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William Vestey, 1st Baron Vestey

William Vestey, 1st Baron Vestey (21 January 1859 – 10 December 1940), was an English shipping magnate.

Early life
William Vestey was born on 21 January 1859. He came from an old Liverpool family of traders. In 1876, at the age of seventeen, he was sent to Chicago by his father Samuel Vestey, a provisioner of Liverpool.

Career
He first managed a meat canning factory that was financed by his father. Together with his younger brother Edmund, he established Vestey Brothers (which later became the Vestey Group) in 1897 from a family butchery business in Liverpool. They were pioneers of refrigeration, opening a cold store in London in 1895.

The Vestey brothers then went to South America in an attempt to make a fortune because the economy there was booming. They started by buying game birds and storing them in the cold stores of American companies before shipping them to Liverpool. These early activities soon developed into importing beef and beef products into the UK, which in turn led to them owning cattle ranches in Brazil, Venezuela and Australia, and their own meat processing factories in Argentina, Uruguay (Frigorífico Anglo del Uruguay), New Zealand and Australia. In 1914, they built a meat processing works at Bullocky Point, Darwin, Australia, but closed its operations in 1920 after the Darwin Rebellion.

They acquired the 3000 km2 Wave Hill Station in the Northern Territory of Australia, in 1914. At that time, legislation permitted Aboriginal Australian workers to be paid in tea, tobacco and other rations. The Vesteys refused to pay their workers in wages, leading to tensions and arguments from the beginning, which continued until the Wave Hill walk-off, a strike beginning in 1967 and lasting eight years.

In 1915, the brothers, after being refused a request for income tax exemption made to David Lloyd George, moved to Buenos Aires to avoid paying income tax in the UK. The family later administered the business through a Paris trust that enabled it to legally avoid UK tax until the loophole was closed in 1991. From 1915 to 1918, they moved to Chicago then to Argentina and back to England. Lord Vestey later became an important benefactor to Liverpool Cathedral, where he funded the building of the bell tower.

First World War and peerage
During the First World War another Vestey company, the Blue Star Line (now part of P&O Nedlloyd), was a major supplier of Argentine beef to England, and it was for this service to the wartime provisioning of England that William Vestey was later raised to the peerage. He was made a Baronet of Bessémer House in the Metropoliton Borough of Camberwell on 21 June 1913, and Baron of Kingswood in the County of Surrey on 20 June 1922.

His appointment occurred at the height of the honours scandal surrounding the sale of peerage, which implicated the Prime Minister, David Lloyd George. William Vestey was not consider worthy of ennoblement because the Vestey brothers had moved their meatpacking business out of the country during the First World War, to avoid paying millions of pounds in tax. Even though King George V opposed Sir William becoming a baron, he received the title from Lloyd George after paying a £20,000 political donation.

Personal life
His first wife died in 1923 and was buried in Liverpool Cathedral. He then married Evelene Brodstone of Superior, Nebraska, on 1 August 1924. She had been working as a stenographer with the Vestey Meat Packing Plant in Chicago, where she was spotted by his brother. She would rise through the company, eventually becoming the highest paid female executive in the world. She survived him following his death aged 81 in December 1940. His ashes were buried in Liverpool Cathedral. On 24 July 1941, the 2nd Lady Vestey was buried at Evergreen Cemetery of Superior in Nebraska. Each spring during memorial weekend, Superior holds the annual Lady Vestey Festival in her honour. This is the town's largest annual celebration and it attracts many people from around the area.

Literature

 * Phillip Knightley The Rise and Fall of the House of Vestey, on the business empire established by William Vestey in 1897;

title: William Vestey, 1st Baron Vestey


text: Category:Islands of California

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Category:Islands of California


text:
User talk:JaneM~enwiki

Hi Jane, this is Nathan from ANI-L (well, I'm trying to rejoin now that my school account is closed). Small world, I guess. Anyhow, if you want advice re contraversies in Autism please let me know - I've been around here awhile. Pakaran. 00:34, 17 Oct 2004 (UTC)

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Clebsch–Gordan coefficients

In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory.

From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in quantum-mechanical terms can be read directly from this approach as spherical harmonics are eigenfunctions of total angular momentum and projection thereof onto an axis, and the integrals correspond to the Hilbert space inner product. From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also exist complicated explicit formulas for their direct calculation.

The formulas below use Dirac's bra–ket notation and the Condon–Shortley phase convention is adopted.

Review of the angular momentum operators
Angular momentum operators are self-adjoint operators $j_{x}$, $j_{y}$, and $j_{z}$ that satisfy the commutation relations $$ \begin{align} &[\mathrm{j}_k, \mathrm{j}_l] \equiv \mathrm{j}_k \mathrm{j}_l - \mathrm{j}_l \mathrm{j}_k = i \hbar \varepsilon_{klm} \mathrm{j}_m & k, l, m &\in \{ \mathrm{x, y,  z}\}, \end{align} $$ where $ε_{klm}$ is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator, $$ \mathbf j = (\mathrm{j_x}, \mathrm{j_y}, \mathrm{j_z}). $$ It is also known as a spherical vector, since it is also a spherical tensor operator. It is only for rank one that spherical tensor operators coincide with the Cartesian tensor operators.

By developing this concept further, one can define another operator $j^{2}$ as the inner product of $j$ with itself: $$ \mathbf j^2 = \mathrm{j_x^2} + \mathrm{j_y^2} + \mathrm{j_z^2}. $$ This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra $$\mathfrak{so}(3,\mathbb{R}) \cong \mathfrak{su}(2)$$. This is physically interpreted as the square of the total angular momentum of the states on which the representation acts.

