Ermakov–Lewis invariant

Many quantum mechanical Hamiltonians 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 it is possible to write several invariants, among them, the Ermakov–Lewis invariant which is developed below.

The time dependent harmonic oscillator Hamiltonian reads

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

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

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

where ${\displaystyle \rho }$ obeys the Ermakov equation

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

The above invariant is the so-called Ermakov–Lewis invariant.[1] It is easy to show that ${\displaystyle {\hat {I}}}$ may be related to the time independent harmonic oscillator Hamiltonian via a unitary transformation of the form [2]

${\displaystyle {\hat {T}}=e^{i{\frac {\ln \rho }{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}e^{-i{\frac {\dot {\rho }}{2\rho }}{\hat {q}}^{2}}=e^{i{\frac {\ln \rho }{2}}{\frac {d{\hat {q}}^{2}}{dt}}}e^{-i{\frac {{\hat {q}}^{2}}{2}}{\frac {d\ln \rho }{dt}}},}$

as

${\displaystyle {\frac {1}{2}}\left[{\hat {p}}^{2}+{\hat {q}}^{2}\right]={\hat {T}}{\hat {I}}{\hat {T}}^{\dagger }.}$

This allows an easy form to express the solution of the Schrödinger equation for the time dependent Hamiltonian.

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

This approach may allow to simplify problems such as the Quadrupole ion trap, 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. [3]

History

It was proposed in 1880 by Vasilij Petrovich Ermakov (1845-1922).[4] The paper is translated in.[5]

In 1966, Ralph Lewis rediscovered the invariant using Kruskal's asymptotic method.[6] He published the solution in 1967.[1]

References

1. Lewis, H. R. (1967-03-27). "Classical and Quantum Systems with Time-Dependent Harmonic-Oscillator-Type Hamiltonians". Physical Review Letters. 18 (13). American Physical Society (APS): 510–512. doi:10.1103/physrevlett.18.510. ISSN 0031-9007.
2. Moya-Cessa, H.; Guasti, M. Fernández (2003). "Coherent states for the time dependent harmonic oscillator: the step function". Physics Letters A. 311 (1): 1–5. arXiv:quant-ph/0301111. Bibcode:2003PhLA..311....1M. doi:10.1016/S0375-9601(03)00461-4. S2CID 54634409.
3. Yeh, L. (1993). "Ermakov-Lewis invariant from the Wigner function of a squeezed coherent state". Phys. Rev. A. 47 (5): 3587–3592. doi:10.1103/PhysRevA.47.3587.
4. Ermakov, V. "Second-order differential equations." Conditions of complete integrability, Universitetskie Izvestiya, Kiev 9 (1880): 1-25.
5. Ermakov, Vasilij Petrovich (2008). "Second-Order Differential Equations: Conditions of Complete Integrability". Applicable Analysis and Discrete Mathematics. 2 (2): 123–145. ISSN 1452-8630.
6. Leach, P. G. L.; Andriopoulos, K. (2008). "The Ermakov Equation: A Commentary". Applicable Analysis and Discrete Mathematics. 2 (2): 146–157. ISSN 1452-8630.