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Turboencabulator

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{{infobox graph
| name = Path graph
| image = [[Image:Path-graph.svg|250px]]
| image_caption = A path graph on 6 vertices
| vertices = ''n''
| edges = ''n - 1''
| automorphisms = 2
| diameter = ''n - 1''
| radius = ''⌊n/2⌋''
| chromatic_number = 2
| chromatic_index = 2
| spectrum = {2 cos(''k'' π / (''n'' + 1))1; ''k''=1,...,''n''}
| properties = [[Unit distance graph|Unit distance]] [[Bipartite graph]] [[tree (graph theory)|Tree]]
|notation = <math>P_n</math>
}}
{{confused|line graph}}
In the [[Mathematics|mathematical]] field of [[graph theory]], a '''path graph''' or '''linear graph''' is a particularly simple example of a [[tree (graph theory)|tree]], namely a tree with two or more [[vertex (graph theory)|vertices]] that is not branched at all, that is, contains only vertices of [[degree (graph theory)|degree]] 2 and 1. In particular, it has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2.

A '''path''' in a [[graph (mathematics)|graph]] is a [[sequence]] of [[vertex (graph theory)|vertices]] such that from each of its vertices there is an [[edge (graph theory)|edge]] to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its ''start vertex'', and a last vertex, called its ''end vertex''. Both of them are called ''end or terminal vertices'' of the path. The other vertices in the path are ''internal vertices''. A '''cycle''' is a path such that the start vertex and end vertex are the same. Note that the choice of the start vertex in a cycle is arbitrary.

[[Image:Directed cycle.svg|frame|A directed cycle. Without the arrows, it is just a cycle. This is not a simple cycle, since the blue vertices are used twice.]]

Paths and cycles are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced [[algorithmic]] topics concerning paths in graphs.

== Different types of path graphs ==
The same concepts apply both to [[undirected graph]]s and [[directed graph]]s, with the edges being directed from each vertex to the following one. Often the terms ''directed path'' and ''directed cycle'' are used in the directed case.

A path with no repeated vertices is called a '''simple path''', and a cycle with no repeated vertices or edges aside from the necessary repetition of the start and end vertex is a '''simple cycle'''. In modern [[graph theory]], most often "simple" is implied; i.e., "cycle" means "simple cycle" and "path" means "simple path", but this convention is not always observed, especially in applied graph theory. Some authors (e.g. Bondy and Murty 1976) use the term "walk" for a path in which vertices or edges may be repeated, and reserve the term "path" for what is here called a simple path.

A path such that no graph edges connect two nonconsecutive path vertices is called an [[induced path]].

A simple cycle that includes every vertex, without repetition, of the graph is known as a [[Hamiltonian cycle]].

A cycle with just one edge removed in the corresponding [[spanning tree]] of the original graph is known as a Fundamental cycle.


Two paths are ''independent'' (alternatively, ''internally vertex-disjoint'') if they do not have any internal vertex in common.

The ''length'' of a path is the number of edges that the path uses, counting multiple edges multiple times. The length can be zero for the case of a single vertex.

A [[weighted graph]] associates a value (''weight'') with every edge in the graph. The ''weight of a path'' in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight.

== See also ==

*[[Glossary of graph theory]]
*[[Shortest path problem]]
*[[Traveling salesman problem]]
*[[Cycle space]]
*[[Path (graph theory)]]
*[[Caterpillar tree]]
*[[Cycle graph]]
*[[Complete graph]]
*[[Null graph]]
*[[Path decomposition]]

== References ==

*{{cite book
| author = [[John Adrian Bondy|Bondy, J. A.]]; [[U. S. R. Murty|Murty, U. S. R.]]
| title = Graph Theory with Applications
| year = 1976
| publisher = North Holland
| isbn = 0-444-19451-7
| pages = 12–21
| url=http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html}}
*{{cite book
| author = Diestel, Reinhard
| title = Graph Theory
| edition = 3rd ed.
| url = http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
| publisher = Graduate Texts in Mathematics, vol. 173, Springer-Verlag
| year = 2005
| isbn = 3-540-26182-6
| pages = 6–9}}
*{{cite book
| author = Gibbons, A.
| title = Algorithmic Graph Theory
| year = 1985
| publisher = Cambridge University Press
| pages = 5–6
| isbn = 0-521-28881-9}}
*{{cite book
| author = [[Bernhard Korte|Korte, Bernhard]]; [[László Lovász|Lovász, László]]; Prömel, Hans Jürgen; [[Alexander Schrijver|Schrijver, Alexander]] (Eds.)
| title = Paths, Flows, and VLSI-Layout
| publisher = Algorithms and Combinatorics 9, Springer-Verlag
| year = 1990
| isbn = 0-387-52685-4}}

==External links==
* {{MathWorld | urlname=PathGraph | title=Path Graph }}

[[Category:Graph theory objects]]
[[Category:Graph connectivity]]
[[Category:Trees (graph theory)]]
[[Category:Parametric families of graphs]]

[[cs:Cesta (graf)]]
[[es:Camino (teoría de grafos)]]
[[fr:Chaîne (graphe)]]
[[he:מסלול (תורת הגרפים)]]
[[ja:道 (グラフ理論)]]
[[pl:Ścieżka (teoria grafów)]]
[[pt:Caminho (teoria dos grafos)]]
[[ru:Путь (теория графов)]]
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Lorenz gauge condition

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{{Refimprove|date=December 2009}}
[[File:Stern-Gerlach experiment.PNG|right|200px|thumb|Passing electrically neutral silver atom particles beam through [[Stern–Gerlach experiment]]'s inhomogeneous magnetic field will split the beam into two beams, each corresponding to one of the possible spin values of the outermost electron of the silver atom .]]

In [[quantum mechanics]], a '''two-state system''' (also known as a '''two-level system''') is a system which can exist in any [[quantum superposition]] of two independent (physically distinguishable) [[quantum state]]s. The [[Hilbert space]] describing such a system is two [[dimension]]al. Therefore, a complete [[Basis (linear algebra)|basis]] spanning the space will consist of two independent states.

Two-state systems are the simplest quantum systems that can exist, since the dynamics of a one-state system is trivial (i.e. there is no other state the system can exist in). The mathematical framework required for the analysis of two-state systems is that of [[linear differential equation]]s and [[linear algebra]] of two-dimensional spaces. As a result, the dynamics of a two can be solved analytically without any approximation.

A very well known example of a two-state system is the [[Spin (physics)|spin]] of a [[spin-1/2]] particle such as an electron, whose spin can have values +''ħ''/2 or −''ħ''/2, where ''ħ'' is the [[reduced Planck constant]]. Another example, frequently studied in [[atomic physics]], is the transition of an atom to or from an [[excited state]]; here the two-state formalism is used to quantitatively explain [[stimulated emission|stimulated]] and [[spontaneous emission]] of [[photons]] from excited atoms.

