![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg2c.png\")
<\/p>\n\n\n\nThe Stern\u2013Gerlach setup of Figure 2.1 with the direction of inhomogeneity<\/p>\n\n\n\n
of the magnetic \ufb01eld de\ufb01ned as the z-axis is going to be the basis for de\ufb01ning<\/p>\n\n\n\n
2<\/p>\n\n\n\n
The standard unit for atomic magnetic moment is the Bohr Magneton, given by<\/p>\n\n\n\n
e~<\/p>\n\n\n\n
2m<\/p>\n\n\n\n
e<\/p>\n\n\n\n
c<\/p>\n\n\n\n
,<\/p>\n\n\n\n
numerically equal to about 5.8\u00d710<\/p>\n\n\n\n
\u22125<\/p>\n\n\n\n
eV\/T, where m<\/p>\n\n\n\n
e<\/p>\n\n\n\n
is the mass of the electron, c is the<\/p>\n\n\n\n
speed of light, and ~ = h\/2\u03c0 is the (reduced) Planck constant.<\/p>\n\n\n\n
<\/p>\n\n\n\n
A Simple Quantum System 19<\/p>\n\n\n\n
and measuring electron spin. Let\u2019s put it in a box and abbreviate as SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
. The<\/p>\n\n\n\n
incident beam consists of unpolarized electrons. The machine SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
produces<\/p>\n\n\n\n
as output two beams, one above the z = 0 axis with electrons of \u201cup\u201d spin<\/p>\n\n\n\n
and the other below z = 0 with electrons of \u201cdown\u201d spin. The SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
machine<\/p>\n\n\n\n
thus manufactures de\ufb01nite quantum states out of an arbitrary beam. If we<\/p>\n\n\n\n
isolate or \u201c\u2018ilter out\u201d either of these two states by blocking the other beam,<\/p>\n\n\n\n
the surviving beam is said to be polarized, and each electron in that beam is in<\/p>\n\n\n\n
a speci\ufb01c quantum state, called a basis state. A quantum state is represented<\/p>\n\n\n\n
as an angular bracket with a descriptive label inside: a very versatile and<\/p>\n\n\n\n
useful notation due to Dirac [29]. The two basis states are represented as |\u2191i<\/p>\n\n\n\n
and |\u2193i. (Note that these states are de\ufb01ned with respect to a direction of<\/p>\n\n\n\n
inhomogeneity of an applied magnetic \ufb01eld.) We can thus use the SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
\ufb01lter<\/p>\n\n\n\n
to prepare electrons in a prede\ufb01ned quantum state.<\/p>\n\n\n\n
|\u2191i |\u2193i<\/p>\n\n\n\n
FIGURE 2.3: The SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
\ufb01lters (the paths of the beams are bent back to z = 0<\/p>\n\n\n\n
using suitable magnets).<\/p>\n\n\n\n
The two SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
\ufb01lters, producing the two basis states, are illustrated in Figure<\/p>\n\n\n\n
2.3. (The paths of the beams can be bent back to the z = 0 axis by using<\/p>\n\n\n\n
appropriate magnets.)<\/p>\n\n\n\n
We thus not only use the SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
as a measuring tool for determining the<\/p>\n\n\n\n
state of an electron, but also as a factory for preparing a known state. This<\/p>\n\n\n\n
state will be labelled by the spin component along the z-direction.<\/p>\n\n\n\n
Suppose a beam of electrons in an unknown state is analyzed using an SG<\/p>\n\n\n\n
machine. The intensity of a particular output beam can be thought of as the<\/p>\n\n\n\n
number of electrons in the input beam that are in the corresponding output<\/p>\n\n\n\n
state. However there are subtleties here. A particular electron in the input<\/p>\n\n\n\n
beam randomly chooses the up or down output port of the machine. From<\/p>\n\n\n\n
the fraction of the total number of electrons that exit from a particular port,<\/p>\n\n\n\n
we can deduce the probability of the incident electron being in that particular<\/p>\n\n\n\n
state. This is how quantum mechanics works. We collect a set of statistics of<\/p>\n\n\n\n
probabilities from measurements and then infer the properties of the system<\/p>\n\n\n\n
and its state. This is the reason quantum mechanics is often described as a<\/p>\n\n\n\n
probabilistic theory.<\/p>\n\n\n\n
A system could be in a purely quantum mechanical state, with quantum<\/p>\n\n\n\n
probabilities, and is said to be a pure quantum state. However, classical un-<\/p>\n\n\n\n
certainties could also be present in a given system, in which case the system<\/p>\n\n\n\n
is said to be in a mixed quantum state. For example, the unpolarized beam<\/p>\n\n\n\n
of silver atoms from the oven in the Stern\u2013Gerlach experiment is actually in<\/p>\n\n\n\n
a mixed state. We will see more of this distinction in later chapters. For our<\/p>\n\n\n\n
present introduction, however, we will assume that our systems are always in<\/p>\n\n\n\n
pure quantum states only<\/p>\n","protected":false},"excerpt":{"rendered":"
The Stern\u2013Gerlach setup of Figure 2.1 with the direction of inhomogeneity of the magnetic \ufb01eld de\ufb01ned as the z-axis is going to be the basis for de\ufb01ning 2 The standard unit for atomic magnetic moment is the Bohr Magneton, given by e~ 2m e c , numerically equal to about 5.8\u00d710 \u22125 eV\/T, where m […]<\/p>\n","protected":false},"author":1,"featured_media":3940,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[488],"tags":[],"class_list":["post-3959","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-3-the-quantum-postulates"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/science-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3959","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=3959"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3959\/revisions"}],"predecessor-version":[{"id":3960,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3959\/revisions\/3960"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3940"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3959"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3959"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}