The state space may be said to be defined by its basis states. How do we
identify the basis? We have said that when we measure a physical quantity,
the state corresponding to the value measured is a basis state for the sys-
tem. This brings us directly to the question: which physical quantity shall we
The Essentials of Quantum Mechanics 41
choose to measure? Well, the choice is entirely ours. But a certain amount of
scientific acumen is necessary to identify the relevant one! In any case, all the
physically measurable properties of the system are important: these are called
observables.
Measurement of a particular observable O yields a set of possible values, in
suitable units, that the observable could take. This set of real numbers charac-
terizes the observable. In mathematical language, these numbers are regarded
as the characteristic values or eigenvalues of an operator representing the
observable. The set of characteristic values is called the spectrum of the ob-
servable. This is indeed a full specification of the observable. But in quantum
mechanics, we try to attach the notion of an operator to the observable. What
is an operator?
3.2.1 Operators
An operator
ˆ
O, formally, is a method for transforming a vector |vi into
another, |v
0
i. Expressed mathematically, the operator acts from the left of the
vector:
ˆ
O|vi = |v
0
i.
In the language of linear algebra, operators are represented as matrices. For
example:
“
0 1
1 0
#”
1
0
#
=
“
0
1
#
.
Box 3.3: Diagonalizable Operators and the Spectral Theorem
An operator
ˆ
A is said to satisfy an eigenvalue equation if there exist
some vectors |
i
i that are transformed into multiples of themselves:
ˆ
A|
i
i = a
i
|
i
i.
The number a
i
is called an eigenvalue corresponding to the vector |
i
i, which
is called an eigenvector. The eigenvalues may be distinct (simple) or some
of them may be equal (multiple). In the latter case they are said to be de-
generate. Not all matrices satisfy eigenvalue equations. Those that do are
called diagonalizable. This name is due to the spectral theorem which
says that such operators can be expressed as diagonal matrices in the basis of
their eigenvectors with the eigenvalues as the diagonal elements:
ˆ
N =
X
i
n
i
|
i
ih
i
|.
This statement essentially means that for a diagonalizable matrix, we can
change basis to one in which the matrix is diagonal. In other words, there
42 Introduction to Quantum Physics and Information Processing
exists a non-singular matrix
ˆ
S bringing
ˆ
A to diagonal form
ˆ
N by a similarity
transformation:
ˆ
S
ˆ
A
ˆ
S
−1
=
ˆ
N.
A special class of diagonalizable operators is important in quantum mechanics,
those that commute with their adjoint:
ˆ
N
†
ˆ
N =
ˆ
N
ˆ
N
†
.
Such an operator is called a normal operator. Some kinds of normal operators
especially relevant to us are:
1. Unitary operators:
ˆ
U
†
=
ˆ
U
−1
2. Hermitian operators:
ˆ
H
†
=
ˆ
H
(Also anti-Hermitian operators:
ˆ
A
†
= −
ˆ
A
3. Positive operators:
ˆ
P =
ˆ
M
ˆ
M
†
. These operators are also Hermitian.
The following important properties of eigenvalues are to be noted
1. A Hermitian operator has real eigenvalues.
2. A positive operator has positive eigenvalues.
3. A unitary operator has eigenvalues of unit modulus, i.e., of the form e
iθ
for real θ.
Suppose we find the dual of the transformed vector |v
0
i. What is the op-
erator in dual space that would take hv| to hv
0
|? The answer is the adjoint
operator
ˆ
O
†
(‘O-dagger’), defined by the equation
hv|
ˆ
O
†
= hv
0
|.
We can also compare the transformed and original vectors by their inner
product with another vector |wi, and thus define the adjoint by
(|wi,
ˆ
O|vi) = (
ˆ
O
†
|wi, |vi). (3.6)
We can see that each side of Equation 3.6 is equivalent to hw|
ˆ
O|vi = hw|v
0
i =
hw
0
|vi, where hw
0
| = hw|
ˆ
O
†
. Notice that the action of the adjoint is from the
right.
The matrix representation of
ˆ
O
†
is the complex conjugate transpose of the
matrix for
ˆ
O. Thus the dual vector hv| is sometimes also called the adjoint of
the ket vector |vi
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