From the components of two vectors, we can construct a matrix by the<\/p>\n\n\n\n
outer product. For vectors |v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i = [a<\/p>\n\n\n\n
1<\/p>\n\n\n\n
a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
…a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
]<\/p>\n\n\n\n
T<\/p>\n\n\n\n
and |v<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i = [b<\/p>\n\n\n\n
1<\/p>\n\n\n\n
b<\/p>\n\n\n\n
2<\/p>\n\n\n\n
…b<\/p>\n\n\n\n
n<\/p>\n\n\n\n
]<\/p>\n\n\n\n
T<\/p>\n\n\n\n
, this<\/p>\n\n\n\n
is denoted by |v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
ihv<\/p>\n\n\n\n
2<\/p>\n\n\n\n
| and represented by a matrix given by<\/p>\n\n\n\n
|v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
ihv<\/p>\n\n\n\n
2<\/p>\n\n\n\n
| =<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
a<\/p>\n\n\n\n
1<\/p>\n\n\n\n
a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
h<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
1<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
2<\/p>\n\n\n\n
… b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
a<\/p>\n\n\n\n
1<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
1<\/p>\n\n\n\n
a<\/p>\n\n\n\n
1<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
2<\/p>\n\n\n\n
… a<\/p>\n\n\n\n
1<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
n<\/p>\n\n\n\n
a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
1<\/p>\n\n\n\n
a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
2<\/p>\n\n\n\n
… a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
n<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
. …<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
1<\/p>\n\n\n\n
a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
2<\/p>\n\n\n\n
… a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
b<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
(3.12)<\/p>\n\n\n\n
Operators on a Hilbert space can be represented in terms of outer products<\/p>\n\n\n\n
of the basis vectors of the space: a matrix A with matrix elements A<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
is the<\/p>\n\n\n\n
expansion<\/p>\n\n\n\n
A \u2261<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,j<\/p>\n\n\n\n
A<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
|iihj|.<\/p>\n\n\n\n
Conversely, in the above basis, a matrix A has elements<\/p>\n\n\n\n
A<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
= hi|A|ji,<\/p>\n\n\n\n
<\/p>\n\n\n\n
The Essentials of Quantum Mechanics 47<\/p>\n\n\n\n
where i is the row index and j is the column index. The space of matrices<\/p>\n\n\n\n
is thus a linear vector space with basis \u201cvectors\u201d given by the matrices |iihj|<\/p>\n\n\n\n
composed of the outer products of the basis vectors of the Hilbert space. For<\/p>\n\n\n\n
example, in 2 dimensions, the basis matrices will be<\/p>\n\n\n\n
|0ih0| =<\/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
h<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
,<\/p>\n\n\n\n
|0ih1| =<\/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
h<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
,<\/p>\n\n\n\n
|1ih0| =<\/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
h<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
,<\/p>\n\n\n\n
|1ih1| =<\/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
h<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
.<\/p>\n\n\n\n
(3.13)<\/p>\n\n\n\n
A 2\u00d72 matrix is represented as<\/p>\n\n\n\n
A = A<\/p>\n\n\n\n
00<\/p>\n\n\n\n
|0ih0| + A<\/p>\n\n\n\n
01<\/p>\n\n\n\n
|0ih1| + A<\/p>\n\n\n\n
10<\/p>\n\n\n\n
|1ih0| + A<\/p>\n\n\n\n
11<\/p>\n\n\n\n
|1ih1|<\/p>\n\n\n\n
= A<\/p>\n\n\n\n
00<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
+ A<\/p>\n\n\n\n
01<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
+ A<\/p>\n\n\n\n
10<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
+ A<\/p>\n\n\n\n
11<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
A<\/p>\n\n\n\n
00<\/p>\n\n\n\n
A<\/p>\n\n\n\n
01<\/p>\n\n\n\n
A<\/p>\n\n\n\n
10<\/p>\n\n\n\n
A<\/p>\n\n\n\n
11<\/p>\n\n\n\n
#<\/p>\n\n\n\n
.<\/p>\n\n\n\n
The spectral theorem can then be expressed in the form<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
iha<\/p>\n\n\n\n
i<\/p>\n\n\n\n
| (3.14)<\/p>\n\n\n\n
where the a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
are the eigenvalues of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A corresponding to its eigenvectors |a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
Note how we use the eigenvalue itself as the label for the corresponding eigen-<\/p>\n\n\n\n
state! The set of eigenvalues is called the spectrum of the operator and this<\/p>\n\n\n\n
equation is called the spectral resolution of the operator.<\/p>\n\n\n\n
Example 3.2.3. Matrix representation for the spin operators in the compu-<\/p>\n\n\n\n
tational basis:{|0i, |1i} = {|\u2191i<\/p>\n\n\n\n
z<\/p>\n\n\n\n
, |\u2193i<\/p>\n\n\n\n
z<\/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
|0ih0| \u2212<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1ih1| =<\/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
.<\/p>\n\n\n\n
To construct the representation for<\/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 we \ufb01rst note that it is<\/p>\n\n\n\n
diagonal in the basis of its own eigenvectors:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
=<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|\u2191i<\/p>\n\n\n\n
x<\/p>\n\n\n\n
h\u2191|<\/p>\n\n\n\n
x<\/p>\n\n\n\n
\u2212<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|\u2193i<\/p>\n\n\n\n
x<\/p>\n\n\n\n
h\u2193|<\/p>\n\n\n\n
x<\/p>\n","protected":false},"excerpt":{"rendered":"
From the components of two vectors, we can construct a matrix by the outer product. For vectors |v 1 i = [a 1 a 2 …a n ] T and |v 2 i = [b 1 b 2 …b n ] T , this is denoted by |v 1 ihv 2 | and represented by […]<\/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-3993","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\/3993","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=3993"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3993\/revisions"}],"predecessor-version":[{"id":4558,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3993\/revisions\/4558"}],"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=3993"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3993"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3993"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}