We saw in the previous chapter how to describe the spin state of an elec-<\/p>\n\n\n\n
tron.<\/p>\n\n\n\n
2<\/p>\n\n\n\n
The \u201csystem\u201d in this case is just that property of an electron that<\/p>\n\n\n\n
responds to a gradient in an applied magnetic \ufb01eld. The state of this system<\/p>\n\n\n\n
is a member of a 2-dimensional vector space. This is because this spin can take<\/p>\n\n\n\n
one of only two possible values, \u00b1~\/2. An electron in either of these states is<\/p>\n\n\n\n
described by the basis vectors<\/p>\n\n\n\n
|0i = |+<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i, |1i = |\u2212<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
A general state |\u03c8i is a linear combination of these basis vectors with complex<\/p>\n\n\n\n
coe\ufb03cients:<\/p>\n\n\n\n
|\u03c8i = \u03b1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|0i + \u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|1i.<\/p>\n\n\n\n
As we will show, these basis states are mutually orthogonal and are nor-<\/p>\n\n\n\n
malized, so that they form an orthonormal basis. This is similar to repre-<\/p>\n\n\n\n
senting a physical 2-dimensional vector in terms of its components along two<\/p>\n\n\n\n
orthogonal directions. This vector is represented as the column matrix of its<\/p>\n\n\n\n
components: [<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
]<\/p>\n\n\n\n
T<\/p>\n\n\n\n
.<\/p>\n\n\n\n
We can easily generalize this to higher dimensions. Such a picture is rel-<\/p>\n\n\n\n
evant when the set of basis states for the system is larger. For example, the<\/p>\n\n\n\n
system may be the magnetic moment of a spin-<\/p>\n\n\n\n
3<\/p>\n\n\n\n
2<\/p>\n\n\n\n
atomic nucleus. This object<\/p>\n\n\n\n
would have four possible states distinguished in a non-uniform magnetic \ufb01eld:<\/p>\n\n\n\n
{|ji} = {|<\/p>\n\n\n\n
3<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i, |<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i, |\u2212<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i, |\u2212<\/p>\n\n\n\n
3<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i}.<\/p>\n\n\n\n
Another example is the electronic energy of the hydrogen atom. This system<\/p>\n\n\n\n
actually has a countable in\ufb01nity of possible energy states labelled by the so-<\/p>\n\n\n\n
called \u201cprincipal quantum number\u201d n:<\/p>\n\n\n\n
{|ni}, n = 0, 1, 2 . . . .<\/p>\n\n\n\n
This Hilbert space is actually in\ufb01nite dimensional, though we might say the<\/p>\n\n\n\n
dimensionality is \u201ccountable.\u201d If we were concentrating on the position states<\/p>\n\n\n\n
of a particle con\ufb01ned to a line then the possible states are a continuous in\ufb01nity<\/p>\n\n\n\n
labelled by the values of the position x:<\/p>\n\n\n\n
{|xi}, \u2212 \u221e \u2264 x \u2264 +\u221e.<\/p>\n\n\n\n
This Hilbert space is also in\ufb01nite dimensional, and the dimensionality is con-<\/p>\n\n\n\n
tinuous and uncountable.<\/p>\n\n\n\n
2<\/p>\n\n\n\n
The spin space is a subspace of the total state space of an electron, which contains<\/p>\n\n\n\n
descriptors of all possible compatible measurable properties of the electron. This Hilbert<\/p>\n\n\n\n
space can be expressed as a direct product of the independent subspaces.<\/p>\n\n\n\n
<\/p>\n\n\n\n
The Essentials of Quantum Mechanics 37<\/p>\n\n\n\n
Box 3.2: Hilbert Space<\/p>\n\n\n\n
The linear vector space of Box 3.1 turns into something rich enough to<\/p>\n\n\n\n
represent states of a physical system if a little more structure is added to it.<\/p>\n\n\n\n
We now have a Hilbert space H<\/p>\n\n\n\n
n<\/p>\n\n\n\n
which is de\ufb01ned as a complex vector space<\/p>\n\n\n\n
with an inner product (., .) \u2208 which satis\ufb01es<\/p>\n\n\n\n
(I1) (v, v) \u2265 0, (v, v) = 0 i\ufb00 v = 0.<\/p>\n\n\n\n
(I2) (u, v) = (v, u)<\/p>\n\n\n\n
\u2217<\/p>\n\n\n\n
(I3) (u, \u03b1v) = \u03b1(u, v)<\/p>\n\n\n\n
(I4) (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
3<\/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
With this structure in place, a vector space becomes a pre-Hilbert space,<\/p>\n\n\n\n
and is a Hilbert space if the dimension is \ufb01nite. For in\ufb01nite-dimensional Hilbert<\/p>\n\n\n\n
spaces, one needs the additional criterion of the space being complete under<\/p>\n\n\n\n
the inner product, which we will not discuss here<\/p>\n","protected":false},"excerpt":{"rendered":"
We saw in the previous chapter how to describe the spin state of an elec- tron. 2 The \u201csystem\u201d in this case is just that property of an electron that responds to a gradient in an applied magnetic \ufb01eld. The state of this system is a member of a 2-dimensional vector space. This is because […]<\/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-3979","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\/3979","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=3979"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3979\/revisions"}],"predecessor-version":[{"id":4554,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3979\/revisions\/4554"}],"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=3979"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3979"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3979"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}