How can we make the energy levels of an one-dimensional infinite square well potential equispaced? We know that the energy values for an one-dimensional infinite square well potential is given by $$E_n = \frac{n^2{\pi}^2{\hbar}^2}{2ma^2}$$ where $a$ is the width of the well. Now, as we can see that the difference in energy of two consecutive levels is given by $$E_{n+1} - E_n = (2n+1)E_1.$$
In my PhD interview, it was asked that how can we make these energy levels equispaced?
I know that the quantum harmonic oscillator energy levels are equispaced, but how can we make the levels equispaced for one-dimensional infinite square well potential? Is this related to Bohr Correspondence Principle somehow?
 A: They just wanted you to use the formula for the energy and adapt it very likely. This means if you know the formula for $E_n$, you can see that a cuadratic dependence on $n$ will not lead to equispaced energy levels. So which parameter could you play with so that $E_n$ does not scale cuadratically with $n$?
You might think of $m$ or $a$ but, let us say the particle you are given has a fixed mass, so the only thing remaining is $a$, you could then require that  $a$ is not constant and you make it such that $a^2 \propto n$ then
$$\Delta E_n = E_{n+1}-E_n = \frac{\pi^2\hbar^2}{2m}.$$
This turns out is achieved by the potential not being a square well, but a parabolic well, i.e. a harmonic oscillator.
A: the question is not very well defined because it is not clear what we are allowed to change. One thing I can think of is if we make the particle massless, then the energy disperssion becomes linear. We don't have $E_p \propto p^2$ but rather $E_p \propto p$. So if we have $H=\sqrt{c^2p^2 + m^2c^4} + V(x)$ and set $m=0$ then we get the same eigenstates of the standard version (sine and cosine) but the energy is $E \propto n$, making it linear and equally spaced.