One can also define raising ($j_{+}$) and lowering ($j_{−}$) operators, the so-called ladder operators, $$ \mathrm {j_\pm} = \mathrm{j_x} \pm i \mathrm{j_y}. $$

Spherical basis for angular momentum eigenstates
It can be shown from the above definitions that $j^{2}$ commutes with $j_{x}$, $j_{y}$, and $j_{z}$: $$ \begin{align} &[\mathbf j^2, \mathrm {j}_k] = 0 & k &\in \{\mathrm x, \mathrm y, \mathrm z\}. \end{align} $$

When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, $j^{2}$ and $j_{z}$ are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted $|j m\rangle$ where $j$ is the angular momentum quantum number and $m$ is the angular momentum projection onto the z-axis.

They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations, $$ \begin{align} \mathbf j^2 |j \, m\rangle &= \hbar^2 j (j + 1) |j \, m\rangle, & j &\in \{0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots\} \\ \mathrm{j_z} |j \, m\rangle &= \hbar m |j \, m\rangle, & m &\in \{-j, -j + 1, \ldots, j\}. \end{align} $$

The raising and lowering operators can be used to alter the value of $m$, $$ \mathrm j_\pm |j \, m\rangle = \hbar C_\pm(j, m) |j \, (m \pm 1)\rangle, $$ where the ladder coefficient is given by:

In principle, one may also introduce a (possibly complex) phase factor in the definition of $$C_\pm(j, m)$$. The choice made in this article is in agreement with the Condon–Shortley phase convention. The angular momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized, $$ \langle j \, m | j' \, m' \rangle = \delta_{j, j'} \delta_{m, m'}. $$

Here the italicized $j$ and $m$ denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman $j_{x}$, $j_{y}$, $j_{z}$, $j_{+}$, $j_{−}$, and $j^{2}$ denote operators. The $$\delta$$ symbols are Kronecker deltas.

Tensor product space
We now consider systems with two physically different angular momenta $j_{1}$ and $j_{2}$. Examples include the spin and the orbital angular momentum of a single electron, or the spins of two electrons, or the orbital angular momenta of two electrons. Mathematically, this means that the angular momentum operators act on a space $$V_1$$ of dimension $$2j_1+1$$ and also on a space $$V_2$$ of dimension $$2j_2 + 1$$. We are then going to define a family of "total angular momentum" operators acting on the tensor product space $$V_1 \otimes V_2$$, which has dimension $$(2j_1+1)(2j_2+1)$$. The action of the total angular momentum operator on this space constitutes a representation of the SU(2) Lie algebra, but a reducible one. The reduction of this reducible representation into irreducible pieces is the goal of Clebsch–Gordan theory.

Let $V_{1}$ be the $(2 j_{1} + 1)$-dimensional vector space spanned by the states $$ \begin{align} &|j_1 \, m_1\rangle, & m_1 &\in \{-j_1, -j_1 + 1, \ldots, j_1\} \end{align}, $$ and $V_{2}$ the $(2 j_{2} + 1)$-dimensional vector space spanned by the states $$ \begin{align} &|j_2 \, m_2\rangle, & m_2 &\in \{-j_2, -j_2 + 1, \ldots, j_2\} \end{align}. $$

The tensor product of these spaces, $V_{3} ≡ V_{1} ⊗ V_{2}$, has a $(2 j_{1} + 1) (2 j_{2} + 1)$-dimensional uncoupled basis $$ |j_1 \, m_1 \, j_2 \, m_2\rangle \equiv |j_1 \, m_1\rangle \otimes |j_2 \, m_2\rangle, \quad m_1 \in \{-j_1, -j_1 + 1, \ldots, j_1\}, \quad m_2 \in \{-j_2, -j_2 + 1, \ldots, j_2\}. $$ Angular momentum operators are defined to act on states in $V_{3}$ in the following manner: $$ (\mathbf j \otimes 1) |j_1 \, m_1 \, j_2 \, m_2\rangle \equiv \mathbf j |j_1 \, m_1\rangle \otimes |j_2 \, m_2\rangle $$ and $$ (1 \otimes \mathrm \mathbf j) |j_1 \, m_1 \, j_2 \, m_2\rangle \equiv |j_1 \, m_1\rangle \otimes \mathbf j |j_2 \, m_2\rangle, $$ where $1$ denotes the identity operator.

The total angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on $j_{1}$,

The total angular momentum operators can be shown to satisfy the very same commutation relations, $$ [\mathrm{J}_k, \mathrm{J}_l] = i \hbar \varepsilon_{k l m} \mathrm{J}_m  ~, $$ where $j_{2}$. Indeed, the preceding construction is the standard method for constructing an action of a Lie algebra on a tensor product representation.

Hence, a set of coupled eigenstates exist for the total angular momentum operator as well, $$ \begin{align} \mathbf{J}^2 |[j_1 \, j_2] \, J \, M\rangle &= \hbar^2 J (J + 1) |[j_1 \, j_2] \, J \, M\rangle \\ \mathrm{J_z} |[j_1 \, j_2] \, J \, M\rangle &= \hbar M |[j_1 \, j_2] \, J \, M\rangle \end{align} $$ for $V_{1}⊗V_{2}$. Note that it is common to omit the $k, l, m ∈ \{x, y, z\}$ part.