==Representation of the Two-state quantum system==
The state of a two-state quantum system can be represented as vectors of a two-dimensional [[Hilbert Space#Definition|complex Hilbert space]], this means every [[Quantum state|state vector]] <math>\vert\psi\rangle</math> is represented by two [[complex numbers|complex]] coordinates.

:<math> |\psi\rangle = \begin{pmatrix} c_1 \\ c_2\end{pmatrix} = c_1\begin{pmatrix} 1 \\ 0\end{pmatrix} + c_2\begin{pmatrix} 0 \\ 1\end{pmatrix} ;</math> where, <math>c_1</math> and <math>c_2</math> are the coordinates.<ref name="griffiths353">{{cite book|last=Griffiths|first=David|title=Introduction to Quantum Mechanics|edition=2nd|year=2005|page=353}}</ref>

If the vectors are normalized, <math>c_1</math> and <math>c_2</math> are related by <math>{|c_1|}^2+{|c_2|}^2 = 1</math> . the basis vectors will be represented as <math>|0\rangle = \begin{pmatrix} 1 \\ 0\end{pmatrix}</math> and <math>|1\rangle = \begin{pmatrix} 0 \\ 1\end{pmatrix}</math>

All [[Observable|observable physical quantities]] associated with this systems are 2 <math>\times</math> 2 [[Hermitian matrices]], this means the [[Hamiltonian]] of the system is also a similar matrix.

===The Two-state Hamiltonian===
The most general form of the Hamiltonian of a two-state system is given
:<math> \mathbf{H} = \begin{pmatrix} a_1 & c-id\\ c+id & a_2\end{pmatrix}</math>

here, <math> a_1, a_2, c</math> and <math>d</math> are real numbers. This matrix can be decomposed as,

:<math> \mathbf{H} = a\cdot\sigma_0 + c\cdot\sigma_1 + d\cdot\sigma_2 + b\cdot\sigma_3 ;</math>

Here, <math>a = \frac{(a_1+a_2)}{2}</math> and <math>b = \frac{(a_1 - a_2)}{2}</math> are real numbers. The matrix <math>\sigma_0</math> is the 2 <math>\times</math> 2 identity matrix and the matrices <math> \sigma_k (k = 1,2,3)</math> are the [[Pauli matrices]]. This decomposition simplifies the analysis of the system especially in the time-independent case where the values of <math> a,b,c</math> and <math>d</math> are constants.

The Hamiltonian can be written (in a slightly different vector form) as:

:<math> \mathbf{H} = a\cdot\sigma_0 + \mathbf{r}\cdot\mathbf{\sigma} ;</math>

The vector <math>r</math> is given by <math>(c,d,b)</math> and <math>\sigma</math> is given by <math>(\sigma_1,\sigma_2,\sigma_3)</math>. This representation simplifies the analysis of the time evolution of the system and is easier to use with other specialized representations such as the [[Bloch sphere]].

===Eigenvalues of the Hamiltonian, Basis vectors and Time evolution===

Let <math>H</math> be the time-independent Hamiltonian of a two-state system, the Eigenvalues are given by <math>E_{\pm} = a \pm |\mathbf{r}|</math> and the eigenvectors corresponding to them are given as <math>|E_{+}\rangle</math> and <math>|E_{-}\rangle</math> corresponding to their respective energies. When one changes the basis to the eigenvetors the Hamiltonian is diagonal and is of the form,

: <math>H = \begin{pmatrix} E_{+} & 0\\ 0 & E_{-}\end{pmatrix}; </math>

the unitary time evolution operator <math>U</math> is given by is given by:

:<math>U(t) = e^{\frac{-iHt}{\hbar}} = e^{\frac{-iat}{\hbar}} (cos(|\mathbf{r}|)\sigma_0 + sin(|\mathbf{r}|)\hat{r}\cdot\mathbf{\sigma});</math>

where, <math>\hat{r} = \frac{\mathbf{r}}{|\mathbf{r}|}. </math> . This is the case when H is in the vector form (i.e. in the <math>|0\rangle , |1\rangle</math> basis); in the eigenvector basis <math>U</math> is diagonal and is given by:

:<math> U(t) = \begin{pmatrix} e^{\frac{-iE_{+}t}{\hbar}} & 0 \\ 0 & e^{\frac{-iE{-}t}{\hbar}}\end{pmatrix}</math>

It is to be noted that the <math>e^{\frac{-iat}{\hbar}}</math> factor only contributes to the overall phase of the operator and can therefore be ignored to yield a new time evolution operator that is physically indistinguishable from the original operator. Moreover, any [[Perturbation theory (quantum mechanics)|perturbation]] to the system (which will be of the same form as the Hamiltonian) can be added to the system in the eigenbasis of the unperturbed Hamiltonian and analysed in the same way as above, this means that for any perturbation the new eigenvectors of the perturbed system can be solved exactly (as mentioned in the introduction).

===Dynamics of the Two-state System : The Rabi formula ===
If <math>H</math> is the time-independent Hamiltonian, and let <math>|+\rang</math> and <math>|-\rang</math> denote the two energy eigenstates of the system, with respective [[eigenvalue]]s <math>E_+</math> and <math>E_-</math>. Any state <math>|\psi(t)\rang</math> of the two-level system can be written as a [[quantum superposition|superposition]] of the energy eigenstates; in particular, at time <math>t=0</math> we can write,

:<math>|\psi(0)\rang = c_+ |+\rang + c_- |-\rang,</math>

the above vector is assumed to be normalized. The time evolution of the state <math>|\psi(0)\rang</math> is given by the relation

:<math>|\psi(t)\rang = U(t)|\psi(0)\rang = c_+ e^{\frac{-iE_{+}t}{\hbar}} |+\rang + c_- e^{\frac{-iE_{-}t}{\hbar}} |-\rang,</math>

further eliminating an overall phase factor of <math>e^{\frac{-iE_{+}t}{\hbar}}</math> the time evolved state can be represented as,

:<math>|\psi(t)\rang = c_+ |+\rang + c_- e^{\frac{-i(E_{-}-E_{+})t}{\hbar}} |-\rang,</math>

It is easy to infer that if the system was initially in one of the eigestates (<math>|+\rangle</math> or <math>|-\rangle</math>) it will continue to remain in the same state, however in a general state as shown above the time evolution is non-trivial. Upon calculating the probability of the state returning to the initial state at a given time <math>t</math> is given by

:<math> {|\langle\psi(0)|\psi(t)\rangle|}^2 = {||c_+|^2 + |c_-|^2 e^{\frac{-i(E_{-}-E_{+})t}{\hbar}}|}^2 = 1 - 4|c_+c_-|^2 {\sin}^2(\frac{\omega t}{2})</math>

Where <math>\omega</math> is a characteristic [[angular frequency]] given by
:<math> \omega = \frac{E_+ - E_-}{\hbar},</math>
where it has been assumed that <math>E_+ \geq E_- </math>.<ref name="griffths343">Griffiths, p. 343.</ref>

It can be seen that the probability of finding the system in its initial quantum state oscillates between <math>1 - 4|c_+c_-|^2</math> and <math>1</math> this formula is called the Rabi oscillation formula. In the case <math>E_+ = E_- </math>, that is when the Hamiltonian is degenerate there is no oscillation.

