Postulates of quantum mechanics 369A more general inner product between two Hilbert-space vectors|φ〉=
φ 1
φ 2
φ 3
·
·
·
|ψ〉=
ψ 1
ψ 2
ψ 3
·
·
·
(9.2.8)
is defined to be
〈ψ|φ〉=
∑
kψ∗kφk. (9.2.9)Note that〈φ|ψ〉=〈ψ|φ〉∗.
9.2.2 Representation of physical observables
In quantum mechanics, physical observables are represented bylinear Hermitianop-
erators, which act on the vectors of the Hilbert space (we will see shortly why the
operators must be Hermitian). When the vectors ofHand its dual space are repre-
sented as ket and bra vectors, respectively, such operators are represented by matrices.
Thus, ifAˆis an operator corresponding to a physical observable, we can represent it
as
Aˆ=
A 11 A 12 A 13 ···
A 21 A 22 A 13 ···
A 31 A 32 A 33 ···
..
.
..
.
..
.
..
.
. (9.2.10)
(The overhat notation is commonly used in quantum mechanics to denote Hilbert-
space operators.) ThatAˆmust be a Hermitian operator means that its matrix elements
satisfy
A∗ji=Aij. (9.2.11)
The Hermitian conjugate ofAˆis defined as
Aˆ†=
A∗ 11 A∗ 21 A∗ 31 ···
A∗ 12 A∗ 22 A∗ 31 ···
A∗ 13 A∗ 23 A∗ 33 ···
..
.
..
.
..
.
..
.
, (9.2.12)
and the requirement thatAˆbe Hermitian meansAˆ†=Aˆ. Since the vectors ofHare
column vectors, it is clear that an operatorAˆcan act on a vector|φ〉to yield a new
vector|φ′〉viaAˆ|φ〉=|φ′〉, which is a simple matrix-vector product.
9.2.3 Possible outcomes of a physical measurement
Quantum mechanics postulates that if a measurement is performedon a physical
observable represented by an operatorAˆ, the result must be one of the eigenvalues of
Aˆ. From this postulate, we now see why observables must be represented by Hermitian