Many quantum mechanical

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

s are time dependent. Methods to solve problems where there is an explicit time dependence is an open subject nowadays. It is important to look for constants of motion or invariants for problems of this kind. For the (time dependent)

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...

it is possible to write several invariants, among them, the Ermakov–Lewis invariant which is developed below. The

time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

dependent

Hamiltonian reads :

\hat =\frac\left hat^2+\Omega^2(t)\hat^2\right

It is well known that an invariant for this type of interaction has the form :

\hat=\frac\left \left( \frac\right)
^+(\rho\hat-\dot\hat)^\right

where

\rho

obeys the Ermakov equation :

\ddot+\Omega^\rho=\rho^.

The above invariant is the so-called Ermakov–Lewis invariant. It is easy to show that

\hat

may be related to the time independent harmonic oscillator Hamiltonian via a

unitary transformation In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precise ...

of the form :

\hat=e^e^=
e^
e^,

as :

\hat\hat\hat^.

This allows an easy form to express the solution of the

Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...

for the time dependent

. The first

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: * Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value * Ex ...

in the transformation is the so-called squeeze operator. This approach may allow to simplify problems such as the

Quadrupole ion trap In experimental physics, a quadrupole ion trap or paul trap is a type of ion trap that uses dynamic electric fields to trap charged particles. They are also called radio frequency (RF) traps or Paul traps in honor of Wolfgang Paul, who invented ...

, where an ion is trapped in a harmonic potential with time dependent frequency. The transformation presented here is then useful to take into account such effects. The geometric meaning of this invariant can be realized within the quantum phase space.

History

It was proposed in 1880 by Vasilij Petrovich Ermakov (1845-1922). The paper is translated in. In 1966, Ralph Lewis rediscovered the invariant using Kruskal's asymptotic method. He published the solution in 1967.

References

{{DEFAULTSORT:Ermakov-Lewis invariant Quantum mechanics