==Analysis of some important Two-state systems==

===Precession in a field===

Consider the case of a [[spin-1/2]] particle in a magnetic field <math>\mathbf{B} = B\mathbf{\hat n}</math>. The interaction Hamiltonian for this system is

:<math> H=-\boldsymbol{\mu}\cdot\mathbf{B}=-\mu\boldsymbol{\sigma}\cdot\mathbf{B}</math>

where <math>\mu</math> is the magnitude of the particle's [[magnetic moment]] and <math>\boldsymbol{\sigma}</math> is the vector of [[Pauli matrices]]. Solving the time dependent Schrödinger equation <math> H\psi = i\hbar \partial_t \psi</math> yields

:<math> \psi(t) = e^{i\omega t \boldsymbol{\sigma}\cdot \mathbf{\hat{n}}} \psi(0),</math>

where <math> \omega = \mu B/\hbar</math> and <math> e^{i\omega t \boldsymbol{\sigma}\cdot \mathbf{\hat{n}}} = \cos{\left(\omega t\right)} I + i \mathbf{\hat{n}}\cdot\boldsymbol{\sigma} \sin{\left(\omega t\right)} </math>. Physically, this corresponds to the [[Bloch vector]] precessing around <math>\mathbf{\hat{n}} </math> with angular frequency <math> 2\omega </math>. Without loss of generality, assume the field is uniform points in <math>\mathbf{\hat{z}} </math>, so that the time evolution operator is given as

:<math> e^{i\omega t \boldsymbol{\sigma}\cdot \mathbf{\hat{n}}} =
\begin{pmatrix}
e^{i\omega t} & 0 \\
0 & e^{-i\omega t}
\end{pmatrix}.</math>

It can be seen that such a time evolution operator acting on a general spin state of a spin-1/2 particle will lead to the precession about the axis defined by the applied magnetic field (this is the quantum mechanical equivalent of [[Larmor precession]])<ref name ="feynman_1">{{cite book|last=Feynman|first=R.P.|title=The Feynman Lectures on Physics: Volume 3|year=1965|publisher=Addison Wesley|chapter=7-5 and 10-7}}</ref>

The above method can however be applied to the analysis of any generic two-state system that is interacting with some field (equivalent to the magnetic filed in the previous case) the interaction is given by an appropriate coupling term thatbis analogous to the magnetic moment. The precession of the state vector (which need not be a physical spinning as in the previous case) can be viewed as the precession of the state vector on the [[Bloch sphere]]

The representation on the Bloch sphere for a state vector <math>\psi(0)</math> will simply be the vector of expectation values <math> \mathbf{R}=\left(\langle \sigma_x \rangle,\langle \sigma_y \rangle,\langle \sigma_z \rangle \right) </math>. As an example, consider a state vector <math> \psi(0) </math> that is a normalized superposition of <math> |\uparrow\rangle </math> and <math> |\downarrow\rangle </math>, that is, a vector that can be represented in the <math> \sigma_z</math> basis as

:<math> \psi(0) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}</math>

The components of <math>\psi(t)</math> on the Bloch sphere will simply be <math>\mathbf{R}=\left(\cos{2\omega t},-\sin{2\omega t},0\right)</math>. This is a unit vector that begins pointing along <math>\mathbf{\hat{x}}</math> and precesses around <math>\mathbf{\hat{z}}</math> in a left-handed manner. In general, by a rotation around <math>\mathbf{\hat{z}}</math>, any state vector <math>\psi(0)</math> can be represented as <math> a|\uparrow\rangle + b|\downarrow\rangle</math> with real coefficients <math>a</math> and <math>b</math>. Such a state vector corresponds to a [[Bloch vector]] in the ''xz''-plane making an angle <math> \tan(\theta/2) = b/a </math> with the ''z''-axis. This vector will proceed to precess around <math>\mathbf{\hat{z}}</math>. In theory, by allowing the system to interact with the field of a particular direction and strength for precise durations, it is possible to obtain any orientation of the [[Bloch vector]], which is equivalent to obtaining any complex superposition. This is the basis for numerous technologies including [[quantum computing]] and [[MRI]].

===Evolution in a Time-dependent Field: Nuclear magnetic resonance===

[[Nuclear magnetic resonance]] (NMR) is an important example in the dynamics of two-state systems because it is involves the exact solution to a time dependent Hamiltonian. The NMR phenomenon is achieved by placing a nucleus in a strong, static field '''B'''0 (the "holding field") and then applying a weak, transverse field '''B'''1 that oscillates at some radiofrequency ''ω''r.<ref name="griffiths377">Griffiths, p. 377.</ref> Explicitly, consider a [[spin-1/2]] particle in a holding field <math>B_0 \mathbf{\hat z}</math> and a transverse rf field '''B'''1 rotating in the ''xy''-plane in a right-handed fashion around '''B'''0:

::<math> \mathbf{B} = \begin{pmatrix}
B_1 \cos\omega_\mathrm{r} t \\
B_1 \sin\omega_\mathrm{r} t\\
B_0 \end{pmatrix}.</math>

As in the free precession case, the Hamiltonian is <math>H = -\mu\boldsymbol{\sigma}\cdot\mathbf{B}</math>, and the evolution of a state vector <math>\psi(t)</math> is found by solving the time-dependent Schrödinger equation <math>H\psi = i\hbar\,\partial \psi/\partial t</math>. After some manipulation (given in the collapsed section below), it can be shown that the Schrödinger equation becomes

:<math> \frac{\partial \psi}{\partial t}=i\left(\omega_1 \sigma_x + \left(w_0+\frac{\omega_r}{2}\right)\sigma_z\right)\psi,</math>
where <math> \omega_0 = \mu B_0/\hbar</math> and <math>\omega_1 = \mu B_1/\hbar</math>.

As per the previous section, the solution to this equation has the [[Bloch vector]] precessing around <math> (\omega_1,0,\omega_0 + \omega_r/2) </math> with a frequency that is twice the magnitude of the vector. If <math> \omega_0 </math> is sufficiently strong, some proportion of the spins will be pointing directly down prior to the introduction of the rotating field. If the angular frequency of the rotating magnetic field is chosen such that <math> \omega_r = -2 \omega_0 </math>, in the rotating frame the state vector will precess around <math> \hat{x} </math> with frequency <math> 2 \omega_1 </math>, and will thus flip from down to up releasing energy in the form of detectable photons. This is the fundamental basis for [[NMR]], and in practice is accomplished by scanning <math> \omega_r </math> until the resonant frequency is found at which point the sample will emit light. Similar calculations are done in atomic physics, and in the case that the field is not rotating, but oscillating with a complex amplitude, use is made of the [[rotating wave approximation]] in deriving such results.

{| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
!Derivation of above expression for the NMR Schrödinger equation
|-
|Here the Schrödinger equation reads
:<math> -\mu\boldsymbol{\sigma}\cdot\mathbf{B} \psi = i\hbar \frac{\partial \psi}{\partial t}.</math>

Expanding the dot product and dividing by <math>i\hbar</math> yields

:<math> \frac{\partial\psi}{\partial t} = i\left(\omega_1\sigma_x \cos{\omega_r t} + \omega_1\sigma_y \sin{\omega_r t} + \omega_0 \sigma_z\right)\psi.</math>

To remove the time dependence from the problem, the wave function is transformed according to <math> \psi \rightarrow e^{-i \sigma_z \omega_r t/2}\psi </math>. The time dependent Schrödinger equation becomes

:<math> -i \sigma_z \frac{\omega_r}{2}e^{-i \sigma_z \omega_r t/2}\psi + e^{-i \sigma_z \omega_r t/2}\frac{\partial \psi}{\partial t}=i\left(\omega_1\sigma_x \cos{\omega_r t} + \omega_1\sigma_y \sin{\omega_r t} + \omega_0 \sigma_z\right)e^{-i \sigma_z \omega_r t/2}\psi </math>

Which after some rearrangement yields

:<math> \frac{\partial \psi}{\partial t}=ie^{i \sigma_z \omega_r t/2}\left(\omega_1\sigma_x \cos{\omega_r t} + \omega_1\sigma_y \sin{\omega_r t} + \left(\omega_0+\frac{\omega_r}{2}\right) \sigma_z\right)e^{-i \sigma_z \omega_r t/2}\psi</math>

Evaluating each term on the right hand side of the equation

:<math> e^{i \sigma_z \omega_r t/2}\sigma_x e^{-i \sigma_z \omega_r t/2} = \begin{pmatrix}
e^{i\omega_r t/2} & 0 \\
0 & e^{-i\omega_r t/2} \end{pmatrix}
\begin{pmatrix}
0 & 1 \\
1 & 0 \end{pmatrix}
\begin{pmatrix}
e^{-i\omega_r t/2} & 0 \\
0 & e^{i\omega_r t/2} \end{pmatrix}=
\begin{pmatrix}
0 & e^{i\omega_r t} \\
e^{-i\omega_r t} & 0 \end{pmatrix}
</math>

:<math> e^{i \sigma_z \omega_r t/2}\sigma_y e^{-i \sigma_z \omega_r t/2} = \begin{pmatrix}
e^{i\omega_r t/2} & 0 \\
0 & e^{-i\omega_r t/2} \end{pmatrix}
\begin{pmatrix}
0 & -i \\
i & 0 \end{pmatrix}
\begin{pmatrix}
e^{-i\omega_r t/2} & 0 \\
0 & e^{i\omega_r t/2} \end{pmatrix}=
\begin{pmatrix}
0 & -i e^{i\omega_r t} \\
i e^{-i\omega_r t} & 0 \end{pmatrix}
</math>

:<math> e^{i \sigma_z \omega_r t/2}\sigma_z e^{-i \sigma_z \omega_r t/2} = \begin{pmatrix}
e^{i\omega_r t/2} & 0 \\
0 & e^{-i\omega_r t/2} \end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1 \end{pmatrix}
\begin{pmatrix}
e^{-i\omega_r t/2} & 0 \\
0 & e^{i\omega_r t/2} \end{pmatrix}=\sigma_z
</math>

The equation now reads

:<math> \frac{\partial \psi}{\partial t} = i\left(\omega_1 \begin{pmatrix}
0 & e^{i\omega_r t}\left(\cos{\omega_r t} - i\sin{\omega_r t}\right) \\
e^{-i\omega_r t}\left(\cos{\omega_r t} + i\sin{\omega_r t}\right) & 0 \end{pmatrix}+\left(w_0+\frac{\omega_r}{2}\right)\sigma_z\right)\psi </math>

Which by [[Euler's identity]] becomes

:<math> \frac{\partial \psi}{\partial t}=i\left(\omega_1 \sigma_x + \left(w_0+\frac{\omega_r}{2}\right)\sigma_z\right)\psi </math>
|}

===Relation to Bloch equations===
The [[optical Bloch equations]] for a collection of [[spin-1/2]] particles can be derived from the time dependent Schrödinger equation for a two level system. Starting with the previously stated Hamiltonian <math> i\hbar\partial_t\psi=-\mu\vec{\sigma}\cdot\vec{B}\psi </math>, it can be written in summation notation after some rearrangement as

::<math> \frac{\partial\psi}{\partial t}=i\frac{\mu}{\hbar}\sigma_i B_i \psi </math>

Multiplying by a [[Pauli matrix]] <math> \sigma_i </math> and the conjugate transpose of the wavefunction, and subsequently expanding the product of two Pauli matrices yields

::<math> \psi^\dagger \sigma_j \frac{\partial\psi}{\partial t}=i\frac{\mu}{\hbar}\psi^\dagger\sigma_j\sigma_i B_i \psi =i\frac{\mu}{\hbar}\psi^\dagger \left(I\delta_{ij} - i \sigma_k \varepsilon_{ijk}\right)B_i\psi=\frac{\mu}{\hbar}\psi^\dagger \left(iI\delta_{ij} + \sigma_k \varepsilon_{ijk}\right)B_i\psi </math>

Adding this equation to its own conjugate transpose yields a left hand side of the form

:: <math> \psi^\dagger \sigma_j \frac{\partial\psi}{\partial t} + \frac{\partial\psi^\dagger}{\partial t} \sigma_j \psi = \frac{\partial \left( \psi^\dagger \sigma_j \psi\right)}{\partial t} </math>

And a right hand side of the form

:: <math> \frac{\mu}{\hbar}\psi^\dagger \left(iI\delta_{ij} + \sigma_k \varepsilon_{ijk}\right)B_i\psi + \frac{\mu}{\hbar}\psi^\dagger \left(-iI\delta_{ij} + \sigma_k \varepsilon_{ijk}\right)B_i\psi = \frac{2\mu}{\hbar} \left( \psi^\dagger\sigma_k\psi\right) B_i \varepsilon_{ijk} </math>

