When you describe the state of a physical system, you collect all the pa-<\/p>\n\n\n\n
rameters required to fully specify it: for instance, the state of a ball may be<\/p>\n\n\n\n
speci\ufb01ed by its position in space, its velocity, and maybe its rate of spin; the<\/p>\n\n\n\n
state of a volume of gas by its temperature and its pressure. If you are trying<\/p>\n\n\n\n
to describe a quantum system like a hydrogen atom, you may think specifying<\/p>\n\n\n\n
the position and velocity of the atom and its constituents, the nucleus and the<\/p>\n\n\n\n
electron would give the quantum state. Whether this is true, or even possible<\/p>\n\n\n\n
in principle, depends on how you are trying to see the atom: which properties<\/p>\n\n\n\n
you are trying to measure and what experiments you are using to measure its<\/p>\n\n\n\n
properties,<\/p>\n\n\n\n
So we \ufb01rst identify a system, an isolated set of physical properties that we<\/p>\n\n\n\n
have experimental access to and are trying to describe. The quantum state of<\/p>\n\n\n\n
the system, denoted by the notation |statei,<\/p>\n\n\n\n
1<\/p>\n\n\n\n
is represented by measured values<\/p>\n\n\n\n
1<\/p>\n\n\n\n
The notation due to Dirac that we use in quantum mechanics may need some more<\/p>\n\n\n\n
clari\ufb01cation. A state is labelled abstractly as |\u03c8i, or as |0i, or as |x<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i. The labels are just<\/p>\n\n\n\n
mnemonics to tag the state. They may be numbers but are not the components of the<\/p>\n\n\n\n
vector in any basis. For instance, |0i does not mean the zero vector, for which we will<\/p>\n\n\n\n
use the notation<\/p>\n\n\n\n
~<\/p>\n\n\n\n
0. The 0 used as a label is an indication of a \ufb01rst basis vector in the<\/p>\n\n\n\n
computational basis.<\/p>\n\n\n\n
33<\/p>\n\n\n\n
<\/p>\n\n\n\n
34 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
of the physical properties used to describe it. It is important to know which<\/p>\n\n\n\n
properties are independent of each other, measuring which do not interfere<\/p>\n\n\n\n
with the other properties. The outcome of the measurement could be one of<\/p>\n\n\n\n
many possibilities. Each possibility labels a di\ufb00erent state. The set of all these<\/p>\n\n\n\n
forms the state space of the system.<\/p>\n\n\n\n
In the last chapter, the particular property of spin of an electron was<\/p>\n\n\n\n
targeted for study, by designing the Stern\u2013Gerlach experiment. This led to a<\/p>\n\n\n\n
state space of two states.<\/p>\n\n\n\n
The properties of a quantum state turn out to conform to those of a vector<\/p>\n\n\n\n
in the mathematical sense: a member of a complex vector space (see Box 3.1),<\/p>\n\n\n\n
with a notion of norm or inner product de\ufb01ned on it. Such a vector space<\/p>\n\n\n\n
is called a Hilbert space (see Box 3.2). The vector describing a state must<\/p>\n\n\n\n
also have unit norm, since we will be attaching a notion of probabilities to<\/p>\n\n\n\n
the state. The complex vector may also have imaginary components, but an<\/p>\n\n\n\n
overall phase factor is unimportant since we have no means of measuring it.<\/p>\n\n\n\n
Thus a state is unit vector in complex space, modulo an overall phase factor.<\/p>\n\n\n\n
Postulate 1. The state of an isolated quantum mechanical system is a unit<\/p>\n\n\n\n
vector in Hilbert space.<\/p>\n\n\n\n
Box 3.1: Linear Vector Space<\/p>\n\n\n\n
A vector space V is a set of objects v, called vectors, that abstractly<\/p>\n\n\n\n
satisfy the properties of closure under an operation of addition, and under<\/p>\n\n\n\n
multiplication by a scalar which belongs to a \ufb01eld F, that for example could<\/p>\n\n\n\n
be real or complex numbers. In what follows, a vector is designated by a<\/p>\n\n\n\n
boldface, such as v, while a scalar is not.<\/p>\n\n\n\n
The axioms de\ufb01ning a vector space are<\/p>\n\n\n\n
1. Addition: one can de\ufb01ne an operation \u201c+\u201d such that for any vectors<\/p>\n\n\n\n
v<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2208 V<\/p>\n\n\n\n
n<\/p>\n\n\n\n
,<\/p>\n\n\n\n
(A1) V<\/p>\n\n\n\n
n<\/p>\n\n\n\n
is closed under +: v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ v<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2208 V<\/p>\n\n\n\n
n<\/p>\n\n\n\n
,<\/p>\n\n\n\n
(A2) + is commutative: 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
2<\/p>\n\n\n\n
+ v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
,<\/p>\n\n\n\n
(A3) + is associative: (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
3<\/p>\n\n\n\n
= 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
3<\/p>\n\n\n\n
) .<\/p>\n\n\n\n
(A4) \u2203 a zero vector or additive identity 0 \u2208 V<\/p>\n\n\n\n
n<\/p>\n\n\n\n
such that v + 0 = v.<\/p>\n\n\n\n
