A Note on the Adiabatic Theorem of Quantum Mechanics∗

Folkmar Bornemann† August 1997

The adiabatic theorem in quantum theory refers to a situation in which the original Hamiltonian of a system is gradually changed into a new Hamil- tonian. Roughly speaking, the theorem states that an eigenstate for the original energy becomes approximately an eigenstate for the new energy if the switch-on of the energy difference is sufficiently slow. The model for this situation is given by a time-dependent Schrödinger equation with slowness parameter 1, ˙ iψ = H(t)ψ, ψ(0) = ψ∗.

The switch-on of the change takes place at time t0 = 0, the switch-off at time t1 = T/. We are interested in the limit situation → 0 of an “infinitely slow” change. It is convenient to transform the time variable linearly onto the fixed interval [0,T ], yielding the singularly perturbed equation ˙ iψ = H(t)ψ, ψ(0) = ψ∗. (1)

We will address the ﬁnite dimensional setting ψ(t) ∈ Cd by using pertur- bation theory of integrable Hamiltonian systems. The key point is to observe that the time-dependent Schr¨odinger equation has a canonical structure. To this end, we use phase-space coordinates † (i ψ, ψ ; E, t), with time t being the canonical momentum corresponding to the energy E, the symplectic two-form

† σ = i dψ ∧ dψ + dE ∧ dt and the Hamiltonian function‡

Z = hH(t)ψ, ψi − E.

∗The work of the ﬁrst author was supported in part by the U.S. Department of Energy under contract DE-FG02-92ER25127. †Konrad-Zuse-Zentrum, Takustr. 7, 14195 Berlin, Germany ([email protected]) ‡To get an autonomous system

1 2 Folkmar Bornemann

In fact, using Wirtinger derivatives, the Schr¨odinger equation Eq. (1) is equivalent to both of the equations§

˙ ∂Z ˙ † 1 ∂Z iψ = † , ψ = − . ∂ψ i ∂ψ The other two canonical equations are just ∂Z ∂Z E˙ = = hH˙ (t)ψ, ψi, t˙ = − = 1. ∂t ∂E We choose the additional initial values

E(0) = E∗ = hH(0)ψ∗, ψ∗i, t∗ = 0. Hence, the value of the invariant of motion Z is ﬁxed to be zero.¶. We will assume right from the beginning that all eigenvalues ωλ(t) of the d- dimensional hermitian matrix H(t) are simple and that there are no resonances of order two,

ωλ(t) 6= ωµ(t), t ∈ R, λ 6= µ.

There is a family of orthonormal eigenvectors (e1(y), . . . , er(y)),

H(y)eλ(y) = ωλ(y)eλ(y), heλ(y), eµ(y)i = δλµ. This normalization yields an important anti-hermitian relation of the time- derivativese ˙λ, speciﬁcally † heλ, e˙µi = −heµ, e˙λi . (2)

We introduce particular action-angle variables (θ, φ), q X λ −1 λ ψ = θ exp(−i φ ) eλ. λ This transformation yields the one-form X q 1 dψ = θλ exp(−i−1φλ) −i−1e dφλ + e dθλ +e ˙ dt . λ 2θλ λ λ λ

Hence, by using the normalization heλ, eµi = δλµ and the anti-hermitian relation Eq. (2), we obtain

† X λ λ i dψ ∧ dψ = dφ ∧ dθ λ q X λ µ −1 λ µ λ + 2 θ θ < exp −i (φ − φ ) heλ, e˙µi dφ ∧ dt λ,µ s µ X θ − = exp −i−1(φλ − φµ) he , e˙ i dθλ ∧ dt. θλ λ µ λ,µ

§Thus, the real dimension of the phase space is eﬀectively 2d + 2, eliminating the † duplication of information in using both ψ and ψ ¶Which explains the choice of the letter Z A Note on the Adiabatic Theorem of Quantum Mechanics 3

However, for obtaining a transformation being symplectic on the phase- space as a whole, we additionally have to transform the energy variable E, q X λ µ −1 λ µ E = P + θ θ = exp −i (φ − φ ) heλ, e˙µi . (3) λ,µ

By the anti-hermitian relation Eq. (2), this transformation results in

dE ∧ dt = dP ∧ dt q X λ µ −1 λ µ λ − 2 θ θ < exp −i (φ − φ ) heλ, e˙µi dφ ∧ dt λ,µ s µ X θ + = exp −i−1(φλ − φµ) he , e˙ i dθλ ∧ dt. θλ λ µ λ,µ Altogether, these lengthy but straightforward calculations have proven that † the transformation (ψ, ψ ; E, t) 7→ (φ, θ; t, P) is symplectic indeed,