As previously mentioned, the expectation value of each [[Pauli matrix]] is a component of the [[Bloch vector]], <math> \langle \sigma_i \rangle = \psi^\dagger\sigma_i\psi = R_i</math>. Equating the left and right hand sides, and noting that <math> \frac{2\mu}{\hbar} </math> is the [[gyromagnetic ratio]] <math> \gamma </math>, yields another form for the equations of motion of the [[Bloch vector]]

::<math> \frac{\partial R_j}{\partial t} = \gamma R_k B_i \varepsilon_{kij} </math>

where the fact that <math> \varepsilon_{ijk} = \varepsilon_{kij} </math> has been used. In vector form these three equations can be expressed in terms of a [[cross product]]

::<math> \frac{\partial \vec{R}}{\partial t} = \gamma \vec{R} \times \vec{B} </math>
Classically, this equation describes the dynamics of a spin in a magnetic field. An ideal magnet consists of a collection of identical spins behaving independently, and thus the total [[magnetization]] <math> \vec{M}</math> is proportional to the [[Bloch vector]] <math> \vec{R} </math>. All that is left to obtain the final form of the [[optical Bloch equations]] is the inclusion of the phenomenological [[Relaxation (NMR)|relaxation]] terms.

As a final aside, the above equation can be derived by considering the time evolution of the [[angular momentum operator]] in the [[Heisenberg picture]].

:: <math> i\hbar\frac{d\sigma_j}{dt}=[\sigma_j,H]=[\sigma_j, -\mu \sigma_i B_i]=-\mu\left(\sigma_j\sigma_i B_i - \sigma_i\sigma_j B_i\right)=\mu[\sigma_i,\sigma_j]B_i = 2\mu i \varepsilon_{ijk}\sigma_k B_i </math>

Which, when coupled with the fact that <math> \vec{R_i} = \langle \sigma_i \rangle </math>, is the same equation as before.

==The Validity of the Two-state formalism==

Two-state systems are the simplest non-trivial quantum systems that occur in nature however in should be noted that the above mentioned methods of analysis are not just valid for simple two-state systems. Any general multi-state quantum system can be effectively treated as two-state system as long as a particular property of is being considered (which behaves as a two-state system) an example of this that of a spin-1/2 particle which may have additional translational or even rotational degrees of freedom, however in the preceding analysis, the additional degrees freedom are ignored.

Another case where the effective two-state formalism is valid is when the system under consideration has two levels that are effectively decoupled from the system, this is the case in the analysis of the spontaneous or stimulated emission of light by atoms and that of [[Charge qubit]]s. In this case it should be kept in mind that the perturbations (interactions with an external field) are in the right range and do not cause transitions to states other than the ones of interest.

==Some more examples and the significance of the Two-state formalism==

Pedagogically, the two-state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems. The most fundamental quantum mechanical phenomenon such as the [[Double-slit experiment#Quantum version of experiment|Interference]] exhibited by particles of the polarization states of the photon.<ref name ="feynman_2">{{cite book|last=Feynman|first=R.P.|title=The Feynman Lectures on Physics: Volume 3|year=1965|publisher=Addison Wesley|chapter=11-4}}</ref> but also more complex phenomenon such as [[neutrino oscillation]] or the neutral K-meson oscillation.

Two-state formalism can be used to describe simple mixing of states which leads to phenomenon such as [[resonance (chemistry)|resonance]] stabilization and other [[Avoided crossing|level crossing]] related symmetries. Such phenomenon have a wide variety of application in chemistry. Phenomena with tremendous industrial applications such as [[Maser]] and [[laser]] can be explained using the two-state formalism.

The two-state formalism forma the basis of Quantum computing. [[Qubit]]s which are the building blocks of a Quantum computer are nothing but two-state systems. Any [[Quantum computer|quantum computational]] operation is a unitary operation that rotates the state vector on the Bloch sphere.

==Further reading==

* An excellent treatment of the two-state formalism and its application to almost all the applications mentioned in this article is presented in the third volume of [[The Feynman Lectures on Physics]]
* The following set of lecture notes covers the necessary mathematics and also treats a few examples in some detail:
** from the ''Quantum mechanics II'' course offered at [[MIT]], http://web.mit.edu/8.05/handouts/Twostates_03.pdf
** from the same course dealing with neutral particle oscillation, http://web.mit.edu/8.05/handouts/nukaon_07.pdf
** from the ''Quantum mechanics I'' course offered at [[TIFR]], http://theory.tifr.res.in/~sgupta/courses/qm2013/hand4.pdf covers the essential mathematics
** http://theory.tifr.res.in/~sgupta/courses/qm2013/hand5.pdf ; from the same course deals with the some physical two-state systems and other important aspects of the formalism
** the mathematical in the initial section is done in a manner similar to these notes http://www.math.columbia.edu/~woit/QM/qubit.pdf which are form the ''Quantum Mechanics for Mathematicians'' course offered at University of Columbia.
** a book version of the same ; http://www.math.columbia.edu/~woit/QM/qmbook.pdf

==See also==
* [[Rabi cycle]]
* [[Doublet (physics)|Doublet]]
* [[NMR|Nuclear Magnetic Resonance]]
* [[Quantum Optics]]

==References==
{{reflist}}

{{DEFAULTSORT:Two-State Quantum System}}
[[Category:Quantum models]]

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The promotion under former West Ham The boss is desired resolution [http://www.jamonjamon.eu/node/162?page=123#comment-6786 BwLGV cheap soccer jerseys 2014 Arsenal 3-0 Bolton Arsene Wenger heaps praise on] with Barton United Club - - and the Magpies' home game against Arsenal in the first game only 10 days, but he stressed that the midfielder will need to show some contrition.

 QUOT, which is used as a football club, you all pulling in the same direction cheap soccer jerseys 2014 is very important - at this moment of time where Joey is not pulling in the same direction.

Barton training - and quotes George Orwell

 [I never received anyone's door, I'm not closing Joey, and QUOT; he said on Sky [http://ausslife.com/mediawiki/index.php?title=User:Ui0aak6p#oV4qF_cheap_soccer_jerseys_2014_Arsenal_and_Barcelona_confir oV4qF cheap soccer jerseys 2014 Arsenal and Barcelona confir] Sports News.

 QUOT; in (from McIntosh), these views and I do not intend to go there in terms. Say £ 40,000 is a ridiculous offer - which is absurd and QUOT; who will look to leave the club 

Two players, however, Jose Enrique and Wayne Routledge.

 QUOT; problem with Twitter - we need to get a grasp this.

Pardew believes that the situation may have been removed immediately, if not because of the Twitter.

Newcastle, can Let free transfer

 QUOT, which is a Under the circumstances, I did not expect was, and QUOT; he added.

Today development, following the Magpies were told they had successfully bid for midfielder Tranquillo slim hope Barnetta.

 Reporter conference QUOT shown, he trained with the team today, who knows, he can quickly and QUOT with the first team training; 

 [I do not see the problem, he will not start. I know there is an interest in San Jose, but no concrete interest. We are ready to respond that.

