Someone gives you an electron and asks you: what is the spin? How will<\/p>\n\n\n\n
you answer? If you measure S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
, S<\/p>\n\n\n\n
y<\/p>\n\n\n\n
, or S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
you will get one of two answers, at<\/p>\n\n\n\n
random. Any observable you measure gives one of its eigenvalues at random.<\/p>\n\n\n\n
The state has probabilistic information about each eigenvalue. The meaning<\/p>\n\n\n\n
of this is statistical: (i) take a very large number of identical copies of the<\/p>\n\n\n\n
state |\u03c8i: a statistical ensemble, (ii) perform the measurement of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A on each<\/p>\n\n\n\n
copy, then if you expand \u03c8 in the basis of eigenvectors a of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A,<\/p>\n\n\n\n
|\u03c8i =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
a<\/p>\n\n\n\n
c<\/p>\n\n\n\n
a<\/p>\n\n\n\n
|ai<\/p>\n\n\n\n
then a fraction |c<\/p>\n\n\n\n
a<\/p>\n\n\n\n
|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
times you will obtain a as the result of the measure-<\/p>\n\n\n\n
ment. The value of the physical observable A is the statistical average value<\/p>\n\n\n\n
obtained by all these measurements. This is called the expectation value of<\/p>\n\n\n\n
the operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A in the state |\u03c8i denoted by<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Ai<\/p>\n\n\n\n
\u03c8<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
a<\/p>\n\n\n\n
P(a)a =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
a<\/p>\n\n\n\n
|c<\/p>\n\n\n\n
a<\/p>\n\n\n\n
|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
a<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
a<\/p>\n\n\n\n
h\u03c8|aiha|\u03c8ia =<\/p>\n\n\n\n
X<\/p>\n\n\n\n
a<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A|aiha|\u03c8i<\/p>\n\n\n\n
= h\u03c8|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A|\u03c8i,<\/p>\n\n\n\n
where in the last step we have used the resolution of the identity.<\/p>\n\n\n\n
We can thus de\ufb01ne statistically the mean value of an observable. The<\/p>\n\n\n\n
statistics of measurement is incomplete without the notion of the variance<\/p>\n\n\n\n
about the mean. We de\ufb01ne the variance \u2206<\/p>\n\n\n\n
2<\/p>\n\n\n\n
A as<\/p>\n\n\n\n
\u2206<\/p>\n\n\n\n
2<\/p>\n\n\n\n
A = hA<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i \u2212 hAi<\/p>\n\n\n\n
2<\/p>\n\n\n\n
The square root of the variance, the standard deviation, is called the error or<\/p>\n\n\n\n
uncertainty in the value of A.<\/p>\n\n\n\n
<\/p>\n\n\n\n
54 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
Example 3.3.1. Expectation value of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
in the state |0i:<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
i<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= h0|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
|0i = h0|<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
=<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= h0|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|0i = h0|<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1i<\/p>\n\n\n\n
= 0.<\/p>\n\n\n\n
Box 3.7: The Uncertainty Principle<\/p>\n\n\n\n
This principle is one of the foundation pillars of quantum mechanics, \ufb01rst<\/p>\n\n\n\n
enunciated by Werner Heisenberg. More accurately called the indeterminacy<\/p>\n\n\n\n
principle, this states that some physical observables are \u201cincompatible\u201d with<\/p>\n\n\n\n
each other, in the sense that on measurement in a given state, it is not pos-<\/p>\n\n\n\n
sible to get sharp values of both. In fact, the uncertainty in one observable<\/p>\n\n\n\n
is inversely related to that in the other. Classic examples are position and<\/p>\n\n\n\n
momentum, and also the three components of the spin vector.<\/p>\n\n\n\n
Mathematically, compatibility is related to the commutation of the opera-<\/p>\n\n\n\n
tors: whether the order of operation of two operators matters or not. For two<\/p>\n\n\n\n
operators<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B the commutator is de\ufb01ned as the operator expressing this<\/p>\n\n\n\n
di\ufb00erence in ordering:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
C = [<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B] \u2261<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B \u2212<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A.<\/p>\n\n\n\n
It can be shown that the product of uncertainties of two operators measured<\/p>\n\n\n\n
in a state |\u03c8i is related to their commutator:<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
