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 Ω : R → R {\displaystyle \Omega :\mathbb {R} \to \mathbb {R} } be a function. It defines a time dependent harmonic oscillator Hamiltonian reads
H ^ = 1 2 [ p ^ 2 + Ω 2 ( t ) q ^ 2 ] . {\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
I ^ = 1 2 [ ( q ^ ρ ) 2 + ( ρ p ^ − ρ ˙ q ^ ) 2 ] , {\displaystyle {\hat {I}}={\frac {1}{2}}\left[\left({\frac {\hat {q}}{\rho }}\right)^{2}+(\rho {\hat {p}}-{\dot {\rho }}{\hat {q}})^{2}\right],}
where ρ : R → R {\displaystyle \rho :\mathbb {R} \to \mathbb {R} } is a solution to the Ermakov equation
ρ ¨ + Ω 2 ρ = ρ − 3 . {\displaystyle {\ddot {\rho }}+\Omega ^{2}\rho =\rho ^{-3}.}
I ^ {\displaystyle {\hat {I}}} is a unitary transformation of the time independent harmonic oscillator Hamiltonian: 1 2 [ p ^ 2 + q ^ 2 ] = T ^ I ^ T ^ † , T ^ = e i ln ρ 2 ( q ^ p ^ + p ^ q ^ ) e − i ρ ˙ 2 ρ q ^ 2 = e i ln ρ 2 d q ^ 2 d t e − i q ^ 2 2 d ln ρ d t , {\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 e i ln ρ 2 ( q ^ p ^ + p ^ q ^ ) {\displaystyle e^{i{\frac {\ln \rho }{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}} is a squeeze operator. The other exponential term e − i ρ ˙ 2 ρ q ^ 2 {\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.
Phase-space geometry
This invariant has a geometrically intuitive interpretation in Wigner's form of phase-space quantum mechanics. For any quadratic Hamiltonian the phase-space flow is linear and symplectic. Writing x = ( q , p ) T {\displaystyle \mathbf {x} =(q,p)^{\mathsf {T}}} , the classical evolution is
x ( t ) = S ( t ) x ( 0 ) , S ( t ) ∈ S p ( 2 , R ) . {\displaystyle \mathbf {x} (t)=S(t)\,\mathbf {x} (0),\qquad S(t)\in \mathrm {Sp} (2,\mathbb {R} ).}
Introduce the Ermakov–Lewis scaling
( Q P ) = L ( t ) ( q p ) , L ( t ) = ( ρ − 1 ( t ) 0 − ρ ˙ ( t ) ρ ( t ) ) , {\displaystyle {\binom {Q}{P}}=L(t){\binom {q}{p}},\qquad L(t)={\begin{pmatrix}\rho ^{-1}(t)&0\\[2pt]-{\dot {\rho }}(t)&\rho (t)\end{pmatrix}},}
for which the invariant becomes 1 2 ( Q 2 + P 2 ) {\displaystyle {\tfrac {1}{2}}\!\left(Q^{2}+P^{2}\right)} and the dynamics is a pure rotation. In these variables
( Q ( t ) P ( t ) ) = R ( θ ( t ) ) ( Q ( 0 ) P ( 0 ) ) , R ( θ ) = ( cos θ sin θ − sin θ cos θ ) , θ ( t ) = ∫ 0 t d τ ρ 2 ( τ ) , {\displaystyle {\binom {Q(t)}{P(t)}}=R(\theta (t)){\binom {Q(0)}{P(0)}},\quad R(\theta )={\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}},\quad \theta (t)=\int _{0}^{t}\!{\frac {d\tau }{\rho ^{2}(\tau )}},}
with ρ {\displaystyle \rho } solving the Ermakov equation (see above). Transforming back gives
S ( t ) = L ( t ) − 1 R ( θ ( t ) ) L ( 0 ) = ( ρ 0 ρ ˙ ρ − 1 ) R ( θ ( t ) ) ( ρ ( 0 ) − 1 0 − ρ ˙ ( 0 ) ρ ( 0 ) ) . {\displaystyle S(t)=L(t)^{-1}R(\theta (t))L(0)={\begin{pmatrix}\rho &0\\{\dot {\rho }}&\rho ^{-1}\end{pmatrix}}R(\theta (t)){\begin{pmatrix}\rho (0)^{-1}&0\\-{\dot {\rho }}(0)&\rho (0)\end{pmatrix}}.}
Because the Wigner function evolves by the pullback of the classical flow for quadratic Hamiltonians,
W ( x , t ) = W 0 ( S ( t ) − 1 x ) , {\displaystyle W(\mathbf {x} ,t)=W_{0}\!\left(S(t)^{-1}\mathbf {x} \right),}
any initial Gaussian remains Gaussian. For an initial coherent state of the unit oscillator with mean μ 0 {\displaystyle {\boldsymbol {\mu }}_{0}} and covariance Σ 0 = 1 2 I {\displaystyle \Sigma _{0}={\tfrac {1}{2}}I} ,
μ ( t ) = S ( t ) μ 0 , Σ ( t ) = S ( t ) Σ 0 S ( t ) T = 1 2 S ( t ) S ( t ) T . {\displaystyle {\boldsymbol {\mu }}(t)=S(t){\boldsymbol {\mu }}_{0},\qquad \Sigma (t)=S(t)\Sigma _{0}S(t)^{\mathsf {T}}={\tfrac {1}{2}}\,S(t)S(t)^{\mathsf {T}}.}
Geometrically, the contours of W ( ⋅ , t ) {\displaystyle W(\cdot ,t)} are ellipses whose axes "breathe" via ρ ( t ) {\displaystyle \rho (t)} and whose orientation rotates by θ ( t ) {\displaystyle \theta (t)} . In the scaled coordinates ( Q , P ) {\displaystyle (Q,P)} , or equivalently under the unitary T ^ {\displaystyle {\hat {T}}} given above, the state is a circular Gaussian rotating at constant angular velocity, so all deformation in the laboratory ( q , p ) {\displaystyle (q,p)} plane is captured by the time-dependent squeeze L ( t ) {\displaystyle L(t)} and variable-velocity rotation R ( θ ) {\displaystyle R(\theta )} .
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.
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