 QUOT, we are following the guidelines, Ferguson has said that they have done (at Manchester United) 0.

 QUOT, perhaps If not for the micro-blog and this instant media it may have been resolved, on Monday morning I and Joey in my office, and QUOT; he added.

Barton been available on a free transfer in the Magpies club confirmed Barton his recent criticism of social site.

Why Barton is about to exit symptomatic of deeper problems in the Newcastle 

Pardew hope fullback Enrique on Saturday play friendlies against Fiorentina, despite being closely associated with Liverpool, while Routledge looks closer to finalizing the transfer Swansea.

 QUOT I interviewed Ferguson today and he used to have this problem, microblogging, and QUOT; he added.

 QUOT; As for Rooney, I do not know if the final completion of the deal, but I believe it is very close to the QUOT; 

 QUOT, does anyone have the number of Isambard Kingdom Brunel University and QUOT? ; He wrote.

Barton morning before training and preparation Team Pardew appeared at a news conference, he announced that he will QUOT; not close QUOT; Barton's time at St James Park.

 QUOT, Enrique is still here and will be played on Saturday and QUOT; he added.

Barnetta British representative Barry McIntosh told they have clubs and QUOT; hell with QUOT no hope; attracted Leverkusen winger claimed that they have provided after each of his week.

The Magpies boss even asked Manchester United manager Sir Alex Ferguson lawyer Mr. regarding this issue, as he looks to provide guidelines to guide his players to use website.

Manager Alan Pardew has hinted that Barton Newcastle career may not be over after this week, despite the microblogging controversy.

 QUOT, Joey is a great player and you want great players in your team.

At that time, he will certainly be looked at leave the club this summer, with his agent Willie McKay said several clubs have shown their interest, but there have been hints of compromise between the parties be allowed to strike a balance today.

 QUOT; This is not to say that in time.

 QUOT short time can not be changed, I still reflect the situation, as is Joey At this moment, on how best to address this issue.

 QUOT, we got nothing from the Premier League on how to deal with this and QUOT; 

 QUOT, we need one or two signature. I have not taken into account (Barton's replacement). Like I said, this is a time reflection.

 [I think it is important and QUOT; 

Read More: 

Pardew reluctant to respond to these comments and £ 40,000 salary, while acknowledging that he needs to introduce new players, denied that he had begun looking for alternative Barton goods. 45,452,323 and QUOT, obviously, we need to solve Joey. We also need to bring some players into this club, and QUOT; he added.

Tottenham and then the Celtics leading Barton chase

== BM2cT cheap soccer jerseys 2014 Alan Shearer interview Liver ==

 'He is a Liverpoool player, through a grand ball and Andy would like to play with him.' 

The most cheap soccer jerseys 2014 abundant when

 ', of course, you do not want to lose your top players and Andy - Carol and Kevin Snow Portland is likely to be both Newcastle's best players.

 'if they are able to further increase next season - which is all the fans want.

 'I would be surprised that they did not go out and spend more money. I hope they do, because they have a decent season, they return to the Premiership League.

He said: 'Everyone was disappointed that Andy Carroll left. But it is understandable when you see them get him 35 million pounds of money is a huge lot cash.

Shearer sell top scorer Kevin - Nolan £ 4,000,000 believes the Magpies must recruit other goal scorer, if they want to improve on last season 0.

 'Andy is fatal appropriate services. get the right ball to him, he is unplayable.

' Last season was good, and now supporters Alan Pardew will be looking to take them into the top 10. 'Alan Shearer 45,452,323 exclusive of all time Prem XI: Who? ...... And who in 

 'Kenny is at the right club. He loves Liverpool, they are born for each other. '

Shearer said:' Andy can not have a better manager. Kenny has extensive experience and expertise in order to bring out the best him.

But Shearer do not think Liverpool are a strong contender for the championship, however, despite an increase spree.

it

' I think it would be a great Downing signing too.

Free transfer Demba Ba is the only In addition to this summer's compelling manager Alan Pardew, who also gave West Ham.

The Premiership striker Andy Carroll think will be the next Kenny Dalglish.

 This is a great move,' in Downing, Charlie - Adam and Henderson, Kenny brought three players who have a pass, cross and dead ball skills to provide Andy.

Newcastle legend Alan - Shearer admitted he would be surprised if his former club did not spend some 35 million pounds, they sell Carroll's Liverpool got the 

He a huge expenditure: 'I think this may be a bit too [http://www.fengleibafs.cn/forum.php?mod=viewthread&tid=25905&fromuid=2091 AcPCr cheap soccer jerseys 2014 039Time to impress039 for Aston Villa039s young] much looking forward to Liverpool won the league title this season.

 *** 

Alan Shearer confident of a great player, Dalglish will remove the rough edges of Carroll's £ 35 million and make sure he gets the service so that he 'unplayable'.

' but Anfield fans [http://bulutaren.com/mw/index.php?title=User:Ve9q2k2b#42iKV_cheap_soccer_jerseys_2014_Alan_Shearer_interview_Liver 42iKV cheap soccer jerseys 2014 Alan Shearer interview Liver] are looking progression.

 'Andy [http://www.france-wojtkowiak.com/galerie/picture.php?/86/list/63,52,84,7,92,67,20,64,71,51,43,86,25,87,50&comments_order=desc#comments&comments_order=desc#comments&comments_order=desc#comments&comments_order=DESC#comments 9qoEH cheap soccer jerseys 2014 Alex Ferguson deli] Carroll.

 'They spent a lot of money, and they in Kenny A great manager fans expect their place in the Champions League this year.

 Exclusive Interview Shearer: Manchester United and Chelsea will contest the championship, but you can forget about Arsenal 

== fyysi cheap soccer jerseys 2014 Alan Shearer Premier League ==

' had a lot of new boys expectations, but beyond that, I think Sir Alex Ferguson's team will be as strong as ever. '

The last seven titles between crown-to-head battles have experienced either Old Trafford and Stamford Bridge, the former England captain does not believe Manchester City's astonishing spending power and rebirth Liverpool will destroy the balance of power in the new campaign.

 success' Liverpool have spent an absolute fortune, and has been a large-scale improvements, I think they will challenge one of the top three or top four spot.

' But if I was pushed I would probably go to Manchester United only leading Chelsea.

'If you ask me now, 'in order to win the league title? '

' the city [http://www.7th.com.cn/plus/view.php?aid=739870 Gorz4 cheap soccer jerseys 2014 Alex Ferguson My 2] could lose a great player, but they have sufficient financial capacity, everyone is blown out of the water to let [http://plataformaisonomia.uji.es/campanyabonrotllo/picture.php?/13/list/22,9,12,30,24,33,26,19,25,28,21,35,2,13,1&comments_order=desc#comments&comments_order=desc#comments&comments_order=DESC#comments wF6TX cheap soccer jerseys 2014 Anton Ferdinand ma] everyone else as good.

