Quantum revival

In quantum mechanics, the quantum revival is a periodic recurrence of the quantum wave function from its original form during the time evolution either many times in space as the multiple scaled fractions in the form of the initial wave function (fractional revival) or approximately or exactly to its original form from the beginning (full revival). The quantum wave function periodic in time exhibits therefore the full revival every period. The phenomenon of revivals is most readily observable for the wave functions being well localized wave packets at the beginning of the time evolution for example in the hydrogen atom. For Hydrogen the fractional revivals show up as multiple Gaussian bumps around the circle and the full revival as the original Gaussian . The full revivals are exact for the infinite quantum well, harmonic oscillator or the hydrogen atom, while for shorter times are approximate for hydrogen atom and a lot of quantum systems.

Example - arbitrary truncated wave function of the quantum system with rational energies

Consider a quantum system with the energies $E_{i}$ and the eigenstates $\psi _{i}$

$H\psi _{i}=E_{i}\psi _{i}$

and let the energies be the rational fractions of some constant $C$

$E_{i}=C{M_{i} \over N_{i}}$

(for example for hydrogen atom $M_{i}=1$, $N_{i}=i^{2}$, $C=-13.6eV$.

Then the truncated (till $\mathbb {N} _{max}$ of states) solution of the time dependent Schrödinger equation is

$\Psi (t)=\sum _{i=0}^{\mathbb {N} _{max}}a_{i}e^{-i{{E_{i}} \over \hbar }t}\psi _{i}$

Let $L_{cm}$ be to lowest common multiple of all $N_{i}$ then for each $N_{i}$ the ${L_{cm}}/N_{i}$ is an integer, $2\pi M_{i}{L_{cm}}/N_{i}$ is the full multiple of $2\pi$ angle and

$\Psi (t)=\Psi (t+T)$

after the full revival time time

$T={2\pi \hbar \over C}L_{cm}$.

For the quantum system as small as Hydrogen and $\mathbb {N} _{max}$ as small as 100 it may take quadrillions of years till it will fully revive.