Quantum revival

In quantum mechanics, quantum revival is a periodic recurrence of the quantum wave function during its time-evolution. This can be either many times in space as multiple scaled copies of the initial wave function (fractional revival), or approximately or exactly to its original form (full revival). A quantum wave function that is periodic in time therefore exhibits a full revival every period. The phenomenon of revival is most readily observable in wave functions that are well-localized wave packets at the beginnings of their time-evolutions, such as in the hydrogen atom. For hydrogen, fractional revivals show up as multiple angular Gaussian bumps around the circle drawn by the radial maximum of the leading circular-state component (that with the highest amplitude in the eigenstate expansion) of the original localized state, and the full revival as the original Gaussian. Full revivals are exact for the infinite quantum well, harmonic oscillator, or hydrogen atom, while for shorter times are approximate for the hydrogen atom and many other quantum systems.

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

Consider a quantum system with the energies

E

i

{\displaystyle E_{i}}

and the eigenstates

ψ

i

{\displaystyle \psi _{i}}

H

ψ

i

=

E

i

ψ

i

,

{\displaystyle H\psi _{i}=E_{i}\psi _{i},}

and let the energies be the rational fractions of some constant

C

{\displaystyle C}

:

E

i

=

C

M

i

N

i

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

(for example, for the hydrogen atom,

M

i

=

1

{\displaystyle M_{i}=1}

,

N

i

=

i

2

{\displaystyle N_{i}=i^{2}}

, and

C

=

−

13.6

e

V

{\displaystyle C=-13.6eV}

).

Then the truncated (till

N

m

a

x

{\displaystyle \mathbb {N} _{max}}

of states) solution of the time-dependent Schrödinger equation is

Ψ

(

t

)

=

∑

i

=

0

N

m

a

x

a

i

e

−

i

E

i

ℏ

t

ψ

i

.

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

Let

L

c

m

{\displaystyle L_{cm}}

be the least common multiple of all

N

i

{\displaystyle N_{i}}

, and

L

c

d

{\displaystyle L_{cd}}

be the greatest common divisor of all

M

i

{\displaystyle M_{i}}

. Then for each

N

i

{\displaystyle N_{i}}

, the quantity

L

c

m

/

N

i

{\displaystyle {L_{cm}}/N_{i}}

is an integer, for each

M

i

{\displaystyle M_{i}}

the quantity

M

i

/

L

c

d

{\displaystyle {M_{i}}/L_{cd}}

is an integer,

2

π

M

i

L

c

m

/

(

N

i

L

c

d

)

{\displaystyle 2\pi M_{i}{L_{cm}}/(N_{i}L_{cd})}

is the full multiple of

2

π

{\displaystyle 2\pi }

angle, and

Ψ

(

t

)

=

Ψ

(

t

+

T

)

{\displaystyle \Psi (t)=\Psi (t+T)}

after the full revival time

T

=

2

π

ℏ

L

c

d

C

L

c

m

{\displaystyle T={2\pi \hbar \over {L_{cd}C}}L_{cm}}

.