- \n
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<\/li>\n<\/ul>\n<\/div>\n<\/div>\n\n\n\n16 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
to measure the magnetic moment of atoms gave unexpected results. This was<\/p>\n\n\n\n
the classic Stern\u2013Gerlach experiment [36], designed to measure the magnetic<\/p>\n\n\n\n
moments of atoms. The results brought out a new quantum property of an<\/p>\n\n\n\n
electron, called intrinsic spin, which could take on quantized values, i.e., one<\/p>\n\n\n\n
of two values only.<\/p>\n\n\n\n
We can get a feel for the physics by looking at the classical de\ufb01nition of the<\/p>\n\n\n\n
magnetic moment. The revolution of electrons around the nucleus of an atom<\/p>\n\n\n\n
is like a circulating current, and a circulating current is a magnetic dipole.<\/p>\n\n\n\n
The dipole moment ~\u00b5 equals the current times the area of the current loop,<\/p>\n\n\n\n
with direction given by the axis about which the current circulates. When a<\/p>\n\n\n\n
magnetic dipole is subjected to a non-uniform magnetic \ufb01eld<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
B (~r), it feels a<\/p>\n\n\n\n
force along the direction of change of the \ufb01eld:<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
F =<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
\u2207 (~\u00b5 \u00b7<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
B ), and will be<\/p>\n\n\n\n
de\ufb02ected. Measuring the de\ufb02ection in a known magnetic \ufb01eld, the value of the<\/p>\n\n\n\n
magnetic moment can be calculated.<\/p>\n\n\n\n
A schematic setup of the Stern\u2013Gerlach type is shown in Figure 2.1.<\/p>\n\n\n\n
N<\/p>\n\n\n\n
N<\/p>\n\n\n\n
SS<\/p>\n\n\n\n
FIGURE 2.1: (a) The Stern\u2013Gerlach Setup. (b) The inhomogeneous magnetic<\/p>\n\n\n\n
\ufb01eld between asymmetric pole pieces.<\/p>\n\n\n\n
The arrangement is such that in the region the electron beam passes<\/p>\n\n\n\n
through, magnetic \ufb01eld is nearly constant in direction (taken to be \u02c6z)<\/p>\n\n\n\n
1<\/p>\n\n\n\n
but<\/p>\n\n\n\n
has a strong z-dependent change in magnitude, i.e.,<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
B \u2248 B(z)\u02c6z. The force<\/p>\n\n\n\n
on the dipole when placed in this \ufb01eld is<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
F =<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
\u2207 (~\u00b5 \u00b7<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
B ) =<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
\u2207 (\u00b5<\/p>\n\n\n\n
z<\/p>\n\n\n\n
B(z)) = \u00b5<\/p>\n\n\n\n
z<\/p>\n\n\n\n
\u2202B(z)<\/p>\n\n\n\n
\u2202z<\/p>\n\n\n\n
\u02c6z. (2.1)<\/p>\n\n\n\n
Thus the atom is de\ufb02ected along the z-axis by an amount proportional to the<\/p>\n\n\n\n
z-component of its magnetic moment. Remember: since a magnetic \ufb01eld can<\/p>\n\n\n\n
de\ufb02ect magnetic moments depending on their magnitudes and directions, it<\/p>\n\n\n\n
1<\/p>\n\n\n\n
By convention, in physics experiments, the coordinate system is aligned to the direction<\/p>\n\n\n\n
of the magnetic \ufb01eld, which is always taken to be the z-axis.<\/p>\n\n\n\n
![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg2a.png\")
<\/p>\n\n\n\nA Simple Quantum System 17<\/p>\n\n\n\n
can be used to select particular magnetic moments. The net magnetic moment<\/p>\n\n\n\n
of a collection of atoms is just the vector sum of all the individual atomic<\/p>\n\n\n\n
magnetic moments. A beam of atoms having a speci\ufb01c constant magnetic<\/p>\n\n\n\n
moment along a particular direction is said to be polarized. It is possible to<\/p>\n\n\n\n
produce a beam of polarized atoms by speci\ufb01c procedures.<\/p>\n\n\n\n
The beam of silver atoms used in the original Stern\u2013Gerlach experiment<\/p>\n\n\n\n
was produced by heating silver in an oven. Each atom emerges with a random<\/p>\n\n\n\n
direction of magnetic moment and the net magnetic moment is zero. If such<\/p>\n\n\n\n
an unpolarized beam is sent into the non-uniform magnetic \ufb01eld, then since<\/p>\n\n\n\n
each atomic magnetic moment is arbitrarily oriented, the z-component of the<\/p>\n\n\n\n
magnetic moment could vary between \u00b1\u00b5. So we expect the beam to spread<\/p>\n\n\n\n
between two extreme limits, which de\ufb01ne the value \u00b5 of the magnetic moment<\/p>\n\n\n\n
(see Figure 2.2(a)).<\/p>\n\n\n\n
N<\/p>\n\n\n\n
N<\/p>\n\n\n\n
SS<\/p>\n\n\n\n
N<\/p>\n\n\n\n
N<\/p>\n\n\n\n
SS<\/p>\n\n\n\n
FIGURE 2.2: The Stern Gerlach experiment: (a) The classically expected<\/p>\n\n\n\n
result. (b) What was actually observed.<\/p>\n\n\n\n
Suppose, for simplicity, that this experiment is performed with a beam of<\/p>\n\n\n\n
hydrogen atoms. The hydrogen atom consists of a single electron and a proton.<\/p>\n\n\n\n
Classically one can think of the electron as orbiting the proton. The atom has<\/p>\n\n\n\n
associated with it energy that has various contributions. At the \ufb01rst level, the<\/p>\n\n\n\n
contribution depends on the electrostatic energy, determined by the distance<\/p>\n\n\n\n
between the positive nucleus and the negative electron, i.e., the orbit radius.<\/p>\n\n\n\n
