Restricted Boltzmann machine

Full and exact revival of the semi-Gaussian wave function in an infinite two-dimensional potential well during its time evolution. In between the fractional revivals occur when the scaled shape of the wave function replicates itself integer number of times over the well area.

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 $C$

$E_i=C\frac{M_i}{N_i}$

(for example for hydrogen atom $M_i=1$, $N_i=i^2$, $C=-13.6eV$.

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

$\mathrm{\Psi }(t)=\sum _{i=0}^{{\mathbb{\mathbb{N}}}_{max}}{a}_i{e}^{-i\frac{E_i}{\hslash }t}{\psi }_i$

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

$\mathrm{\Psi }(t)=\mathrm{\Psi }(t+T)$

after the full revival time time

$T=\frac{2\pi \hslash }{C}{L}_{cm}$.

For the quantum system as small as Hydrogen and ${\mathbb{\mathbb{N}}}_{max}$ as small as 100 it may take quadrillions of years till it will fully revive. Especially once created by fields the Trojan wave packet in a hydrogen atom exists without any external fields stroboscopically and eternally repeating itself after sweeping almost the whole hypercube of quantum phases exactly every full revival time.

The striking consequence is that no finite-bit computer can propagate the numerical wave function accurately for the arbitrarily long time. If the processor number is n-bit long floating point number then the energy is for example 2.34576893 = 234576893/100000000 and is exactly rational and the full revival occurs for any wave function of any quantum system after the time $t/2\pi =100000000$ which is its maximum exponent and so on that may not be true for all quantum systems or all stationary quantum systems undergo the full and exact revival numerically.

In the system with the rational energies i.e. where the quantum exact full revival exists its existence immediately proves the quantum Poincaré recurrence theorem and the time of the full quantum revival equals to the Poincaré recurrence time. While the rational numbers are dense in real numbers and the arbitrary function of the quantum number can be approximated arbitrarily exactly with Padé approximants with the coefficients of arbitrary decimal precision for the arbitrarily long time each quantum system therefore revives almost exactly.

Despite of the general theory of the relativity and the cosmological models if there is a universal wavefunction of the universe quantum theory predicts therefore the universe evolution as the repetitive and infinite appearance of the big bang and big crunch (Big bounce and cyclic model) and so on as the global Poincaré recurrence.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.