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10 Time-Periodic Quantum Systems
ˆ
H (t) = ˆ
H 0 + ˆ
V G(t).
(10.2)
In Eq. (10.2), ˆ
H 0 is a time-independent contribution to the Hamiltonian, is a small
parameter, ˆ
V is a potential energy operator, and G(t) is a time-periodic function,
G(t) = G(t + T 0 ), where T 0 is the period of G(t). The Hamiltonian, ˆ
H (t), satisfies
the condition ˆ
H (t) = ˆ
H (t + T 0 ).
Let us now assume that Eq. (10.1) has a solution of the form
| α (t) = exp
−i
α
¯
h
t
| α (t),
(10.3)
where | α (t) = | α (t + T 0 ) is the αth Floquet eigenstate and α is the
corresponding Floquet eigenvalue. The Floquet eigenvalues (quasienergies) are only
defined modulus
2π
T 0
, because of the periodicity of | α (t). It is easy to see that
ˆ
H F (t)| α (t) = α | α (t),
(10.4)
where the Floquet Hamiltonian, ˆ
H F (t), is defined as
ˆ
H F (t) = ˆ
H (t) − i ¯
h
∂
∂t
.
(10.5)
ˆ
H F (t) is a Hermitian operator with eigenvalues α and eigenstates | α (t).
Let us now expand the full solution of the Schrödinger equation, | in terms
of Floquet states, | α (t). We assume that the Floquet eigenstates form a complete
orthonormal set since they are eigenstates of a Hermitian operator. We write
| =
α
A α exp
−i
α
¯
h
t
| α (t).
(10.6)
The coefficients, A α , are determined in terms of the initial condition, |(0), and
are given by
A α = = α (0)|(0),
(10.7)
where we have used the orthonormality of states | α (t). Thus
| =
α
exp
−i
α
¯
h
t
| α (t) α (0)|(0).
(10.8)
If we use the periodicity of the states | α (t), we find that after one period, T 0 , of
the external field we can write
10 Time-Periodic Quantum Systems
ˆ
H (t) = ˆ
H 0 + ˆ
V G(t).
(10.2)
In Eq. (10.2), ˆ
H 0 is a time-independent contribution to the Hamiltonian, is a small
parameter, ˆ
V is a potential energy operator, and G(t) is a time-periodic function,
G(t) = G(t + T 0 ), where T 0 is the period of G(t). The Hamiltonian, ˆ
H (t), satisfies
the condition ˆ
H (t) = ˆ
H (t + T 0 ).
Let us now assume that Eq. (10.1) has a solution of the form
| α (t) = exp
−i
α
¯
h
t
| α (t),
(10.3)
where | α (t) = | α (t + T 0 ) is the αth Floquet eigenstate and α is the
corresponding Floquet eigenvalue. The Floquet eigenvalues (quasienergies) are only
defined modulus
2π
T 0
, because of the periodicity of | α (t). It is easy to see that
ˆ
H F (t)| α (t) = α | α (t),
(10.4)
where the Floquet Hamiltonian, ˆ
H F (t), is defined as
ˆ
H F (t) = ˆ
H (t) − i ¯
h
∂
∂t
.
(10.5)
ˆ
H F (t) is a Hermitian operator with eigenvalues α and eigenstates | α (t).
Let us now expand the full solution of the Schrödinger equation, | in terms
of Floquet states, | α (t). We assume that the Floquet eigenstates form a complete
orthonormal set since they are eigenstates of a Hermitian operator. We write
| =
α
A α exp
−i
α
¯
h
t
| α (t).
(10.6)
The coefficients, A α , are determined in terms of the initial condition, |(0), and
are given by
A α = = α (0)|(0),
(10.7)
where we have used the orthonormality of states | α (t). Thus
| =
α
exp
−i
α
¯
h
t
| α (t) α (0)|(0).
(10.8)
If we use the periodicity of the states | α (t), we find that after one period, T 0 , of
the external field we can write