The total angular momentum quantum number $M ∈ \{−J, −J + 1, ..., J\}$ must satisfy the triangular condition that $$ |j_1 - j_2| \leq J \leq j_1 + j_2, $$ such that the three nonnegative integer or half-integer values could correspond to the three sides of a triangle.

The total number of total angular momentum eigenstates is necessarily equal to the dimension of $[j_{1} j_{2}]$: $$ \sum_{J = |j_1 - j_2|}^{j_1 + j_2} (2 J + 1) = (2 j_1 + 1) (2 j_2 + 1) ~. $$ As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension $$2J+1$$, where $$J$$ ranges from $$|j_1 - j_2|$$ to $$j_1 + j_2$$ in increments of 1. As an example, consider the tensor product of the three-dimensional representation corresponding to $$j_1 = 1$$ with the two-dimensional representation with $$j_2 = 1/2$$. The possible values of $$J$$ are then $$J = 1/2$$ and $$J = 3/2$$. Thus, the six-dimensional tensor product representation decomposes as the direct sum of a two-dimensional representation and a four-dimensional representation.

The goal is now to describe the preceding decomposition explicitly, that is, to explicitly describe basis elements in the tensor product space for each of the component representations that arise.

The total angular momentum states form an orthonormal basis of $J$: $$ \left\langle J\, M | J'\, M' \right\rangle = \delta_{J, J'}\delta_{M, M'}~. $$

These rules may be iterated to, e.g., combine $$ doublets ($n$=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle), $$ \mathbf{2}^{\otimes n} = \bigoplus_{k=0}^{\lfloor n/2 \rfloor}~ \left(\frac{n + 1 - 2k}{n + 1}{n + 1 \choose k}\right)~(\mathbf{n} + \mathbf{1} - \mathbf{2}\mathbf{k})~, $$ where $$\lfloor n/2 \rfloor$$ is the integer floor function; and the number preceding the boldface irreducible representation dimensionality ($V_{3}$) label indicates multiplicity of that representation in the representation reduction. For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, $${\mathbf 2}\otimes{\mathbf 2}\otimes{\mathbf 2} = {\mathbf 4}\oplus{\mathbf 2}\oplus{\mathbf 2}$$.

Formal definition of Clebsch–Gordan coefficients
The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis

The expansion coefficients

$$ \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle $$

are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as $V_{3}$. Another common notation is $2j+1$.

Applying the operators

$$ \begin{align} \mathrm J&=\mathrm j \otimes 1+1\otimes\mathrm j \\ \mathrm J_{\mathrm z}&=\mathrm j_{\mathrm z}\otimes 1+1\otimes\mathrm j_{\mathrm z} \end{align} $$

to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when

$$ \begin{align} |j_1 - j_2| \leq J &\leq j_1 + j_2  \\ M &= m_1 + m_2. \end{align} $$

Recursion relations
The recursion relations were discovered by physicist Giulio Racah from the Hebrew University of Jerusalem in 1941.

Applying the total angular momentum raising and lowering operators$$ \mathrm J_\pm = \mathrm j_\pm \otimes 1 + 1 \otimes \mathrm j_\pm $$ to the left hand side of the defining equation gives$$ \begin{align} \mathrm J_\pm |[j_1 \, j_2] \, J \, M\rangle &= \hbar C_\pm(J, M) |[j_1 \, j_2] \, J \, (M \pm 1)\rangle \\ &= \hbar C_\pm(J, M)     \sum_{m_1, m_2} |j_1 \, m_1 \, j_2 \, m_2\rangle \langle j_1 \, m_1 \, j_2 \, m_2 | J \, (M \pm 1)\rangle \end{align} $$ Applying the same operators to the right hand side gives$$ \begin{align} \mathrm J_\pm &\sum_{m_1, m_2} |j_1 \, m_1 \, j_2 \, m_2\rangle \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M\rangle \\ = \hbar &\sum_{m_1, m_2} \Bigl(       C_\pm(j_1, m_1) |j_1 \, (m_1 \pm 1) \, j_2 \, m_2\rangle      + C_\pm(j_2, m_2) |j_1 \, m_1 \, j_2 \, (m_2 \pm 1)\rangle    \Bigr) \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M\rangle \\ = \hbar &\sum_{m_1, m_2} |j_1 \, m_1 \, j_2 \, m_2\rangle \Bigl(       C_\pm(j_1, m_1 \mp 1) \langle j_1 \, (m_1 \mp 1) \, j_2 \, m_2 | J \, M\rangle      + C_\pm(j_2, m_2 \mp 1) \langle j_1 \, m_1 \, j_2 \, (m_2 \mp 1) | J \, M\rangle    \Bigr). \end{align} $$

Combining these results gives recursion relations for the Clebsch–Gordan coefficients, where $⟨j_{1} j_{2}; m_{1} m_{2}$ was defined in $s$: $$   C_\pm(J, M) \langle j_1 \, m_1 \, j_2 \, m_2 | J \, (M \pm 1)\rangle = C_\pm(j_1, m_1 \mp 1) \langle j_1 \, (m_1 \mp 1) \, j_2 \, m_2 | J \, M\rangle + C_\pm(j_2, m_2 \mp 1) \langle j_1 \, m_1 \, j_2 \, (m_2 \mp 1) | J \, M\rangle. $$

Taking the upper sign with the condition that $J M⟩ = CJM j_{1}m_{1}j_{2}m_{2}$ gives initial recursion relation:$$ 0 = C_+(j_1, m_1 - 1) \langle j_1 \, (m_1 - 1) \, j_2 \, m_2 | J \, J\rangle + C_+(j_2, m_2 - 1) \langle j_1 \, m_1 \, j_2 \, (m_2 - 1) | J \, J\rangle. $$In the Condon–Shortley phase convention, one adds the constraint that
 * $$\langle j_1 \, j_1 \, j_2 \, (J - j_1) | J \, J\rangle > 0$$

(and is therefore also real). The Clebsch–Gordan coefficients $C_{±}$ can then be found from these recursion relations. The normalization is fixed by the requirement that the sum of the squares, which equivalent to the requirement that the norm of the state $M = J$ must be one.

