Inner product of vectors gives the component of one vector in the direction<\/p>\n\n\n\n
of the other. Similarly for quantum states, the inner product h\u03c8|\u03c6i is the<\/p>\n\n\n\n
probability amplitude that one state is along the other. For example,<\/p>\n\n\n\n
|\u03c8i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
3<\/p>\n\n\n\n
|0i +<\/p>\n\n\n\n
r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
3<\/p>\n\n\n\n
|1i.<\/p>\n\n\n\n
Then the inner product<\/p>\n\n\n\n
h0|\u03c8i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
3<\/p>\n\n\n\n
is the probability amplitude that the state |\u03c8i has spin up.<\/p>\n\n\n\n
As another example, the Hilbert space of position states of a particle along<\/p>\n\n\n\n
the x-axis would have an in\ufb01nite set of basis states |xi. A general state |\u03c8i<\/p>\n\n\n\n
has a probability amplitude hx|\u03c8i = \u03c8(x) of being found at the location x.<\/p>\n\n\n\n
This probability amplitude as a function of position is better known as the<\/p>\n\n\n\n
wave function of the particle.<\/p>\n\n\n\n
The inner product also comes in when describing the outcome of a process<\/p>\n\n\n\n
that transforms a system from initial state |\u03c8<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i to \ufb01nal state |\u03c8<\/p>\n\n\n\n
f<\/p>\n\n\n\n
i. The mod-<\/p>\n\n\n\n
square of the probability amplitude for this process is then the probability<\/p>\n\n\n\n
that such an event can occur:<\/p>\n\n\n\n
P(|\u03c8<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i \u2192 |\u03c8<\/p>\n\n\n\n
f<\/p>\n\n\n\n
i) = |h\u03c8<\/p>\n\n\n\n
f<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. (3.3)<\/p>\n\n\n\n
This statement, one of the underpinnings of quantum mechanics, is known as<\/p>\n\n\n\n
the Born rule after Max Born,<\/p>\n\n\n\n
3<\/p>\n\n\n\n
who \ufb01rst postulated it.<\/p>\n\n\n\n
3.1.3 Phases<\/p>\n\n\n\n
The coe\ufb03cients in the expansion of a state in terms of the basis states are<\/p>\n\n\n\n
complex numbers in general. We saw one reason for this in the last chapter:<\/p>\n\n\n\n
3<\/p>\n\n\n\n
In a 1926 paper in a German journal, Born mentioned the probability interpretation in<\/p>\n\n\n\n
a footnote.<\/p>\n\n\n\n
<\/p>\n\n\n\n
40 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
we need to account for interference when probability amplitudes are added.<\/p>\n\n\n\n
Now a complex number has a modulus and a phase: z = x + iy has magnitude<\/p>\n\n\n\n
r =<\/p>\n\n\n\n
p<\/p>\n\n\n\n
x<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ y<\/p>\n\n\n\n
2<\/p>\n\n\n\n
and a phase \u03c6 = tan<\/p>\n\n\n\n
\u22121<\/p>\n\n\n\n
y\/x. r and \u03c6 are real numbers and we<\/p>\n\n\n\n
express the same complex number in modular form as z = re<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
. Suppose we<\/p>\n\n\n\n
write<\/p>\n\n\n\n
|\u03c8i = r<\/p>\n\n\n\n
1<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|0i + r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1i. (3.4)<\/p>\n\n\n\n
Di\ufb00erent values of r<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
and r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
give di\ufb00erent vectors. For a given vector |\u03c8i,<\/p>\n\n\n\n
we can factor out one of the phases to write<\/p>\n\n\n\n
|\u03c8i = e<\/p>\n\n\n\n
i\u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
r<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|0i + r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i(\u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2212\u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
)<\/p>\n\n\n\n
|1i<\/p>\n\n\n\n
\ue011<\/p>\n\n\n\n
(3.5)<\/p>\n\n\n\n
The factored phase \u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
is called a global phase. This cannot be measured by<\/p>\n\n\n\n
any experiment since experiments only measure probabilities. In other words<\/p>\n\n\n\n
the above state is experimentally indistinguishable from the state<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i = r<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|0i + r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i(\u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2212\u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
)<\/p>\n\n\n\n
|1i,<\/p>\n\n\n\n
since |h\u03c8|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1. What is measurable, however, is the relative phase<\/p>\n\n\n\n
(\u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2212 \u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
), which will show up in an interference experiment. The set of all<\/p>\n\n\n\n
states di\ufb00ering by a global phase is called a ray in Hilbert space.<\/p>\n\n\n\n
Thus the space of quantum states of a system is the space of rays in Hilbert<\/p>\n\n\n\n
space, also called the projective Hilbert space. We will not emphasize this<\/p>\n\n\n\n
di\ufb00erence in what follows, but it is a point to be kept in mind.<\/p>\n\n\n\n
The fact that relative phases between components in a superposition state<\/p>\n\n\n\n
are very important will become more relevant when we consider operations<\/p>\n\n\n\n
on quantum systems that impart selective phases to one basis state, say |1i.<\/p>\n\n\n\n
For instance, consider an operation<\/p>\n\n\n\n
|0i \u2192 |0i; |1i \u2192 e<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
|1i.<\/p>\n\n\n\n
Though such an operation produces indistinguishable states out of basis states,<\/p>\n\n\n\n
the e\ufb00ect will be non-trivial on superposition states, since it would introduce<\/p>\n\n\n\n
a relative phase between the |0i and |1i components:<\/p>\n\n\n\n
|\u03c8i = c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|0i + c<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1i \u2192 c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|0i + e<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
c<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1i<\/p>\n","protected":false},"excerpt":{"rendered":"
Inner product of vectors gives the component of one vector in the direction of the other. Similarly for quantum states, the inner product h\u03c8|\u03c6i is the probability amplitude that one state is along the other. For example, |\u03c8i = 1 \u221a 3 |0i + r 2 3 |1i. Then the inner product h0|\u03c8i = 1 […]<\/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-3985","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\/3985","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=3985"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3985\/revisions"}],"predecessor-version":[{"id":4555,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3985\/revisions\/4555"}],"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=3985"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3985"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}