 'It is difficult to assess Arsenal at the moment, because I believe they will bring players, but we do not know if Fabregas and Nasri will go there to make look very difficult.

 'He will require not only the Premier League title and the Champions League League.

' If so, now I do not think the city can win the league title, but ask me again in six weeks time This may be related to [http://bulutaren.com/mw/index.php?title=User:Ve9q2k2b#OWIK0_cheap_soccer_jerseys_2014_Alex_Ferguson_25_years_at_Ma OWIK0 cheap soccer jerseys 2014 Alex Ferguson 25 years at Ma] their ability to in.

He added:' Chelsea will not spend all the money to attract Boas from Porto, and if they do not believe he can do things, Mourinho made a sort. They will increase between cheap soccer jerseys 2014 now and the end of August, a number of new players. Chelsea went close to last season. They can win the league championship. This is expected Boas. We all know that Abramovich's ambition are.

 'I'm sure they will because of what Wenger has achieved.

He explained:' It will be interesting to see how they cope with the loss of Edwin van der Sar . 'They brought the young Spanish goalkeeper De Gea from Atletico Madrid, but Van der Sar is unmatched and they will also have a job to replace Paul Scholes, who is a off.

' I would say 'no' .

Alan Shearer believes the first 20 Premier League season will once again become the head of Manchester United and Chelsea.

 'Mancini's team will be a big say in what happens, but when push comes to shove I think it will again come down to Manchester United and Chelsea. It will be very nervous, because I believe that this season more teams will beat each other more than ever. '

' big club big loss players.

 'There are about six weeks to go before the transfer window closes, the purchasing power of the city can still change picture.

's 'Their incredible record and experience are likely to make their advantage '

Shearer, who won the title in 1995 Blackburn , admits the FA Cup could be a major force in the city, but think before they surpass champion and runner-up Chelsea.

EXCLUSIVE Alan Shearer's all-time Prem XI: Who is in and who is it ......? 

 'Manchester United have shown over the years, still gone on to be successful.

 totally different players, 'It looks like the city will lose Carlos Tevez, but they will have a plan to take care it.

Shearer not rule out another powerful challenge of the new Chelsea boss Andre Villas-headed Boas.

Shearer forecast, the biggest challenge may be the star of Manchester United's rule greatly from the experience, such as Edwin van der Sar and Paul Scholes losses 0.

EXCLUSIVE Alan Shearer's Carroll, Liverpool's title credentials and Newcastle's transfer policy, 

 is a saying,' but there is no great pressure on them, because they have gone so long no trophy. 

Shearer said the United States, you may need a season: 'I think the city will have something to say Premiership American domination and Chelsea. 

 'in the upcoming championship unbelievable close.

== T2vJG cheap soccer jerseys 2014 Alex Ferguson 25 years at Ma ==

Fergie: Slow Rio must change his game quickly

 ', when you consider the number of times he has gone a bridge too far, he has successfully star said even if people think the players are too young, and he did not. arrival Tom Cleverley in this season, he and all the young lads have begun to bloom. Alex just talks to them, they listen. '

Fergie: Rooney the only Englishman no surprises in the election of 

Charlton Ferguson revealed that this is a long-term vision, the United States persuaded the board to appoint him in the first place.' He gave us a stable and consistent dividend has been paid, 'said Sir Bobby.' When you think the average stay Managers are three and a half years, you know what I mean about stability. '

United director Charlton, who at the club Ferguson from Aberdeen plays a key role in 1986 , said the Scot's appetite for work is as strong as ever, despite turning 70 next month.

Ferguson famous changed his mind, retired in February 2002 and the Reds boss Sir Bobby does not want to retire any time soon,' I do not think he will be another decision again about him, said: 'Charles Dayton (right) [http://www.fengleibafs.cn/forum.php?mod=viewthread&tid=25997&fromuid=2091 aJnhV cheap soccer jerseys 2014 Arsenal Alex Oxlade-Chamberlain hopes Arsene Wen] .

Asked restrictions if a manager will be able to repeat Ferguson 25 years with the United Nations in the modern game, Charlton said:' There are a lot of good young coach, but Alex has a longevity. He was born in it. He has the patience, very talented. '. 

United boss full of praise for top students Solskajer

Charlton, 74, also praised Ferguson's relentless desire to keep the United States at the top, despite the challenges they face, with Manchester City and Barcelona are now down the gauntlet at home and abroad. 

Ruthless Ferguson is not afraid to rebuild his defense 

Sir Bobby Charlton praised Ferguson is a 'genius' as the Manchester United boss approaching his 25th anniversary in the old Trafford.

, 'he done it once, it did not work, I do not think he would do it again. He would go and and. You have to tolerate him for a while [http://www.jamonjamon.eu/node/162?page=122#comment-6783 86pPP cheap soccer jerseys 2014 Anton Ferdinand make statement about John terry] yet.

 'as long as he has his health, he said: He will not retire, I think All Manchester United supporters hope his life the longest possible. '

And Sir Bobby cheap soccer jerseys 2014 Ferguson singled extraordinary ability to constantly reinvent his team, his biggest shareholder strength. 'Alex is [http://www.jhwsjds.com/Review.asp?NewsID=2167 Air Asia and F1 Team Lotus tycoon Tony Fernandes completes QPR takeover] a genius,' Sir Bobby Charlton, Matt Busby's legendary lineup 50-60 60s.

 'I admire the way the sidelines, he still got exactly what he wanted modern affluent players, he is simply fantastic, it is difficult to understand the time, here is a man toward his 70s, he is an inspiration, and his contemporaries, if I used to think I should take it easier, I looked at Alex, age of the head as far as possible to achieve in body.

 'He just continues to be no slowing down, his energy, his drive and his desire to make the best conditions of Manchester United, added:' The World Cup winner.

 'I never cross word he was an absolute joy and cooperate with him in all the time he's been here.' 

Golden Globes D'Or

== Jha4Q cheap soccer jerseys 2014 Alex Ferguson 25 years at Ma ==

 everyone 'This is a great football club, full stop. 45,452,323 and QUOT; gaffer has done is what it should be in a level, winning the championship and the FA Cup, and is participating in the Champions League, the club will be on the biggest stage in the world '

Former Manchester United captain Gary Neville, who retired last season, the 25th anniversary of his former manager, is responsible for Ferguson ahead of Manchester United at Old praised this weekend .

 fruit 'It will not surprise me if he went, or even five years. 45,452,323 and QUOT; who eventually takes over will face a daunting task because there are so many comparisons - possibly more from outside than from the inside '

' We thought he was going to retire a few years ago, but he changed his mind, everyone was surprised I was surprised, the most important. The reason is because he is usually very clear what he does.

