An operator is said to be self-adjoint if it satis\ufb01es<\/p>\n\n\n\n
hv|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A|wi = hv|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
|wi. (3.7)<\/p>\n\n\n\n
The corresponding matrix is said to be Hermitian. An important conse-<\/p>\n\n\n\n
quence of self-adjointness is that the eigenvalues will turn out to be real. A<\/p>\n\n\n\n
self-adjoint operator is thus a good candidate for a physical observable whose<\/p>\n\n\n\n
values are always real.<\/p>\n\n\n\n
Postulate 2. Observables An observable A in quantum mechanics is usually<\/p>\n\n\n\n
represented by a self-adjoint operator<\/p>\n\n\n\n
4<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A. Measurement of A in an experiment<\/p>\n\n\n\n
gives a real number value \u03b1, which is one of the eigenvalues of the operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A.<\/p>\n\n\n\n
By \u201cmeasurement of an observable\u201d we mean the setting up of a suitable<\/p>\n\n\n\n
experiment and determining the value associated with that physical property.<\/p>\n\n\n\n
We will discuss measurements in quantum mechanics in more detail soon.<\/p>\n\n\n\n
For example, the machine SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
of the previous chapter measures the z-<\/p>\n\n\n\n
component of the spin, S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
, and yields two possible values \u00b1~\/2. The operator<\/p>\n\n\n\n
corresponding to this spin observable, denoted by<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
, has eigenvalues \u00b1~\/2<\/p>\n\n\n\n
and corresponding eigenstates |0i and |1i. This means it satis\ufb01es the eigenvalue<\/p>\n\n\n\n
equations<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
|0i =<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|0i,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
|1i = \u2212<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1i.<\/p>\n\n\n\n
Applying the spectral theorem (3.2.1 ), the matrix representation of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
in the<\/p>\n\n\n\n
computational basis is:<\/p>\n\n\n\n
|0i =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
, |1i =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
=<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 \u22121<\/p>\n\n\n\n
#<\/p>\n\n\n\n
(3.8)<\/p>\n\n\n\n
Exercise 3.1. Show that<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
is Hermitian.<\/p>\n\n\n\n
Exercise 3.2. Solve the eigenvalue equation for<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
and show that its eigenvalues<\/p>\n\n\n\n
are \u00b1~\/2.<\/p>\n\n\n\n
3.2.3 Basis transformation<\/p>\n\n\n\n
We have been saying that the choice of basis depends on the observable<\/p>\n\n\n\n
we choose to measure. The Hilbert space must be spanned by the bases corre-<\/p>\n\n\n\n
sponding to the eigenstates of other observables too. This implies a relation-<\/p>\n\n\n\n
ship between di\ufb00erent bases for a given system.<\/p>\n\n\n\n
4<\/p>\n\n\n\n
We need operators with real eigenvalues. In recent times, non-Hermitian operators also<\/p>\n\n\n\n
seem to be relevant under certain special conditions, but these need not concern us here.<\/p>\n\n\n\n
<\/p>\n\n\n\n
44 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
FIGURE 3.1: Experiment for determining the eigenstates of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
in the com-<\/p>\n\n\n\n
putational basis<\/p>\n\n\n\n
Consider the spin observable, related to the magnetic moment, which is a<\/p>\n\n\n\n
vector in 3-dimensional space. The vector spin observable<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
S has the nature<\/p>\n\n\n\n
of angular momentum, and has three spatial components: S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
, S<\/p>\n\n\n\n
y<\/p>\n\n\n\n
, and S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
. The<\/p>\n\n\n\n
machines for measuring these observables would be, respectively, SG<\/p>\n\n\n\n
x<\/p>\n\n\n\n
, SG<\/p>\n\n\n\n
y<\/p>\n\n\n\n
,<\/p>\n\n\n\n
and SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
, each with its B \ufb01eld inhomogeneity at right angles to that of the<\/p>\n\n\n\n
other. But measurement of each of these would give one of two values, \u00b1~\/2.<\/p>\n\n\n\n
This means that in each basis of representation, the eigenstates and the matrix<\/p>\n\n\n\n
for the operator is given by Equation 3.8.<\/p>\n\n\n\n
We would like to represent each of these observables and their eigenstates<\/p>\n\n\n\n
in the common computational basis {|0i, |1i}. This, by convention, is the basis<\/p>\n\n\n\n
of eigenstates of the operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
which we had written as |\u2191<\/p>\n\n\n\n
z<\/p>\n\n\n\n
i and |\u2193<\/p>\n\n\n\n
z<\/p>\n\n\n\n
i. What,<\/p>\n\n\n\n
for instance, is the form of the eigenstates |\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i and |\u2193<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
in this basis?<\/p>\n\n\n\n
Look at the Stern\u2013Gerlach experiments shown in Figure 3.1.<\/p>\n\n\n\n
This says that |\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i and |\u2193<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i are 50-50 superpositions of |0i and |1i.<\/p>\n\n\n\n
|\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i = \u03b1|0i + \u03b2|1i, where |\u03b1|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= |\u03b2|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
.<\/p>\n\n\n\n
A similar equation can be written for |\u2193<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i. In fact a similar equation would<\/p>\n\n\n\n
hold for the eigenstates |\u2191<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i and |\u2193<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
y<\/p>\n\n\n\n
. Each would need to have di\ufb00erent<\/p>\n\n\n\n
complex coe\ufb03cients \u03b1 and \u03b2 to distinguish them. We can \ufb01x these coe\ufb03cients<\/p>\n\n\n\n
up to a relative phase: each has magnitude<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
and some phase which is not<\/p>\n\n\n\n
\ufb01xed experimentally. (See Section 2.3.) By convention, we choose the relative<\/p>\n\n\n\n
phase angle \u03c6 to be zero for |\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i and \u03c0 for |\u2191<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i and \ufb01x the rest by demanding<\/p>\n\n\n\n
orthogonality.<\/p>\n\n\n\n
Example 3.2.1. Basis transformation from<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
to<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
