Ermakov–Lewis invariant

In quantum mechanics, the Ermakov–Lewis invariant is a conserved quantity used to analyze explicitly time-dependent systems, especially the time-dependent harmonic oscillator. Because many quantum Hamiltonians are time-dependent, methods that identify constants of motion or invariants are central; the Ermakov–Lewis invariant provides such an integral of motion and underpins exact or approximate solutions. It is one of several invariants known for this system.

Let ${\displaystyle \Omega :\mathbb {R} \to \mathbb {R} }$ be a function. It defines a time dependent harmonic oscillator Hamiltonian reads

${\displaystyle {\hat {H}}={\frac {1}{2}}\left[{\hat {p}}^{2}+\Omega ^{2}(t){\hat {q}}^{2}\right].}$

The Ermakov–Lewis invariant for this type of interaction is

${\displaystyle {\hat {I}}={\frac {1}{2}}\left[\left({\frac {\hat {q}}{\rho }}\right)^{2}+(\rho {\hat {p}}-{\dot {\rho }}{\hat {q}})^{2}\right],}$

where ${\displaystyle \rho :\mathbb {R} \to \mathbb {R} }$ is a solution to the Ermakov equation

${\displaystyle {\ddot {\rho }}+\Omega ^{2}\rho =\rho ^{-3}.}$

${\displaystyle {\hat {I}}}$ is a unitary transformation of the time independent harmonic oscillator Hamiltonian:$${\displaystyle {\frac {1}{2}}\left[{\hat {p}}^{2}+{\hat {q}}^{2}\right]={\hat {T}}{\hat {I}}{\hat {T}}^{\dagger },\quad {\hat {T}}=e^{i{\frac {\ln \rho }{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}e^{-i{\frac {\dot {\rho }}{2\rho }}{\hat {q}}^{2}}=e^{i{\frac {\ln \rho }{2}}{\frac {d{\hat {q}}^{2}}{dt}}}e^{-i{\frac {{\hat {q}}^{2}}{2}}{\frac {d\ln \rho }{dt}}},}$$This allows an easy form to express the solution of the Schrödinger equation for the time dependent Hamiltonian. The exponential term ${\displaystyle e^{i{\frac {\ln \rho }{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}}$ is a squeeze operator. The other exponential term ${\displaystyle e^{-i{\frac {\dot {\rho }}{2\rho }}{\hat {q}}^{2}}}$ is a shear operator (momentum-dependent phase shift).

This approach simplifies problems such as the Quadrupole ion trap, where an ion is trapped in a harmonic potential with time dependent frequency.