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 Hamiltonian. 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,
iψ̇ = 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
iψ̇ = H(t)ψ ,
ψ (0) = ψ∗ .
(1)
We will address the finite dimensional setting ψ(t) ∈ Cd by using perturbation theory of integrable Hamiltonian systems.
The key point is to observe that the time-dependent Schrödinger 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 , ψ i − 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 Schrödinger equation Eq. (1) is
equivalent to both of the equations§
iψ̇ =
∂Z
∂ψ†
,
ψ̇† = −
1 ∂Z
.
i ∂ψ
The other two canonical equations are just
∂Z
= hḢ(t)ψ , ψ i,
∂t
We choose the additional initial values
Ė =
ṫ = −
E (0) = E∗ = hH(0)ψ∗ , ψ∗ i,
∂Z
= 1.
∂E
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 ddimensional 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 timederivatives ėλ , specifically
heλ , ėµ i = −heµ , ėλ i† .
(2)
We introduce particular action-angle variables (θ , φ ),
Xq
θλ exp(−i−1 φλ ) eλ .
ψ =
λ
This transformation yields the one-form
Xq
1
−1 λ
−1
λ
λ
λ
θ exp(−i φ ) −i eλ dφ + λ eλ dθ + ėλ dt .
dψ =
2θ
λ
Hence, by using the normalization heλ , eµ i = δλµ and the anti-hermitian
relation Eq. (2), we obtain
X
i dψ ∧ dψ† =
dφλ ∧ dθλ
λ
Xq
+2
θλ θµ < exp −i−1 (φλ − φµ ) heλ , ėµ i dφλ ∧ dt
λ,µ
s
−
X
λ,µ
§ Thus,
θµ
= exp −i−1 (φλ − φµ ) heλ , ėµ i dθλ ∧ dt.
θλ
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 phasespace as a whole, we additionally have to transform the energy variable E ,
Xq
E = P + θλ θµ = exp −i−1 (φλ − φµ ) heλ , ėµ i .
(3)
λ,µ
By the anti-hermitian relation Eq. (2), this transformation results in
dE ∧ dt = dP ∧ dt
Xq
−2
θλ θµ < exp −i−1 (φλ − φµ ) heλ , ėµ i dφλ ∧ dt
λ,µ
s
+
X
λ,µ
θµ
= exp −i−1 (φλ − φµ ) heλ , ėµ i dθλ ∧ dt.
λ
θ
Altogether, these lengthy but straightforward calculations have proven that
the transformation (ψ , ψ† ; E , t) 7→ (φ , θ ; t, P ) is symplectic indeed,
X
σ = i dψ ∧ dψ† + dE ∧ dt =
dφλ ∧ dθλ + dP ∧ dt.
λ
The autonomous Hamiltonian function Z transforms to the expression
X
Xq
θλ θµ = exp −i−1 (φλ − φµ ) heλ , ėµ i .
Z=
θλ · ωλ − P − λ
λ,µ
Thus, by the canonical formalism, the equation of motion take the form
φ̇λ =
∂Z
,
∂θλ
θ̇λ = −
∂Z
,
∂φλ
Ṗ =
∂Z
,
∂t
ṫ = −
∂Z
= 1,
∂P
i.e., after some calculation,
s
X θµ
λ
= exp −i−1 (φλ − φµ ) heλ , ėµ i
φ̇ = ωλ − λ
θ
µ
Xq
θλ θµ < exp −i−1 (φλ − φµ ) heλ , ėµ i
θ̇λ = −2
µ6=λ
Ṗ =
X
θλ · ω̇λ
λ
−
Xq
θλ θµ = exp −i−1 (φλ − φµ ) (hėλ , ėµ i + heλ , ëµ i) .
λµ
The initial values transform as follows. Using polar coordinates,
q
hψ∗ , eλ (0)i = θ∗λ · exp −iφλ∗ ,
λ = 1, . . . , r,
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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
X
θλ θµ
λ
λ
Θ = θ − 2
= exp −i−1 (φλ − φµ ) heλ , ėµ i ,
(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
Θ̇ = O(),
yielding the estimate
Θ = θ∗ + O(),
i.e.,
θ = θ∗ + O().
Thus, the energy level probabilities are adiabatic invariants. Likewise, elimination of the O() term in the equations for φ is achieved by introducing
p
X θ∗µ /θλ
∗
< exp −i−1 (φλ − φµ ) heλ , ėµ i
Φλ = φλ + 2
ωλ − ωµ
µ6=λ
with initial value Φ (0) = φ∗ + O(2 ). This transformation is only welldefined, if the energy level λ is initially excited, θ∗λ 6= 0. We denote the set
of all these levels by Λex . For λ ∈ Λex the equation of motion is now given
by
Φ̇λ = ωλ − =heλ , ėλ i + O(2 ),
yielding the estimate
φλ = Φλ + O(2 ) = φλav + φλBerry + O(2 ),
with
φλav (t) =
Z
t
ωλ (τ ) dτ,
φλBerry (t) = φλ∗ + i
0
Z
t
heλ (τ ), ėλ (τ )i dτ.
0
Notice, that because of the anti-hermitian relation Eq. (2) the term heλ , ėλ i
is purely imaginary. Altogether, we have obtained an order O() approximation of the wave function ψ itself,
X q
ψ =
θ∗λ exp(−iφλBerry ) exp(−i−1 φλav ) eλ + O().
λ∈Λex
Finally, there is no difficulty left to prove the energy estimate
X
E =
θ∗λ · ωλ + O().
λ
A Note on the Adiabatic Theorem of Quantum Mechanics
Remarks and Observations.
ing points.
5
We conclude by discussing some interest-
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 θ̇ would have been unknown and
a proof of the adiabatic invariance of θ would have been impossible.
2. Because of the factor −1 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 making 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
∗
ψ * 0
in L∞ ([0, T ], Cd ),
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 = ψ ψ† yields the weak limit
X
∗
ρ * ρ 0 =
θ∗λ · eλ e†λ .
λ∈Λex