Let\u2019s assume that this is the minimum possible, or the \u201cground state\u201d radius.<\/p>\n\n\n\n
Second, the energy depends on the orbital velocity of the electron, contributing<\/p>\n\n\n\n
to the orbital angular momentum of the atom. Finally, if the atom is subjected<\/p>\n\n\n\n
to a magnetic \ufb01eld, its interaction with the \ufb01eld contributes to the energy. The<\/p>\n\n\n\n
classical analogy however, is severely limited, because radius, velocity, and<\/p>\n\n\n\n
component of magnetic moment along the magnetic \ufb01eld direction, all can<\/p>\n\n\n\n
take continuous possible values whereas an atom\u2019s energy is quantized: takes<\/p>\n\n\n\n
on only certain discrete values. Correspondingly, the atom is said to exist in<\/p>\n\n\n\n
possible quantized energy states.<\/p>\n\n\n\n
The lowest energy state (s-state) has a symmetrical distribution of veloci-<\/p>\n\n\n\n
ties such that there is no net circulating velocity. Therefore, in this state, the<\/p>\n\n\n\n
atom is expected to have zero magnetic moment since the average \u201ccurrent\u201d<\/p>\n\n\n\n
is zero. This means that the beam, when it passes through the Stern\u2013Gerlach<\/p>\n\n\n\n
<\/p>\n\n\n\n
18 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
setup, will just proceed without de\ufb02ecting or spreading. The silver atoms used<\/p>\n\n\n\n
in the original experiment also have zero average magnetic moment. The ex-<\/p>\n\n\n\n
periment with hydrogen was also subsequently performed, in 1927 [56].<\/p>\n\n\n\n
When the experiment was actually performed, there were two surprises.<\/p>\n\n\n\n
The beam of atoms did not pass through unde\ufb02ected. Nor did it spread, but<\/p>\n\n\n\n
instead split into two beams symmetrically about the central axis, one up and<\/p>\n\n\n\n
one down. Measuring the positions of the beams indicated a value of \u00b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
,<\/p>\n\n\n\n
in appropriate units,<\/p>\n\n\n\n
2<\/p>\n\n\n\n
for the magnetic moment! The appearance of Planck\u2019s<\/p>\n\n\n\n
constant h (numerically 6.63\u00d710<\/p>\n\n\n\n
\u221234<\/p>\n\n\n\n
Js), in the magnitude, is a signature of the<\/p>\n\n\n\n
quantum nature of this property. Whose magnetic moment? The atom in s-<\/p>\n\n\n\n
state has no net magnetic moment, but has a lone electron. So this had to be an<\/p>\n\n\n\n
intrinsic moment associated with the electron in the atom. Thus, the magnetic<\/p>\n\n\n\n
moment of the electron is allowed only to take one of two discrete values!<\/p>\n\n\n\n
Classically, the magnetic moment is proportional to the angular momentum of<\/p>\n\n\n\n
the system. Here, the electron magnetic moment is proportional to a property<\/p>\n\n\n\n
called intrinsic spin, which mathematically behaves like angular momentum.<\/p>\n\n\n\n
Thus was discovered the spin of the electron, a quantum property that is<\/p>\n\n\n\n
allowed only two possible values, plus or minus a half.<\/p>\n\n\n\n
A word of caution is in place here. In the previous paragraphs we gave a<\/p>\n\n\n\n
description of the atom in a classical way, to help you form a picture of the<\/p>\n\n\n\n
physics involved. However, this description is severely limited. In truth, the<\/p>\n\n\n\n
orbiting of the electrons about the nucleus is not like point particles revolving<\/p>\n\n\n\n
in space. Nor is the electron really spinning about its axis, it is a point particle<\/p>\n\n\n\n
with no extension in space! We want to emphasize that the electron spin is a<\/p>\n\n\n\n
purely quantum mechanical concept, and is physically probed by virtue of its<\/p>\n\n\n\n
interaction with a non-uniform magnetic \ufb01eld.<\/p>\n\n\n\n
You can well imagine that the choice of a particular direction for the<\/p>\n\n\n\n
magnetic \ufb01eld inhomogeneity cannot a\ufb00ect the value of the magnetic moment<\/p>\n\n\n\n
of the electron: so even if the apparatus was tilted along any direction, the<\/p>\n\n\n\n
results would remain the same.<\/p>\n\n\n\n
Thus an atom with a single electron, described on the basis of its spin<\/p>\n\n\n\n
alone, is a 2-state quantum system, well suited as a candidate qubit. We will<\/p>\n\n\n\n
now illustrate the properties of a qubit using this experiment<\/p>\n","protected":false},"excerpt":{"rendered":"
Way back in 1922, when physicists were still studying the new and aston- ishing properties of the basic constituents of matter, an experiment designed 15 16 Introduction to Quantum Physics and Information Processing to measure the magnetic moment of atoms gave unexpected results. This was the classic Stern\u2013Gerlach experiment [36], designed to measure the magnetic […]<\/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-3957","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\/3957","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=3957"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3957\/revisions"}],"predecessor-version":[{"id":4551,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3957\/revisions\/4551"}],"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=3957"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3957"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3957"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}