I was watching a Lecture by Douglas Hofstadter on "Albert Einstein on Light; Light on Albert Einstein". And there was a slide which said that Einstein had idea that
All periodic phenomena should have quantized energy levels.
Can some one explain such idea? Thanks.
"Quantisation" does not actually arise from principle of Quantum Mechanics. Nowhere in the postulates of QM you see that the energy levels are quantised. The quantisation of energy levels arise because of the boundary conditions you impose to solve the Schrodinger Equation. Furthermore, not all systems are quantised (e.g free particle has continuous energy spectra).
To see a classical "quantisation" of spectrum, consider a taut string that is fixed by both ends (say at $x = 0$ and $x = L$). We also impose that the string cannot displace in both ends. This is essentially a boundary condition that imposes $\psi(0,t) = \psi(L,t) = 0$, where $\psi(x,t)$ denotes the vertical displacement of the string. If we are to have waves that in this string, they need to obey the appropriate boundary conditions. Suppose a wave is given by
$$ \psi(x,t)\sim \sin(kx)\sin(\omega t)$$
The boundary conditions imply
$$ \sin(kL) = 0$$
which means that $k = \frac{n\pi}{L}$. Therefore the allowed solutions must have quantised wavelengths. This is true for any valid wave that travels in this rope.
Hope this helps.
It is a general consideration, there is no specific paper. The Fourier transform of a periodic function is discrete. If the wavefunction evolves periodically in time, there must be a discrete energy spectrum.
The paper you want to read is by Sir Nevill Mott:
Mott, Sir Nevill. "On teaching quantum phenomena." Contemporary Physics 5.6 (1964): 401-418
wherein Mott argues that it is an experimental fact that "quantization applies to any movement of particles within a confined space, or any periodic motion, but not to unconfined motion."
Confined motion implies boundary conditions. For instance, since $\psi$ must eventually be $0$ well outside the region where the particle is confined, only some solutions of the Schrodinger equation (usually labelled by discrete indices) will eventually be $0$ in the correct way.
One should be careful here about the use of periodic. It may be that the radial motion (motion in time of the radius) is periodic with a period $T_r$ and that the angular motion (motion in time of the angle $\theta$) is periodic with a period $T_\theta$. This occurs generally in central force problems. However, if the periods are not commensurate, i.e. if $T_r/T_\theta$ is not the ratio of two integers, then the motion in $(r,\theta)$ space will never exactly repeat itself, even if the motions in $r$ and $\theta$ are individually periodic. In your case, I believe "periodic" refers to the periodic motion of the individual variables.
A phenomenon periodic in time clearly has discrete frequencies/energies, cf. the theory of Fourier series.
However, Douglas Hofstadter/Albert Einstein are referring to phenomena periodic in space. A periodic space$^1$ can be viewed as a compact space. A compact region of phase space has a finite number of number of states, cf. e.g. my related Phys.SE answer here.
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$^1$ Here we assume for simplicity that the spatial boundary conditions (BCs) are periodic as well. If the BCs are different (say instead Dirichlet BCs, as one would expect of e.g. a realistic crystal of finite volume), then the above analysis is limited to bulk properties that are not affected by the BCs.