 'He is the man who will not flatter you, but when he came, and said,' Well done, son 'One, this is the most important thing.

Neville, now Sky Sports football analyst, known as Ferguson's only people management skills for long life and his whole reason behind that 69-year-old was a 'father figure' to those who fought him.

And although public perception Ferguson is a manager who is ruled by [http://wiki.crimsonbattalion.info/index.php/User:Hi6apk9p#9M3zy_cheap_soccer_jerseys_2014_Alex_Ferguson_25_years_at_Ma 9M3zy cheap soccer jerseys 2014 Alex Ferguson 25 years at Ma] fear, and his notoriety impeccable hair dryer treatment given to those who reach his standards, Neville revealed his father's side drive Scot.

 'has in the past players Who is the manager at other clubs now who often ask him, he always had lots to talk to.

' I know, from a player. I know, from a fan. This will make him young. Once upon a cheap soccer [http://bbs.ihepeng.com/forum.php?mod=viewthread&tid=119618&fromuid=60167 0RDoJ cheap soccer jerseys 2014 039Time to impress039 for Aston Villa039s young] jerseys 2014 time, people are talking about the time he retired, but has been away.

 years of work,' the problem is that you never want to leave the United States, 'he said. . 'Everything about the club is so professional - even the gaffer not leave.

 'has anyone considered irreplaceable player, but time has proven that they do not, because the other players always want to come to Manchester United, walking in the mantle. '

 'This is an invalid can not fill, whether you leave this club player or manager. 45,452,323 and QUOT, it is impossible to lose the passion, that pleasure. who had left Old Trafford will tell you the same .

Gary Neville Ferguson has estimated that hunger and desire to be responsible for another five years.

', he is certainly not one to dwell on the emotions, this is one thing that has identified him continued success in the past 25 years - to change the team continually.

 'However, the time draws near, what he was thinking, he can not do without it, it would be the same man in his position.

 'people from the outside may think he has a fear exhibited by his players, but there are More, he said, 'said Neville.

 'He has now done 25 years of work, still here, why did he want to leave it? 'Neville said. 'He felt as determined as ever, this is an ongoing challenge in United.

And, far from slowing down, Neville said he could see the Scot - who turn 70 in the New Year's Eve - until he has accumulated 30 years of club.

' He has been a father figure, and he has been at Old Trafford, I would imagine it is the same, when he Aberdeen.

 Ferguson: People want to be considered from Alex Ferguson was sacked early in the Edwards 45,452,323 Ferguson admitted the 25-year United States: Quetzal Dayton reveals the secret behind the genius 'manager's success, 45,452,323 Ferguson's 25-year United States: how to put a truancy Scottish accent helped Manchester United get their man 45,452,323 Ferguson's 25 years of the United States: How to become almost Eriksson Old Trafford boss Sir Alex Ferguson in 2002 45,452,323 25-year United States: Sir Alex Ferguson will see off the city's challenges - Edwards 

 * Neville extract matter by David Mick and Tom 25 years the United States and Manchester [http://iehdi.org/wiki/index.php?title=User:Ae2o3b5y#bqaWp_cheap_soccer_jerseys_2014_Alex_Ferguson_25_years_at_Ma bqaWp cheap soccer jerseys 2014 Alex Ferguson 25 years at Ma] United official story Tyrrell.

Neville forecast, which is under the unique temptations and pull the United Nations, Ferguson will be difficult to finally let go, surrender management team who else.

 'in the percentage of time the past few years, he has let some great players go. Sometimes people have agreed with him, sometimes They haven't.

 ' He almost saw the end of the players only to see himself and he has proved 100% correct.

 'He knows the club, he knows how to make it successful, we have been developing a new side there are a lot of young players into the national team, I hope we can continue to bear that.

== 47Lc4 cheap soccer jerseys 2014 Alex Ferguson 25 years at Ma ==

Ferguson took over Manchester United in 1986, when England star Wayne Rooney is just A one cheap soccer jerseys 2014 year old baby.

Alex team, there are a lot more talent than ever are playing - Dalglish

Wayne Rooney said that Manchester United players were still blown away by Ferguson's inspirational team talks.

MirrorFootball podcast: What is it you want to get as far as possible to find a hair dryer Ferguson Savage, David MacDonald, sponsored Gershon Portnoi participate history? Steve Angell West 

 'His hunger and desire is still the same and I've never seen anyone with his desire and passion.

Ferguson it comes to live in the United States of Contract confrontation October 2010.

 'LT will not surprise me if he was here 10 years and his desire to remain there.

 QUOT; reach 25 years is a huge achievement. This is incredibly long in charge of the same club, and with all his success. This will never happen again.

Power

since 2004

Rooney Ferguson praised the capacity to change the game's staff at Manchester United but still allow them to top.

Rooney has been in the United States, but he says he still gets surprised by Ferguson's dressing room speeches.

No people will comply with Ferguson! Steve Bruce is running forecast 45,452,323 see huge new banner will be unveiled Ferguson's big day in 

The players hinted that there is no other manager has also been able to convince him stay.

 'I do not think this is what they can not replace.' 

 'so good, he understands the players, he can have a go and those who do not take this approach.' [http://www.wikisportsmemorabilia.com/User:B3ebr6n9#ao13N_cheap_soccer_jerseys_2014_Arsenal_3-0_Bolton_match_rep ao13N cheap soccer jerseys 2014 Arsenal 3-0 Bolton match rep] Highly striker Wayne Rooney 

 'some of his Team talks are incredible, said: 'striker. 'You go to court and tell each other,' how good of it? '

' There are a number of other different things - it's not just managers - but he was, why I was still a major part of here.

' I think he'll stay here for a very long time. '

 'All my childhood growing up, I was watching him on TV, doing interviews before, after games, and QUOT; he said.' To play so he is a great honour.

 'in 1994, the team has a lot of English [http://112.126.65.107/read.php?tid=183307&ds=1&page=e#a] AFC Wimbledon boss Terry Brown says only getting started as prepare for Football League debut] and British players now have a lot of foreign players, who if he has a go of their favorite British players might.

 'For me, he is a great coach, a great person,' Rooney said. 'He is a big reason I signed here, the reason why I'm still here.

' For British players, who know him, this is incredible. It is more difficult for foreign lads to understand, but it lets you right up the game.

Ferguson: My 25-year spell in the United States has always been a 'fairy tale' 

David Beckham Interview: I owe it all Fergie

' so he could have been really successful thirty-four team, 'Rooney said.' To lose players , then build a successful team again three or four times more than is incredible.

 'It [http://bp2-team.com:1000/museum/nsfg/gameserver/viewtopic.php?f=2&t=34448 RDG8M cheap soccer jerseys 2014 Alex Ferguson 25 y] may be a bit difficult for them to take may not react the same. This is his man management to in.

' wipe it off, but also to me, I'm sure, all other players. '