basis: to emphasize that<\/p>\n\n\n\n
|\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i and |\u2193<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i are also a di\ufb00erent set of basis vectors, let us denote them by<\/p>\n\n\n\n
|0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i and |1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i. Experiment is consistent with<\/p>\n\n\n\n
|0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i + |1i) .<\/p>\n\n\n\n
<\/p>\n\n\n\n
The Essentials of Quantum Mechanics 45<\/p>\n\n\n\n
We also require h0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i = 0, which is consistent with<\/p>\n\n\n\n
|1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i \u2212 |1i) .<\/p>\n\n\n\n
It is also easy to see that the basis vector transformation can be written in<\/p>\n\n\n\n
matrix form as<\/p>\n\n\n\n
|0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i<\/p>\n\n\n\n
!<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
1 \u22121<\/p>\n\n\n\n
!<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
|1i<\/p>\n\n\n\n
!<\/p>\n\n\n\n
Henceforth, we will switch to a less cumbersome notation for the spin op-<\/p>\n\n\n\n
erators. We consider the following dimensionless operators, each having eigen-<\/p>\n\n\n\n
values \u00b11 and the same eigenstates as those of corresponding spin operators.<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
X =<\/p>\n\n\n\n
2<\/p>\n\n\n\n
~<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
; eigentates |0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i, |1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i, (3.9a)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y =<\/p>\n\n\n\n
2<\/p>\n\n\n\n
~<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
y<\/p>\n\n\n\n
; eigentates |0<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i, |1<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i, (3.9b)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Z =<\/p>\n\n\n\n
2<\/p>\n\n\n\n
~<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
; eigentates |0<\/p>\n\n\n\n
z<\/p>\n\n\n\n
i \u2261 |0i, |1<\/p>\n\n\n\n
z<\/p>\n\n\n\n
i \u2261 |1i. (3.9c)<\/p>\n\n\n\n
Box 3.4: Basis Transformations among the<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
X,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Y and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Z Bases<\/p>\n\n\n\n
|0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i + |1i) (3.10a)<\/p>\n\n\n\n
|1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i \u2212 |1i) (3.10b)<\/p>\n\n\n\n
|0<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i + i|1i) (3.10c)<\/p>\n\n\n\n
|1<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i \u2212 i|1i] (3.10d)<\/p>\n\n\n\n
Exercise 3.3. Verify from these de\ufb01nitions that {|0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i, |1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i} are an orthonormal<\/p>\n\n\n\n
set. Similarly for {|0<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i, |1<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i}.<\/p>\n\n\n\n
Exercise 3.4. Express {|0<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i, |1<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i} in terms of {|0<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i, |1<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i}.<\/p>\n\n\n\n
<\/p>\n\n\n\n
46 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
It is important to realize that a change of basis is e\ufb00ected by a linear trans-<\/p>\n\n\n\n
formation: When a basis {|ii} \u2192 {|ji} then for each |ji we can \ufb01nd a set of<\/p>\n\n\n\n
n complex coe\ufb03cients U<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
such that<\/p>\n\n\n\n
|ji =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
U<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
|ii. (3.11)<\/p>\n\n\n\n
These components U<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
can be shown to form the components of a unitary<\/p>\n\n\n\n
matrix U. The change of basis can be visualized as a sort of rotation of the<\/p>\n\n\n\n
axes that span the Hilbert space.<\/p>\n\n\n\n
Example 3.2.2. Unitarity of the transformation matrix for basis change:<\/p>\n\n\n\n
from Equation 3.11, let us use the orthogonality of the basis {|ji} to write<\/p>\n\n\n\n
hj<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|ji =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
0<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
j<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
0<\/p>\n\n\n\n
hi<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
U<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
|ii = \u03b4<\/p>\n\n\n\n
j<\/p>\n\n\n\n
0<\/p>\n\n\n\n
j<\/p>\n\n\n\n
=\u21d2<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
0<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
j<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
0<\/p>\n\n\n\n
U<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
hi<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|ii = \u03b4<\/p>\n\n\n\n
j<\/p>\n\n\n\n
0<\/p>\n\n\n\n
j<\/p>\n\n\n\n
=\u21d2<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
j<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
U<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
= \u03b4<\/p>\n\n\n\n
j<\/p>\n\n\n\n
0<\/p>\n\n\n\n
j<\/p>\n\n\n\n
But this last equation is exactly the condition U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
U = for unitarity of U.<\/p>\n","protected":false},"excerpt":{"rendered":"
An operator is said to be self-adjoint if it satis\ufb01es hv| \u02c6 A|wi = hv| \u02c6 A \u2020 |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 […]<\/p>\n","protected":false},"author":1,"featured_media":3974,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[489],"tags":[],"class_list":["post-3989","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-4-quantum-mechanics"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-computer-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3989","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=3989"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3989\/revisions"}],"predecessor-version":[{"id":4557,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3989\/revisions\/4557"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3974"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3989"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3989"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3989"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}