† X λ λ σ = i dψ ∧ dψ + dE ∧ dt = dφ ∧ dθ + dP ∧ dt. λ The autonomous Hamiltonian function Z transforms to the expression q X λ X λ µ −1 λ µ Z = θ · ωλ − P − θ θ = exp −i (φ − φ ) heλ, e˙µi . λ λ,µ Thus, by the canonical formalism, the equation of motion take the form

˙λ ∂Z ˙λ ∂Z ˙ ∂Z ˙ ∂Z φ = λ , θ = − λ , P = , t = − = 1, ∂θ ∂φ ∂t ∂P i.e., after some calculation,

s µ X θ φ˙λ = ω − = exp −i−1(φλ − φµ) he , e˙ i λ θλ λ µ µ q ˙λ X λ µ −1 λ µ θ = −2 θ θ < exp −i (φ − φ ) heλ, e˙µi µ6=λ ˙ X λ P = θ · ω˙ λ λ q X λ µ −1 λ µ − θ θ = exp −i (φ − φ ) (he˙λ, e˙µi + heλ, e¨µi) . λµ The initial values transform as follows. Using polar coordinates, q λ λ hψ∗, eλ(0)i = θ∗ · exp −iφ∗ , λ = 1, . . . , r, 4 Folkmar Bornemann we obtain

φ(0) = φ∗, θ(0) = θ∗,P(0) = E∗ + O(). Now, for eliminating the fast dependence on the angle variables of the O(1)-terms we introduce the transformed action variables

p λ µ λ λ X θ θ −1 λ µ Θ = θ − 2 = exp −i (φ − φ ) heλ, e˙µi , (4) ωλ − ωµ µ6=λ with initial value Θ(0) = θ∗ +O(). Since we have excluded any resonance of order two, this transformation is well-deﬁned. For Θ the equation of motion takes the simple form

Θ˙ = O(), yielding the estimate

Θ = θ∗ + O(), i.e., θ = θ∗ + O(). Thus, the energy level probabilities are adiabatic invariants. Likewise, elim- ination of the O() term in the equations for φ is achieved by introducing

p µ λ λ λ 2 X θ∗ /θ∗ −1 λ µ Φ = φ + < exp −i (φ − φ ) heλ, e˙µi ωλ − ωµ µ6=λ

2 with initial value Φ(0) = φ∗ + O( ). This transformation is only well- λ deﬁned, if the energy level λ is initially excited, θ∗ 6= 0. We denote the set of all these levels by Λex. For λ ∈ Λex the equation of motion is now given by ˙ λ 2 Φ = ωλ − =heλ, e˙λi + O( ), yielding the estimate λ λ 2 λ λ 2 φ = Φ + O( ) = φav + φBerry + O( ), with Z t Z t λ λ λ φav(t) = ωλ(τ) dτ, φBerry(t) = φ∗ + i heλ(τ), e˙λ(τ)i dτ. 0 0

Notice, that because of the anti-hermitian relation Eq. (2) the term heλ, e˙λi is purely imaginary. Altogether, we have obtained an order O() approximation of the wave function ψ itself, q X λ λ −1 λ ψ = θ∗ exp(−iφBerry) exp(−i φav) eλ + O().

λ∈Λex Finally, there is no diﬃculty left to prove the energy estimate

X λ E = θ∗ · ωλ + O(). λ A Note on the Adiabatic Theorem of Quantum Mechanics 5

Remarks and Observations. We conclude by discussing some interest- ing points. 1. Using the new action-angle variables, the Hamiltonian function Z had to be expanded including the ﬁrst order term in . Otherwise the ˙ zero order term of the equation for θ would have been unknown and a proof of the adiabatic invariance of θ would have been impossible.

−1 2. Because of the factor multiplying the angle φ in the expression for the wavefunction ψ we had to expand the angle up to an error of second order for obtaining a ﬁrst order approximation of ψ.

3. The occurrence of the Berry-phase φBerry can be understood as mak- ing the zero-order approximation of the wave-function gauge-invariant, i.e., invariant with respect to a phase transformation of the eigenvectors eλ 7→ exp(iγλ) eλ.

4. Using the method of stationary phase, one can prove that the given approximation of ψ directly implies

∗ ∞ d ψ * 0 in L ([0,T ], C ),

provided the eigenvalue families ωλ just have isolated zeroes. 5. Since there are no resonances, the method of stationary phase applied † to the density matrix ρ = ψψ yields the weak limit

∗ X λ † ρ * ρ0 = θ∗ · eλeλ. λ∈Λex