The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with $⟨j_{1} m_{1} j_{2} m_{2} | J M⟩$. Repeated use of that equation gives all coefficients.

This procedure to find the Clebsch–Gordan coefficients shows that they are all real in the Condon–Shortley phase convention.

Orthogonality relations
These are most clearly written down by introducing the alternative notation $$ \langle J \, M | j_1 \, m_1 \, j_2 \, m_2 \rangle \equiv \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle $$

The first orthogonality relation is $$ \sum_{J = |j_1 - j_2|}^{j_1 + j_2} \sum_{M = -J}^J \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle \langle J \, M | j_1 \, m_1' \, j_2 \, m_2' \rangle = \langle j_1 \, m_1 \, j_2 \, m_2 | j_1 \, m_1' \, j_2 \, m_2' \rangle = \delta_{m_1, m_1'} \delta_{m_2, m_2'} $$ (derived from the fact that $\mathbf 1 = \sum_x |x\rangle \langle x|$ ) and the second one is $$ \sum_{m_1, m_2} \langle J \, M | j_1 \, m_1 \, j_2 \, m_2 \rangle \langle j_1 \, m_1 \, j_2 \, m_2 | J' \, M' \rangle = \langle J \, M | J' \, M' \rangle = \delta_{J, J'} \delta_{M, M'}. $$

Special cases
For $|[j_{1} j_{2}] J J⟩$ the Clebsch–Gordan coefficients are given by $$ \langle j_1 \, m_1 \, j_2 \, m_2 | 0 \, 0 \rangle = \delta_{j_1, j_2} \delta_{m_1, -m_2} \frac{(-1)^{j_1 - m_1}}{\sqrt{2 j_1 + 1}}. $$

For $M = J − 1$ and $J = 0$ we have $$ \langle j_1 \, j_1 \, j_2 \, j_2 | (j_1 + j_2) \, (j_1 + j_2) \rangle = 1.$$

For $J = j_{1} + j_{2}$ and $M = J$ we have $$   \langle j_1 \, m_1 \, j_1 \, (-m_1) | (2 j_1) \, 0 \rangle = \frac{(2 j_1)!^2}{(j_1 - m_1)! (j_1 + m_1)! \sqrt{(4 j_1)!}}. $$

For $j_{1} = j_{2} = J / 2$ we have $$   \langle j_1 \, j_1 \, j_1 \, (-j_1) | J \, 0 \rangle = (2 j_1)! \sqrt{\frac{2 J + 1}{(J + 2 j_1 + 1)! (2 j_1 - J)!}}. $$

For $m_{1} = −m_{2}$, $j_{1} = j_{2} = m_{1} = −m_{2}$ we have $$ \begin{align} \langle j_1 \, m \, 1 \, 0 | (j_1 + 1) \, m \rangle &= \sqrt{\frac{(j_1 - m + 1) (j_1 + m + 1)}{(2 j_1 + 1) (j_1 + 1)}} \\ \langle j_1 \, m \, 1 \, 0 | j_1      \, m \rangle &= \frac{m}{\sqrt{j_1 (j_1 + 1)}} \\ \langle j_1 \, m \, 1 \, 0 | (j_1 - 1) \, m \rangle &= -\sqrt{\frac{(j_1 - m) (j_1 + m)}{j_1 (2 j_1 + 1)}} \end{align} $$

For $j_{2} = 1$ we have $$ \begin{align} \left\langle j_1 \, \left( M - \frac{1}{2} \right) \, \frac{1}{2} \, \frac{1}{2} \Bigg| \left( j_1 \pm \frac{1}{2} \right) \, M \right\rangle &= \pm \sqrt{\frac{1}{2} \left( 1 \pm \frac{M}{j_1 + \frac{1}{2}} \right)} \\ \left\langle j_1 \, \left( M + \frac{1}{2} \right) \, \frac{1}{2} \, \left( -\frac{1}{2} \right) \Bigg| \left( j_1 \pm \frac{1}{2} \right) \, M \right\rangle &= \sqrt{\frac{1}{2} \left( 1 \mp \frac{M}{j_1 + \frac{1}{2}} \right)} \end{align} $$

Symmetry properties
$$ \begin{align} \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle &= (-1)^{j_1 + j_2 - J} \langle j_1 \, (-m_1) \, j_2 \, (-m_2) | J \, (-M)\rangle \\ &= (-1)^{j_1 + j_2 - J} \langle j_2 \, m_2 \, j_1 \, m_1 | J \, M \rangle \\ &= (-1)^{j_1 - m_1} \sqrt{\frac{2 J + 1}{2 j_2 + 1}} \langle j_1 \, m_1 \, J \, (-M)| j_2 \, (-m_2) \rangle \\ &= (-1)^{j_2 + m_2} \sqrt{\frac{2 J + 1}{2 j_1 + 1}} \langle J \, (-M) \, j_2 \, m_2| j_1 \, (-m_1) \rangle \\ &= (-1)^{j_1 - m_1} \sqrt{\frac{2 J + 1}{2 j_2 + 1}} \langle J \, M \, j_1 \, (-m_1) | j_2 \, m_2 \rangle \\ &= (-1)^{j_2 + m_2} \sqrt{\frac{2 J + 1}{2 j_1 + 1}} \langle j_2 \, (-m_2) \, J \, M | j_1 \, m_1 \rangle \end{align} $$

A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using $$. The symmetry properties of Wigner 3-j symbols are much simpler.

