2.1 Dirac Notation and Linear Algebra
9
A linear operator A with domain D(A) is called symmetric if
(ψ|A|φ)
∗
= =φ|A|ψ ∀|ψ, |φ ∈ D(A).
(2.3)
The above definition, combined with the definition of adjoint (2.2), implies that
D(A) ⊆ D(A
†
) and that A and A
† act in the same way on the vectors in D(A).
A linear operator A with domain D(A) is called self-adjoint if it is symmetric and
D(A) = D(A
†
). In other words, the operator A and its adjoint A
† coincide: A = A
† .
Finally, a linear operator A is called Hermitian if it is self-adjoint ( A = A
† ) and
bounded. Note that a linear operator A is bounded if there exists a number M ≥ 0
such that:
A|ψ ≤ M |ψ ∀ |ψ ∈ H.
(2.4)
A bounded operator is typically assumed to be defined on the whole Hilbert space
H on which it acts. Therefore a linear operator is Hermitian if A = A
† , D(A) = H
and (2.4) holds.
We remark that if H is finite-dimensional, then any linear operator A is bounded.
This implies that for finite-dimensional Hilbert spaces the definitions of symmetric,
self-adjoint and Hermitian operator are all equivalent.
In this book, unless otherwise specified, we always implicitly consider linear
operators acting on finite-dimensional Hilbert spaces over C.
An operator P is called a projector if P
2
= P. If P is also Hermitian, then it is
called an orthogonal projector. Indeed, an orthogonal projector P and its complement, 1 − P, project the same vector |v onto orthogonal subspaces, as shown by
the inner product of the two projected vectors:
v|P
†
(1 − P)|v = =v|P(1 − P)|v = =v|(P − P
2
)|v = 0.
(2.5)
An example of orthogonal projector is given by the following rank-one orthogonal
projector |ψψ |. Note that any orthogonal projector can be written as:
P =
i∈S
|b i b i | , S ⊆ {1, . . . , d},
(2.6)
i.e. as a sum of rank-one projectors on some elements of an orthonormal basis
{|b i }
d
i=1 ⊂ H.
Importantly, the eigenvalues a i of an Hermitian operator A = A
† on the ddimensional Hilbert space H are real and the eigenvectors form an orthonormal
9
A linear operator A with domain D(A) is called symmetric if
(ψ|A|φ)
∗
= =φ|A|ψ ∀|ψ, |φ ∈ D(A).
(2.3)
The above definition, combined with the definition of adjoint (2.2), implies that
D(A) ⊆ D(A
†
) and that A and A
† act in the same way on the vectors in D(A).
A linear operator A with domain D(A) is called self-adjoint if it is symmetric and
D(A) = D(A
†
). In other words, the operator A and its adjoint A
† coincide: A = A
† .
Finally, a linear operator A is called Hermitian if it is self-adjoint ( A = A
† ) and
bounded. Note that a linear operator A is bounded if there exists a number M ≥ 0
such that:
A|ψ ≤ M |ψ ∀ |ψ ∈ H.
(2.4)
A bounded operator is typically assumed to be defined on the whole Hilbert space
H on which it acts. Therefore a linear operator is Hermitian if A = A
† , D(A) = H
and (2.4) holds.
We remark that if H is finite-dimensional, then any linear operator A is bounded.
This implies that for finite-dimensional Hilbert spaces the definitions of symmetric,
self-adjoint and Hermitian operator are all equivalent.
In this book, unless otherwise specified, we always implicitly consider linear
operators acting on finite-dimensional Hilbert spaces over C.
An operator P is called a projector if P
2
= P. If P is also Hermitian, then it is
called an orthogonal projector. Indeed, an orthogonal projector P and its complement, 1 − P, project the same vector |v onto orthogonal subspaces, as shown by
the inner product of the two projected vectors:
v|P
†
(1 − P)|v = =v|P(1 − P)|v = =v|(P − P
2
)|v = 0.
(2.5)
An example of orthogonal projector is given by the following rank-one orthogonal
projector |ψψ |. Note that any orthogonal projector can be written as:
P =
i∈S
|b i b i | , S ⊆ {1, . . . , d},
(2.6)
i.e. as a sum of rank-one projectors on some elements of an orthonormal basis
{|b i }
d
i=1 ⊂ H.
Importantly, the eigenvalues a i of an Hermitian operator A = A
† on the ddimensional Hilbert space H are real and the eigenvectors form an orthonormal