\u2206<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A\u2206<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\ue011<\/p>\n\n\n\n
\u03c8<\/p>\n\n\n\n
\u2265<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue00a<\/p>\n\n\n\n
[<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B]<\/p>\n\n\n\n
\ue00b<\/p>\n\n\n\n
\u03c8<\/p>\n\n\n\n
. (3.22)<\/p>\n\n\n\n
Note that experimentally uncertainty refers to the standard deviation from<\/p>\n\n\n\n
the mean of a statistically large set of measurements of the observable, made<\/p>\n\n\n\n
on identically prepared states. Physically the meaning of the uncertainty prin-<\/p>\n\n\n\n
ciple is that if we perform a set of measurements of observable A and B in<\/p>\n\n\n\n
an ensemble prepared in a state |\u03c8i, then the products of the uncertainties<\/p>\n\n\n\n
of the two observables is limited by the expression on the right, related to<\/p>\n\n\n\n
their commutator. Experimental uncertainties would add to this limit. Thus<\/p>\n\n\n\n
in principle, the uncertainty in either of a pair of observables that do not<\/p>\n\n\n\n
commute can never be zero.<\/p>\n\n\n\n
<\/p>\n\n\n\n
The Essentials of Quantum Mechanics 55<\/p>\n\n\n\n
3.4 Evolution<\/p>\n\n\n\n
An isolated system is said to evolve when its state changes with time. The<\/p>\n\n\n\n
change in state would take place by the action of an operator on it. This action<\/p>\n\n\n\n
cannot take it out of the Hilbert space, and must preserve its norm. Therefore<\/p>\n\n\n\n
the evolution operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U has to satisfy some conditions.<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2212\u2192 |\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U|\u03c8i<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i = h\u03c8|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U|\u03c8i<\/p>\n\n\n\n
If h\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
i = h\u03c8|\u03c8i<\/p>\n\n\n\n
then<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u22121<\/p>\n\n\n\n
Such an operator is called unitary:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U = .<\/p>\n\n\n\n
In quantum computation, any operation we wish to perform on a qubit must<\/p>\n\n\n\n
be represented by such an operator. One of the important consequences of<\/p>\n\n\n\n
this is that since any unitary operator is invertible, any quantum operation is<\/p>\n\n\n\n
reversible.<\/p>\n\n\n\n
For example, the Pauli spin operators are unitary and are valid evolution<\/p>\n\n\n\n
operators. The operation |0i\u2192 |1iand |1i\u2192 |0iis achieved by the \u03c3<\/p>\n\n\n\n
1<\/p>\n\n\n\n
operator.<\/p>\n\n\n\n
This operation \ufb02ips the bits 0 and 1, and is therefore also called the NOT<\/p>\n\n\n\n
operator X.<\/p>\n\n\n\n
Thus evolution is another application of unitary operators in quantum<\/p>\n\n\n\n
mechanics. The \ufb01rst one we encountered of course was while implementing<\/p>\n\n\n\n
basis change.<\/p>\n\n\n\n
3.4.1 Continuous time evolution<\/p>\n\n\n\n
From the physical viewpoint, evolution in time occurs due to interaction<\/p>\n\n\n\n
of the system with an external force. A characteristic of this \u201cforce\u201d is the<\/p>\n\n\n\n
energy the system has in its presence. This energy is represented by a func-<\/p>\n\n\n\n
tion called the Hamiltonian function H. In a given situation it has to be<\/p>\n\n\n\n
determined experimentally. The quantum version of the Hamiltonian is the<\/p>\n\n\n\n
Hamiltonian operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H. This operator, being an observable, must be Hermi-<\/p>\n\n\n\n
tian. Now it turns out that when the Hamiltonian acts on a state vector, it<\/p>\n\n\n\n
creates an in\ufb01nitesimal time evolution. This gives a di\ufb00erential version of the<\/p>\n\n\n\n
time evolution postulate of which there are two (experimentally equivalent)<\/p>\n\n\n\n
viewpoints or \u201cpictures\u201d:<\/p>\n\n\n\n
<\/p>\n\n\n\n
56 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
3.4.1.1 Schr\u00a8odinger viewpoint<\/p>\n\n\n\n
Postulate 4. The evolution in time of a quantum state vector |\u03c8(t)i is given<\/p>\n\n\n\n
by the Schr\u00a8odinger equation:<\/p>\n\n\n\n
i~<\/p>\n\n\n\n
d|\u03c8i<\/p>\n\n\n\n
dt<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H|\u03c8i. (3.23)<\/p>\n\n\n\n
We can try to understand what this implies by formally integrating this<\/p>\n\n\n\n
equation to solve for |\u03c8(t)i from |\u03c8(t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)i. Assuming that the Hamiltonian func-<\/p>\n\n\n\n
tion is itself explicitly independent of time, we would get<\/p>\n\n\n\n
|\u03c8(t)i = exp<\/p>\n\n\n\n
\ue014<\/p>\n\n\n\n
\u2212<\/p>\n\n\n\n
i<\/p>\n\n\n\n