Rules for phase factors
Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore $m_{2} = 0$ is not necessarily $j_{2} = 1/2$ for a given quantum number $(−1)^{2k}$ unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule: $$(-1)^{4 k} = 1$$ for any angular-momentum-like quantum number $1$.

Nonetheless, a combination of $k$ and $k$ is always an integer, so the stronger rule applies for these combinations: $$(-1)^{2 (j_i - m_i)} = 1$$ This identity also holds if the sign of either $j_{i}$ or $m_{i}$ or both is reversed.

It is useful to observe that any phase factor for a given $j_{i}$ pair can be reduced to the canonical form: $$(-1)^{a j_i + b (j_i - m_i)}$$ where $m_{i}$ and $(j_{i}, m_{i})$ (other conventions are possible too). Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent. (Note that this form is only locally canonical: it fails to take into account the rules that govern combinations of $a ∈ {0, 1, 2, 3}$ pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of $b ∈ {0, 1}$, $(j_{i}, m_{i})$, and $j_{1}$ that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol: $$(-1)^{2 (j_1 + j_2 + j_3)} = 1$$ This identity also holds if the sign of any $j_{2}$ is reversed, or if any of them are substituted with an $j_{3}$ instead.

Relation to Wigner 3-j symbols
Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations.

The factor $j_{i}$ is due to the Condon–Shortley constraint that $m_{i}$, while $(−1)^{2 j_{2}}|undefined$ is due to the time-reversed nature of $⟨j_{1} j_{1} j_{2} (J − j_{1})|J J⟩ > 0$.

This allows to reach the general expression:

\begin{align} \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle & \equiv \delta(m_1+m_2,M) \sqrt{\frac{(2J+1)(j_1+j_2-J)!(j_1-j_2+J)!(-j_1+j_2+J)!}{(j_1+j_2+J+1)!}}\ \times {} \\[6pt] &\times\sqrt{(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!(J-M)!(J+M)!}\ \times {} \\[6pt] &\times\sum_{k=K}^N \frac{(-1)^k}{k!(j_1+j_2-J-k)!(j_1-m_1-k)!(j_2+m_2-k)!(J-j_2+m_1+k)!(J-j_1-m_2+k)!}, \end{align}. $$

The summation is performed over those integer values $$ for which the argument of each factorial in the denominator is non-negative, i.e. summation limits $$ and $$ are taken equal: the lower one $$K=\max(0, j_2-J-m_1, j_1-J+m_2),$$ the upper one $$N=\min(j_1+j_2-J, j_1-m_1, j_2+m_2).$$ Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, $$j_3>j_1+j_2$$ or $$j_1<m_1$$ are automatically set to zero.

Relation to Wigner D-matrices
$$ \begin{align} &\int_0^{2 \pi} d \alpha \int_0^\pi \sin \beta \, d\beta \int_0^{2 \pi} d \gamma \, D^J_{M, K}(\alpha, \beta, \gamma)^* D^{j_1}_{m_1, k_1}(\alpha, \beta, \gamma) D^{j_2}_{m_2, k_2}(\alpha, \beta, \gamma) \\ {}={} &\frac{8 \pi^2}{2 J + 1} \langle j_1 \, m_1 \, j_2 \, m_2 | J \, M \rangle \langle j_1 \, k_1 \, j_2 \, k_2 | J \, K \rangle \end{align} $$

Relation to spherical harmonics
In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics: $$ \int_{4 \pi} Y_{\ell_1}^{m_1}{}^*(\Omega) Y_{\ell_2}^{m_2}{}^*(\Omega) Y_L^M (\Omega) \, d \Omega = \sqrt{\frac{(2 \ell_1 + 1) (2 \ell_2 + 1)}{4 \pi (2 L + 1)}} \langle \ell_1 \, 0 \, \ell_2 \, 0 | L \, 0 \rangle \langle \ell_1 \, m_1 \, \ell_2 \, m_2 | L \, M \rangle $$

It follows from this and orthonormality of the spherical harmonics that CG coefficients are in fact the expansion coefficients of a product of two spherical harmonics in terms of a single spherical harmonic: $$ Y_{\ell_1}^{m_1}(\Omega) Y_{\ell_2}^{m_2}(\Omega) = \sum_{L, M}   \sqrt{\frac{(2 \ell_1 + 1) (2 \ell_2 + 1)}{4 \pi (2 L + 1)}} \langle \ell_1 \, 0 \, \ell_2 \, 0 | L \, 0 \rangle \langle \ell_1 \, m_1 \, \ell_2 \, m_2 | L \, M \rangle Y_L^M (\Omega) $$

Other properties
$$\sum_m (-1)^{j - m} \langle j \, m \, j \, (-m) | J \, 0 \rangle = \delta_{J, 0} \sqrt{2 j + 1}$$

Clebsch–Gordan coefficients for specific groups
For arbitrary groups and their representations, Clebsch–Gordan coefficients are not known in general. However, algorithms to produce Clebsch–Gordan coefficients for the special unitary group SU(n) are known. In particular, SU(3) Clebsch-Gordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists that relates the up, down, and strange quarks. A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.

