discreteness of the energy levels of $E(n)$.
Note that for the case of the particle in a 1D box we have $E(n)=\frac{n^2\pi^2\hbar^2}{2mL^2}$, where $m$ is the particle mass and $L$ the box length. Hence, $\Delta_r(n)=(2n+1)/n^2$, which correctly vanishes when $n$ goes to infinity. In contrast, a physical system with energy dependence $E(n)= E_0 e^n$ would display quantum effects in the macroscopic world, since $\Delta_r(n)=e-1$. As far as I know, however, we still haven't found such a quantum system and the classical limit is always recovered.