(A5) For each v \u2208 V, \u2203 an additive inverse \u2212v \u2208 V<\/p>\n\n\n\n
n<\/p>\n\n\n\n
such that v+(\u2212v) =<\/p>\n\n\n\n
0.<\/p>\n\n\n\n
2. Scalar Multiplication: for any scalar \u03b1 \u2208 F and vector v<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2208 V<\/p>\n\n\n\n
n<\/p>\n\n\n\n
,<\/p>\n\n\n\n
(M1) V is closed under scalar multiplication: \u03b1v \u2208 V<\/p>\n\n\n\n
n<\/p>\n\n\n\n
,<\/p>\n\n\n\n
(M2) For the multiplicative identity 1, we have 1v = v,<\/p>\n\n\n\n
(M3) Multiplication by the scalar 0 gives the zero vector: 0v = 0,<\/p>\n\n\n\n
<\/p>\n\n\n\n
The Essentials of Quantum Mechanics 35<\/p>\n\n\n\n
(M4) Associativity: \u03b1(\u03b2v) = (\u03b1\u03b2)v,<\/p>\n\n\n\n
(M5) Distributivity over vector addition: \u03b1(v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ v<\/p>\n\n\n\n
2<\/p>\n\n\n\n
) = \u03b1v<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ \u03b1v<\/p>\n\n\n\n
2<\/p>\n\n\n\n
,<\/p>\n\n\n\n
(M6) Distributivity over scalar addition: (\u03b1 + \u03b2)v = \u03b1v + \u03b2v<\/p>\n\n\n\n
The element \u2212v = (\u22121)v is the additive inverse of v.<\/p>\n\n\n\n
Vectors can be represented by components if we choose a set of \u201ccoor-<\/p>\n\n\n\n
dinates\u201d or basis vectors for the representation. A basis for a vector space<\/p>\n\n\n\n
consists of a set of vectors {e<\/p>\n\n\n\n
i<\/p>\n\n\n\n
} whose de\ufb01ning properties are:<\/p>\n\n\n\n
1. they are linearly independent: no basis vector can be expressed as a<\/p>\n\n\n\n
linear combination of the other basis vectors; no set of numbers {a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
}<\/p>\n\n\n\n
can be found such that<\/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
e<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= 0.<\/p>\n\n\n\n
2. they span the vector space V: any vector v \u2208 V can be expressed as a<\/p>\n\n\n\n
linear combination of the basis vectors:<\/p>\n\n\n\n
v = c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
e<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ c<\/p>\n\n\n\n
2<\/p>\n\n\n\n
e<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ \u00b7\u00b7\u00b7 + c<\/p>\n\n\n\n
n<\/p>\n\n\n\n
e<\/p>\n\n\n\n
n<\/p>\n\n\n\n
.<\/p>\n\n\n\n
The index i counts the basis vectors: i = 1…n. The total number n of basis<\/p>\n\n\n\n
vectors is the dimension of the vector space. This dimension can be \ufb01nite or<\/p>\n\n\n\n
in\ufb01nite. The index i can be discrete or continuous. We will only be dealing<\/p>\n\n\n\n
here with \ufb01nite-dimensional complex vector spaces.<\/p>\n\n\n\n
A vector space in general has more than one basis. A vector represented<\/p>\n\n\n\n
by its components is also represented as a column matrix:<\/p>\n\n\n\n
v =<\/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
c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
c<\/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
c<\/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
.<\/p>\n\n\n\n
To save space, we will also represent this as the transpose of a row vector<\/p>\n\n\n\n
v =<\/p>\n\n\n\n
h<\/p>\n\n\n\n
c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
c<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. . . c<\/p>\n\n\n\n
n<\/p>\n\n\n\n
i<\/p>\n\n\n\n
T<\/p>\n\n\n\n
.<\/p>\n\n\n\n
This representation is extremely useful when we consider transformations of<\/p>\n\n\n\n
a vector space into another by linear maps, which can be represented by<\/p>\n\n\n\n
matrices.<\/p>\n\n\n\n
Vector spaces are familiar to us from 3-dimensional spacial vectors, but<\/p>\n\n\n\n
the above de\ufb01nitions generalize such properties to a larger class of objects.<\/p>\n\n\n\n
We \ufb01nd that even continuous functions of complex numbers that are in\ufb01nitely<\/p>\n\n\n\n
di\ufb00erentiable and vanish fast at in\ufb01nity form a vector space<\/p>\n","protected":false},"excerpt":{"rendered":"
When you describe the state of a physical system, you collect all the pa- rameters required to fully specify it: for instance, the state of a ball may be speci\ufb01ed by its position in space, its velocity, and maybe its rate of spin; the state of a volume of gas by its temperature and its […]<\/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-3977","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\/3977","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=3977"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3977\/revisions"}],"predecessor-version":[{"id":4553,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3977\/revisions\/4553"}],"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=3977"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3977"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3977"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}