Clebsch–Gordan coefficients for symmetric group are also known as Kronecker coefficients.

title: Clebsch–Gordan coefficients


text:
User talk:Rhobite/Archive 1

Welcome to the Wikipedia
Here are some links I find useful

 * Policy Library
 * Cite your sources
 * Verifiability
 * Wikiquette
 * Civility
 * Conflict resolution
 * Brilliant prose
 * Neutral point of view
 * Pages needing attention
 * Peer review
 * Bad jokes and other deleted nonsense
 * Village pump
 * Boilerplate text

Feel free to ask me anything the links and talk pages don't answer. You can sign your name by typing 4 tildes, likes this: ~.

Cheers, Sam [Spade] 19:47, 30 Jun 2004 (UTC)

 * Ah, thank you. Long-time lurker, short-time contributor. --Rhobite 19:57, Jun 30, 2004 (UTC)

ACLU Page - Patriot II Act
I'm a newcomer to Wikipedia, so forgive me if I'm being erroneous. I made a minor edit on the ACLU page concerning the PATRIOT 2 act... deleting the "proposed" portion and shifting it into the definite. Correctly, you changed this. However, I am wondering if there should be yet another edit to this page, as while the entire act has not been passed yet, major portions of it were signed into law December 13th, 2003, the day Saddam Hussein was kidnapped by coalition forces? The San Antonio Current, the Associated Press (through Boston Globe Link), and the President himself. Your thoughts?

--jackferd 22:26, 6 Jul 2004 (UTC)

 * Don't worry about it, I'm new here too. I wasn't aware that parts of PATRIOT 2 had been signed into law in another bill. You may want to add this information to Domestic_Security_Enhancement_Act_of_2003 but I don't know if it belongs on the ACLU page. Rhobite 15:24, Jul 7, 2004 (UTC)

Sociobiological theories of rape
I've started on reworking this. It needs to be fed through the grinder a few more times before anything acceptable comes out the other end. See the talk page, too. -- The Anome 16:56, 21 Jul 2004 (UTC)

Abandonware
Alright. I see what you mean. Maybe at a later date WinWorld can be referenced in some way, as it continues to grow. Please remember that I don't own WinWorld.--Surfinshell23 04:49, 24 Jul 2004 (UTC)

Re: Commercialism
Hello,

Although I agree that Commercialism is, as you wrote on my talk page, rather vague and somewhat speculative, I don't think the tag was applicable. It's a simple article. It had a good start, but it will take some time to develop into a good article. It was a good idea to post at WP:PNA, hopefully to get the attention of Wiki-veterans knowledgeable in the field.

I don't think we should call for a clean-up on brand new articles, as time should be allowed for an article to grow in length and in depth. IMO, the tag should only be used on unattended (no one working on it the past many hours) articles with serious problems, e.g. nonsense, poor organization, atrocious grammar, non-NPOV/bias, unwikified data-dumping, etc. (Judging from what have been listed at WP:CU these days, there are people who disagree with me .... Oh, well .... )

To some extent, I was also trying to be gentle to a newbie. He just started here a few days ago, and apparently, this article on Commercialism was his first major undertaking ......

If you have ideas on how to improve the article, perhaps you can make suggestions at Talk:Commercialism.

Have a nice day. -- PFHLai 16:00, 2004 Jul 29 (UTC)

 * Thanks, I'm still learning the intricacies of the various "call for help" pages here. Between cleanup, PNA, RFC, VFD, speedy delete there really is a confusing array of choices - too many, in my opinion. Thanks for clearing this up. Unfortunately I don't have too much background in the area but I do think I'll fix the references to "cave men." Rhobite 16:32, Jul 29, 2004 (UTC)

 * You are very welcome. I have to say that I have only been around for a few months, so my opinions should not be taken as any "authority" at any level.  I do think PNA was the right route in this case.  Requests for page expansion is a possibility, but things don't really move there...  If you don't feel like fixing it, let it be.  Someone else may stroll by anytime and fix things to his or her liking.
 * Yeah, there are Wikipedians who prefer different ways of handling problematic articles, and there are articles that require different handling procedures. It IS confusing.  But it's better than having edit wars.
 * Happy editing. :-)
 * -- PFHLai 17:54, 2004 Jul 29 (UTC)

US Mint copyrights
Thanks for the notice. I agree that the images are probably Public domain. Not sure who owns the images of the US currency, but it is probably the Government, and hence PD. At Wikipedia, copies of 2D work are assumed to have the same copyright as the original work, (e.g. a painiting by Rubens has expired copyrights even if it is a modern photo thereof) The arrangement of the bills and the labels are probably original work, but very minor. I think we can also claim if necessary. -- Chris 73 | Talk 02:48, 5 Aug 2004 (UTC)

Windows XP
I don't know what you're talking about removing stuff--I haven't removed a single thing, I promise. As a matter of fact, I created the spyware sections. I also like the fact that the similarites to Mac OS X is gone, but I kind of started to like it, it seemed to bring a little friendship into the mix between the two, which have a history of enmity.--naryathegreat 03:33, Aug 13, 2004 (UTC)

 * Thanks, for a while I was afraid someone might be hijacking my account, i've heard of people changing user names on others, so I got a little worried, anyway, no harm done!

Mises
What's the dealio on Ludwig von Mises? I spoke directly with Jeff Tucker, the Vice-President of the LvMI and he gave me permission to use that text. It was written for ANB (who the LvMI gave permission to use it) by a fellow (Hullsmann) who up until this month was a resident senior fellow there (he has now gone on to a University in France). Anyways, I stated that in the edit log and at the bottom. Although I appreciate the concern, please contact me next time.