~<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H(t \u2212 t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)<\/p>\n\n\n\n
\ue015<\/p>\n\n\n\n
|\u03c8(t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)i.<\/p>\n\n\n\n
So the unitary operator for time evolution is just<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U(t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, t) \u2261 exp<\/p>\n\n\n\n
\ue014<\/p>\n\n\n\n
\u2212<\/p>\n\n\n\n
i<\/p>\n\n\n\n
~<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H(t \u2212 t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)<\/p>\n\n\n\n
\ue015<\/p>\n\n\n\n
. (3.24)<\/p>\n\n\n\n
We can set t<\/p>\n\n\n\n
0<\/p>\n\n\n\n
= 0 and write<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U(t) = e<\/p>\n\n\n\n
\u2212i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Ht\/~<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Here, the exponential of the operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H is understood as the in\ufb01nite sum of<\/p>\n\n\n\n
powers of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H:<\/p>\n\n\n\n
e<\/p>\n\n\n\n
\u2212i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Ht\/~<\/p>\n\n\n\n
\u2261 \u2212<\/p>\n\n\n\n
it<\/p>\n\n\n\n
~<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H +<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
it<\/p>\n\n\n\n
~<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ \u00b7\u00b7\u00b7 ,<\/p>\n\n\n\n
itself an operator that can be expressed as a matrix. You can verify that since<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H is Hermitian,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U(t) is indeed unitary.<\/p>\n\n\n\n
3.4.1.2 Heisenberg viewpoint<\/p>\n\n\n\n
One can focus on the observables being measured instead of the state in<\/p>\n\n\n\n
which they are measured, and think of evolution as a\ufb00ecting the observable<\/p>\n\n\n\n
(operator) instead of the state vector. In this picture, the evolution of an<\/p>\n\n\n\n
observable<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(t) is given by<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(t) =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U(t)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(0)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
(t) (3.25)<\/p>\n\n\n\n
=\u21d2<\/p>\n\n\n\n
d<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
dt<\/p>\n\n\n\n
=<\/p>\n\n\n\n
d<\/p>\n\n\n\n
dt<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(0)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
+<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(0)<\/p>\n\n\n\n
d<\/p>\n\n\n\n
dt<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
=<\/p>\n\n\n\n
i<\/p>\n\n\n\n
~<\/p>\n\n\n\n
\ue010<\/p>\n\n\n\n
\u2212<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(0)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
+<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(0)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
\ue011<\/p>\n\n\n\n
d<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
dt<\/p>\n\n\n\n
=<\/p>\n\n\n\n
i<\/p>\n\n\n\n
~<\/p>\n\n\n\n
[<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(t),<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H], (3.26)<\/p>\n\n\n\n
where the square brackets indicate the commutator AH \u2212HA. Here we have<\/p>\n\n\n\n
assumed that the observable A itself has no explicit time-dependence; that is,<\/p>\n\n\n\n
t does not occur in its form. If it did then we would have to add the partial<\/p>\n\n\n\n
derivative of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A(t) with respect to t. It is straightforward to see that both<\/p>\n\n\n\n
pictures give the same value for the experimentally observed quantities: the<\/p>\n\n\n\n
expectation values of observables<\/p>\n","protected":false},"excerpt":{"rendered":"
Someone gives you an electron and asks you: what is the spin? How will you answer? If you measure S x , S y , or S z you will get one of two answers, at random. Any observable you measure gives one of its eigenvalues at random. The state has probabilistic information about each […]<\/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-3995","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\/3995","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=3995"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3995\/revisions"}],"predecessor-version":[{"id":4559,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3995\/revisions\/4559"}],"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=3995"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3995"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3995"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}