In order to be able to de\ufb01ne orthogonality and the \u201csize\u201d of a vector, we<\/p>\n\n\n\n
need the notion of an inner product. This is just like the dot product of two<\/p>\n\n\n\n
vectors. This is basically a rule for assigning a (complex) number to a pair of<\/p>\n\n\n\n
vectors.<\/p>\n\n\n\n
For this we de\ufb01ne a dual vector space V<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
of same dimensions. Vectors in<\/p>\n\n\n\n
this space are represented by row matrices [\u03b1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
… \u03b1<\/p>\n\n\n\n
n<\/p>\n\n\n\n
]. The dual of the<\/p>\n\n\n\n
vector |vi = [v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
v<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. . . v<\/p>\n\n\n\n
n<\/p>\n\n\n\n
]<\/p>\n\n\n\n
T<\/p>\n\n\n\n
is represented by hv| = [v<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
1<\/p>\n\n\n\n
v<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
2<\/p>\n\n\n\n
… v<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
n<\/p>\n\n\n\n
] where the<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
denotes complex conjugation. Thus the matrix representation of the dual<\/p>\n\n\n\n
vector hv| is the complex conjugate transpose of |vi, denoted by |vi<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
.<\/p>\n\n\n\n
The inner product of vectors |\u03c6i and |\u03c8i is de\ufb01ned as the complex number<\/p>\n\n\n\n
h\u03c6|\u03c8i. (This bracket h\u00b7|\u00b7i for inner product is the origin of the Dirac bra-ket<\/p>\n\n\n\n
notation: the ket vector |\u00b7i has a dual bra vector h\u00b7| and their product gives<\/p>\n\n\n\n
the \u201cbra(c)ket\u201d.)<\/p>\n\n\n\n
If |\u03c8i = [\u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. . . \u03b1<\/p>\n\n\n\n
n<\/p>\n\n\n\n
]<\/p>\n\n\n\n
T<\/p>\n\n\n\n
and |\u03c6i = [\u03b2<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. . . \u03b2<\/p>\n\n\n\n
n<\/p>\n\n\n\n
]<\/p>\n\n\n\n
T<\/p>\n\n\n\n
then their inner<\/p>\n\n\n\n
product is<\/p>\n\n\n\n
h\u03c6|\u03c8i = \u03b2<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ \u03b2<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ … + \u03b2<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
n<\/p>\n\n\n\n
. (3.1)<\/p>\n\n\n\n
Some of the consequences of this de\ufb01nition are:<\/p>\n\n\n\n
\u2022 Norm of a vector is de\ufb01ned as kvk =<\/p>\n\n\n\n
p<\/p>\n\n\n\n
hv|vi. A vector is said to be<\/p>\n\n\n\n
normalized if it has unit norm. An arbitrary vector can be normalized<\/p>\n","protected":false},"excerpt":{"rendered":"
In order to be able to de\ufb01ne orthogonality and the \u201csize\u201d of a vector, we need the notion of an inner product. This is just like the dot product of two vectors. This is basically a rule for assigning a (complex) number to a pair of vectors. For this we de\ufb01ne a dual vector space […]<\/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-3981","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\/3981","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=3981"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3981\/revisions"}],"predecessor-version":[{"id":3982,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3981\/revisions\/3982"}],"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=3981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3981"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}