--Tejano 13:48, Aug 16, 2004 (UTC)

 * Since you posted it anonymously, I had no idea who to contact. If you are User:Tejano you should log in to that account. Anyway, the article says this: "Copyright © 2003 American Council of Learned Societies. Published by Oxford University Press. All rights reserved." Is that wrong? Does Mr. Tucker have the authority to license this under the GFDL? He's not the original author and he doesn't appear to represent the copyright holder. Also, let's move this to Talk:Ludwig von Mises. I'm copying these comments there. Rhobite 22:02, Aug 16, 2004 (UTC)

Windows XP reverts
Hi Rhobite,

Could we work out a compromise on the Windows XP page? I'm kind of wondering if we could cut down on the spyware/malware bit as it isn't specific to just Windows XP. Please don't get me wrong here, I'm not saying that it shouldn't be mentioned, but do we really need to mention the executive from the MSN group saying that half their crashes are caused by malware? That sort of sounds dodgy, even though I know this source is from Microsoft. Sometimes "executives" like to shoot their mouths off. If we could perhaps make the comment on malware/spyware more concise maybe that will appease both sides. It's worth a try before we bring it to the RFC page, anyway (I've kind of already gone there and I think they'll just lock the page for a while so noone will be able to edit it, but it would cool the revert wars that are going on).

What do you say? Sound OK to you?

Ta bu shi da yu 15:56, 17 Aug 2004 (UTC)

 * Hey Rhobite, want to vote in my staw poll? You seem to be involved in the edits on the XP page. See the Talk page for more info. - Ta bu shi da yu 14:36, 20 Aug 2004 (UTC)

K5 and Windows XP edits
You're captainsuperboy on K5? Good to see another K5 member here! BTW, I think your compromise on the XP article was actually pretty good. - Ta bu shi da yu 00:36, 20 Aug 2004 (UTC)
 * Yeah, that's me. I hate that nick. It's good to see you here too. Rhobite 04:28, Aug 20, 2004 (UTC)

O'Reilly change
You reverted a paragraph that I wrote that contained the following sentence: "Despite his self-proclaimed political independence, he has become a controversial figure in American politics."

I did a partial revert and changed it to: "He is a self-proclaimed political independent who has become a controversial figure in American politics."

The original sentence was: "As a self-proclaimed political independent, his social and political views vary widely."

I changed it for the following reason: The phrase, "his social and political views vary widely," is rather meaningnless. How do you measure "widely?" And vary widely from what, exactly? From Stalinism, from Neoliberalism? I can only guess the original author intended to say that he draws from both Conservative and Liberal ideologies. However, this is a judgement call. One man's idea of "liberal" is another man's "conservative" and vice versa. This is a debate best to be avoided in the opening sentences of the article.

title: User talk:Rhobite/Archive 1


text: File:The B-52's - Bouncing Off the Satellites.jpg


text:
Victor Daley

Victor James William Patrick Daley (5 September 1858 – 29 December 1905) was an Australian poet. Daley serves chiefly as an example of the Celtic Twilight in Australian verse. He also serves as a lyrical alternative to his contemporary bush balladists of Henry Lawson, Banjo Paterson, and Will H. Ogilvie.

Life
Daley was born at the Navan, County Meath, Ireland, and was educated at the Christian Brothers at Devonport in England. He arrived in Australia in 1878, and became a freelance journalist and writer in both Melbourne and Sydney. Whilst in Melbourne, he met and became a friend of Marcus Clarke; later, in Sydney, he became acquainted with Henry Kendall. He is notable for becoming the first author in Australia who tried to earn a living from writing alone.

In Sydney in 1898, he founded the bohemian Dawn and Dusk Club, and the later Supper Club, which had many notable members such as writer Henry Lawson. Together with fellow notable poet Louise Mack, he organised the farewell dinner to Scottish-Australian poet and bush balladeer Will H. Ogilvie (1869–1963) in 1901 at the Hotel Australia, Sydney. Banjo Paterson and Roderic Quinn were also present at the send-off.

He used the pseudonym Creeve Roe (Irish for Red Branch, the area next to the Navan where Cú Chulainn trained as a Red Branch Knight), as well as his own name.

Daley died at his home in Waitara, Sydney of tuberculosis and was buried in the Roman Catholic section of Waverley Cemetery.

Legacy
His work was not considered particularly Australian in nature, but quite lyrical, with 'natural delicacy of expression, graceful imagery, and refinement of language'. His Poems (1908) and other collections were published posthumously. Daley's finest Australian work was considered to be A Sunset Fantasy.

When he died, Scottish-Australian poet and bush balladeer Will H. Ogilvie (1869–1963) penned:

 * When 'the little folk' meet by the red rowan tree
 * The dance shall be stayed in the ring on the plot
 * While they twine in his green Irish isle of the sea
 * The wreath we forgot.

A memoir of Daley by Bertram Stevens was published in Wine and Roses.

