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 Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...

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

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

dependent

Hamiltonian reads :

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

It is well known that an

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

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 transformation 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 precisely, ...

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 linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...

for the time dependent

. The first exponential in the transformation is the so-called

squeeze operator In quantum physics, the squeeze operator for a single mode of the electromagnetic field is :\hat(z) = \exp \left ( (z^* \hat^2 - z \hat^) \right ) , \qquad z = r \, e^ where the operators inside the exponential are the ladder operators. It is a ...

. This approach may allow to simplify problems such as the

Quadrupole ion trap 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 the device and shared the N ...

, 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.

References

{{DEFAULTSORT:Ermakov-Lewis invariant Quantum mechanics