An operator is said to be self-adjoint if it satisfies
hv|
ˆ
A|wi = hv|
ˆ
A
†
|wi. (3.7)
The corresponding matrix is said to be Hermitian. An important conse-
quence of self-adjointness is that the eigenvalues will turn out to be real. A
self-adjoint operator is thus a good candidate for a physical observable whose
values are always real.
Postulate 2. Observables An observable A in quantum mechanics is usually
represented by a self-adjoint operator
4
ˆ
A. Measurement of A in an experiment
gives a real number value α, which is one of the eigenvalues of the operator
ˆ
A.
By “measurement of an observable” we mean the setting up of a suitable
experiment and determining the value associated with that physical property.
We will discuss measurements in quantum mechanics in more detail soon.
For example, the machine SG
z
of the previous chapter measures the z-
component of the spin, S
z
, and yields two possible values ±~/2. The operator
corresponding to this spin observable, denoted by
ˆ
S
z
, has eigenvalues ±~/2
and corresponding eigenstates |0i and |1i. This means it satisfies the eigenvalue
equations
ˆ
S
z
|0i =
~
2
|0i,
ˆ
S
z
|1i = −
~
2
|1i.
Applying the spectral theorem (3.2.1 ), the matrix representation of
ˆ
S
z
in the
computational basis is:
|0i =
“
1
0
#
, |1i =
“
0
1
#
,
ˆ
S
z
=
~
2
“
1 0
0 −1
#
(3.8)
Exercise 3.1. Show that
ˆ
S
z
is Hermitian.
Exercise 3.2. Solve the eigenvalue equation for
ˆ
S
z
and show that its eigenvalues
are ±~/2.
3.2.3 Basis transformation
We have been saying that the choice of basis depends on the observable
we choose to measure. The Hilbert space must be spanned by the bases corre-
sponding to the eigenstates of other observables too. This implies a relation-
ship between different bases for a given system.
4
We need operators with real eigenvalues. In recent times, non-Hermitian operators also
seem to be relevant under certain special conditions, but these need not concern us here.
44 Introduction to Quantum Physics and Information Processing
FIGURE 3.1: Experiment for determining the eigenstates of
ˆ
S
x
in the com-
putational basis
Consider the spin observable, related to the magnetic moment, which is a
vector in 3-dimensional space. The vector spin observable
−→
S has the nature
of angular momentum, and has three spatial components: S
x
, S
y
, and S
z
. The
machines for measuring these observables would be, respectively, SG
x
, SG
y
,
and SG
z
, each with its B field inhomogeneity at right angles to that of the
other. But measurement of each of these would give one of two values, ±~/2.
This means that in each basis of representation, the eigenstates and the matrix
for the operator is given by Equation 3.8.
We would like to represent each of these observables and their eigenstates
in the common computational basis {|0i, |1i}. This, by convention, is the basis
of eigenstates of the operator
ˆ
S
z
which we had written as |↑
z
i and |↓
z
i. What,
for instance, is the form of the eigenstates |↑
x
i and |↓
x
i of
ˆ
S
x
in this basis?
Look at the Stern–Gerlach experiments shown in Figure 3.1.
This says that |↑
x
i and |↓
x
i are 50-50 superpositions of |0i and |1i.
|↑
x
i = α|0i + β|1i, where |α|
2
= |β|
2
=
1
2
.
A similar equation can be written for |↓
x
i. In fact a similar equation would
hold for the eigenstates |↑
y
i and |↓
y
i of
ˆ
S
y
. Each would need to have different
complex coefficients α and β to distinguish them. We can fix these coefficients
up to a relative phase: each has magnitude
1
√
2
and some phase which is not
fixed experimentally. (See Section 2.3.) By convention, we choose the relative
phase angle φ to be zero for |↑
x
i and π for |↑
y
i and fix the rest by demanding
orthogonality.
Example 3.2.1. Basis transformation from
ˆ
S
z
to
ˆ
S
x
basis: to emphasize that
|↑
x
i and |↓
x
i are also a different set of basis vectors, let us denote them by
|0
x
i and |1
x
i. Experiment is consistent with
|0
x
i =
1
√
2
(|0i + |1i) .
The Essentials of Quantum Mechanics 45
We also require h0
x
|1
x
i = 0, which is consistent with
|1
x
i =
1
√
2
(|0i − |1i) .
It is also easy to see that the basis vector transformation can be written in
matrix form as
|0
x
i
|1
x
i
!
=
1
√
2
1 1
1 −1
!
|0i
|1i
!
Henceforth, we will switch to a less cumbersome notation for the spin op-
erators. We consider the following dimensionless operators, each having eigen-
values ±1 and the same eigenstates as those of corresponding spin operators.
ˆ
X =
2
~
ˆ
S
x
; eigentates |0
x
i, |1
x
i, (3.9a)
ˆ
Y =
2
~
ˆ
S
y
; eigentates |0
y
i, |1
y
i, (3.9b)
ˆ
Z =
2
~
ˆ
S
z
; eigentates |0
z
i ≡ |0i, |1
z
i ≡ |1i. (3.9c)
Box 3.4: Basis Transformations among the
ˆ
X,
ˆ
Y and
ˆ
Z Bases
|0
x
i =
1
√
2
(|0i + |1i) (3.10a)
|1
x
i =
1
√
2
(|0i − |1i) (3.10b)
|0
y
i =
1
√
2
(|0i + i|1i) (3.10c)
|1
y
i =
1
√
2
(|0i − i|1i] (3.10d)
Exercise 3.3. Verify from these definitions that {|0
x
i, |1
x
i} are an orthonormal
set. Similarly for {|0
y
i, |1
y
i}.
Exercise 3.4. Express {|0
y
i, |1
y
i} in terms of {|0
x
i, |1
x
i}.
46 Introduction to Quantum Physics and Information Processing
It is important to realize that a change of basis is effected by a linear trans-
formation: When a basis {|ii} → {|ji} then for each |ji we can find a set of
n complex coefficients U
ij
such that
|ji =
X
i
U
ij
|ii. (3.11)
These components U
ij
can be shown to form the components of a unitary
matrix U. The change of basis can be visualized as a sort of rotation of the
axes that span the Hilbert space.
Example 3.2.2. Unitarity of the transformation matrix for basis change:
from Equation 3.11, let us use the orthogonality of the basis {|ji} to write
hj
0
|ji =
X
i
0
U
∗
j
0
i
0
hi
0
|
X
i
U
ij
|ii = δ
j
0
j
=⇒
X
i
0
X
i
U
∗
j
0
i
0
U
ij
hi
0
|ii = δ
j
0
j
=⇒
X
i
U
∗
j
0
i
U
ij
= δ
j
0
j
But this last equation is exactly the condition U
†
U = for unitarity of U.
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