The state space may be said to be de\ufb01ned by its basis states. How do we<\/p>\n\n\n\n
identify the basis? We have said that when we measure a physical quantity,<\/p>\n\n\n\n
the state corresponding to the value measured is a basis state for the sys-<\/p>\n\n\n\n
tem. This brings us directly to the question: which physical quantity shall we<\/p>\n\n\n\n
<\/p>\n\n\n\n
The Essentials of Quantum Mechanics 41<\/p>\n\n\n\n
choose to measure? Well, the choice is entirely ours. But a certain amount of<\/p>\n\n\n\n
scienti\ufb01c acumen is necessary to identify the relevant one! In any case, all the<\/p>\n\n\n\n
physically measurable properties of the system are important: these are called<\/p>\n\n\n\n
observables.<\/p>\n\n\n\n
Measurement of a particular observable O yields a set of possible values, in<\/p>\n\n\n\n
suitable units, that the observable could take. This set of real numbers charac-<\/p>\n\n\n\n
terizes the observable. In mathematical language, these numbers are regarded<\/p>\n\n\n\n
as the characteristic values or eigenvalues of an operator representing the<\/p>\n\n\n\n
observable. The set of characteristic values is called the spectrum of the ob-<\/p>\n\n\n\n
servable. This is indeed a full speci\ufb01cation of the observable. But in quantum<\/p>\n\n\n\n
mechanics, we try to attach the notion of an operator to the observable. What<\/p>\n\n\n\n
is an operator?<\/p>\n\n\n\n
3.2.1 Operators<\/p>\n\n\n\n
An operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O, formally, is a method for transforming a vector |vi into<\/p>\n\n\n\n
another, |v<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i. Expressed mathematically, the operator acts from the left of the<\/p>\n\n\n\n
vector:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O|vi = |v<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
In the language of linear algebra, operators are represented as matrices. For<\/p>\n\n\n\n
example:<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
1 0<\/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
=<\/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
Box 3.3: Diagonalizable Operators and the Spectral Theorem<\/p>\n\n\n\n
An operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A is said to satisfy an eigenvalue equation if there exist<\/p>\n\n\n\n
some vectors |\ue00f<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i that are transformed into multiples of themselves:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A|\ue00f<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i = a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\ue00f<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
The number a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
is called an eigenvalue corresponding to the vector |\ue00f<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i, which<\/p>\n\n\n\n
is called an eigenvector. The eigenvalues may be distinct (simple) or some<\/p>\n\n\n\n
of them may be equal (multiple). In the latter case they are said to be de-<\/p>\n\n\n\n
generate. Not all matrices satisfy eigenvalue equations. Those that do are<\/p>\n\n\n\n
called diagonalizable. This name is due to the spectral theorem which<\/p>\n\n\n\n
says that such operators can be expressed as diagonal matrices in the basis of<\/p>\n\n\n\n
their eigenvectors with the eigenvalues as the diagonal elements:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
N =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\ue00f<\/p>\n\n\n\n
i<\/p>\n\n\n\n
ih\ue00f<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|.<\/p>\n\n\n\n
This statement essentially means that for a diagonalizable matrix, we can<\/p>\n\n\n\n
change basis to one in which the matrix is diagonal. In other words, there<\/p>\n\n\n\n
<\/p>\n\n\n\n
42 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
exists a non-singular matrix<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S bringing<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A to diagonal form<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
N by a similarity<\/p>\n\n\n\n
transformation:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
\u22121<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
N.<\/p>\n\n\n\n
A special class of diagonalizable operators is important in quantum mechanics,<\/p>\n\n\n\n
those that commute with their adjoint:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
N<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
N =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
N<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
N<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Such an operator is called a normal operator. Some kinds of normal operators<\/p>\n\n\n\n
especially relevant to us are:<\/p>\n\n\n\n
1. Unitary operators:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u22121<\/p>\n\n\n\n
2. Hermitian operators:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H<\/p>\n\n\n\n
(Also anti-Hermitian operators:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
= \u2212<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
3. Positive operators:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
P =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
. These operators are also Hermitian.<\/p>\n\n\n\n
The following important properties of eigenvalues are to be noted<\/p>\n\n\n\n
1. A Hermitian operator has real eigenvalues.<\/p>\n\n\n\n
2. A positive operator has positive eigenvalues.<\/p>\n\n\n\n
3. A unitary operator has eigenvalues of unit modulus, i.e., of the form e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
for real \u03b8.<\/p>\n\n\n\n
Suppose we \ufb01nd the dual of the transformed vector |v<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i. What is the op-<\/p>\n\n\n\n
erator in dual space that would take hv| to hv<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|? The answer is the adjoint<\/p>\n\n\n\n
operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
(\u2018O-dagger\u2019), de\ufb01ned by the equation<\/p>\n\n\n\n
hv|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
= hv<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|.<\/p>\n\n\n\n
We can also compare the transformed and original vectors by their inner<\/p>\n\n\n\n
product with another vector |wi, and thus de\ufb01ne the adjoint by<\/p>\n\n\n\n
(|wi,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O|vi) = (<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
|wi, |vi). (3.6)<\/p>\n\n\n\n
We can see that each side of Equation 3.6 is equivalent to hw|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O|vi = hw|v<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
hw<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|vi, where hw<\/p>\n\n\n\n
0<\/p>\n\n\n\n
| = hw|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
. Notice that the action of the adjoint is from the<\/p>\n\n\n\n
right.<\/p>\n\n\n\n
The matrix representation of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
is the complex conjugate transpose of the<\/p>\n\n\n\n
matrix for<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
O. Thus the dual vector hv| is sometimes also called the adjoint of<\/p>\n\n\n\n
the ket vector |vi<\/p>\n","protected":false},"excerpt":{"rendered":"
The state space may be said to be de\ufb01ned 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 […]<\/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-3987","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\/3987","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=3987"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3987\/revisions"}],"predecessor-version":[{"id":4556,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3987\/revisions\/4556"}],"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=3987"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3987"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3987"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}