Individual works

 * "The First of May" (1882)
 * "At the Opera" (1883)
 * "Dreams" (1883)
 * "On the River" (1885) (also known as "Years After")
 * "On the Shore" (1885) (also known as "Sunset")
 * "Brunette" (1886)
 * "Poppies" (1886)
 * "The Old Wife and the New" (1887)
 * "A Sunset Fantasy" (1888)
 * "Even So" (1890)
 * "Lachesis" (1891)
 * "A-Roving" (1892)
 * "Cares" (1892)
 * "Correggio Jones" (1898)
 * "The Woods of Dandenong" (1899)
 * "Players" (1900)
 * "Anna" (1902)
 * "The Woman at the Washtub" (1902)
 * "The Night Ride" (1907)

title: Victor Daley


text: Category:Vice presidents of Iran

title: Category:Vice presidents of Iran


text:
Lamb shift

In physics, the Lamb shift, named after Willis Lamb, is an anomalous difference in energy between two electron orbitals in a hydrogen atom. The difference was not predicted by theory and it cannot be derived from the Dirac equation, which predicts identical energies. Hence the Lamb shift is a deviation from theory seen in the differing energies contained by the 2S1/2 and 2P1/2 orbitals of the hydrogen atom.

The Lamb shift is caused by interactions between the virtual photons created through vacuum energy fluctuations and the electron as it moves around the hydrogen nucleus in each of these two orbitals. The Lamb shift has since played a significant role through vacuum energy fluctuations in theoretical prediction of Hawking radiation from black holes.

This effect was first measured in 1947 in the Lamb–Retherford experiment on the hydrogen microwave spectrum and this measurement provided the stimulus for renormalization theory to handle the divergences. It was the harbinger of modern quantum electrodynamics developed by Julian Schwinger, Richard Feynman, Ernst Stueckelberg, Sin-Itiro Tomonaga and Freeman Dyson. Lamb won the Nobel Prize in Physics in 1955 for his discoveries related to the Lamb shift.

Importance
In 1978, on Lamb's 65th birthday, Freeman Dyson addressed him as follows: "Those years, when the Lamb shift was the central theme of physics, were golden years for all the physicists of my generation. You were the first to see that this tiny shift, so elusive and hard to measure, would clarify our thinking about particles and fields."

Derivation
This heuristic derivation of the electrodynamic level shift follows Theodore A. Welton's approach.

The fluctuations in the electric and magnetic fields associated with the QED vacuum perturbs the electric potential due to the atomic nucleus. This perturbation causes a fluctuation in the position of the electron, which explains the energy shift. The difference of potential energy is given by

 * $$\Delta V = V(\vec{r}+\delta \vec{r})-V(\vec{r})=\delta \vec{r} \cdot \nabla V (\vec{r}) + \frac{1}{2} (\delta \vec{r} \cdot \nabla)^2V(\vec{r})+\cdots$$

Since the fluctuations are isotropic,

 * $$\langle \delta \vec{r} \rangle _{\rm vac} =0,$$
 * $$\langle (\delta \vec{r} \cdot \nabla )^2 \rangle _{\rm vac} = \frac{1}{3} \langle (\delta \vec{r})^2\rangle _{\rm vac} \nabla ^2.$$

So one can obtain

 * $$\langle \Delta V\rangle =\frac{1}{6} \langle (\delta \vec{r})^2\rangle _{\rm vac}\left\langle \nabla ^2\left(\frac{-e^2}{4\pi \epsilon _0r}\right)\right\rangle _{\rm at}.$$

The classical equation of motion for the electron displacement (δr)$k$ induced by a single mode of the field of wave vector $k$ and frequency ν is

 * $$m\frac{d^2}{dt^2} (\delta r)_{\vec{k}}=-eE_{\vec{k}},$$

and this is valid only when the frequency ν is greater than ν0 in the Bohr orbit, $$\nu > \pi c/a_0$$. The electron is unable to respond to the fluctuating field if the fluctuations are smaller than the natural orbital frequency in the atom.

For the field oscillating at ν,

 * $$\delta r(t)\cong \delta r(0)(e^{-i\nu t}+e^{i\nu t}),$$

therefore

 * $$(\delta r)_{\vec{k}} \cong \frac{e}{mc^2k^2} E_{\vec{k}}=\frac{e}{mc^2k^2} \mathcal{E} _{\vec{k}} \left (a_{\vec{k}}e^{-i\nu t+i\vec{k}\cdot \vec{r}}+h.c. \right) \qquad \text{with} \qquad \mathcal{E} _{\vec{k}}=\left(\frac{\hbar ck/2}{\epsilon _0 \Omega}\right)^{1/2},$$

where $$\Omega$$ is some large normalization volume (the volume of the hypothetical "box" containing the hydrogen atom), and $$h.c.$$ denotes the hermitian conjugate of the preceding term. By the summation over all $$\vec{k},$$

 * $$\begin{align}

\langle (\delta \vec{r} )^2\rangle _{\rm vac} &=\sum_{\vec{k}} \left(\frac{e}{mc^2k^2} \right)^2 \left\langle 0\left |(E_{\vec{k}})^2 \right |0 \right \rangle \\ &=\sum_{\vec{k}} \left(\frac{e}{mc^2k^2} \right)^2\left(\frac{\hbar ck}{2\epsilon _0 \Omega} \right) \\ &=2\frac{\Omega}{(2\pi )^3}4\pi \int dkk^2\left(\frac{e}{mc^2k^2} \right)^2\left(\frac{\hbar ck}{2\epsilon_0 \Omega}\right) && \text{since continuity of } \vec{k} \text{ implies } \sum_{\vec{k}} \to 2 \frac{\Omega}{(2\pi)^3} \int d^3 k \\ &=\frac{1}{2\epsilon_0\pi^2}\left(\frac{e^2}{\hbar c}\right)\left(\frac{\hbar}{mc}\right)^2\int \frac{dk}{k} \end{align}$$

This result diverges when no limits about the integral (at both large and small frequencies). As mentioned above, this method is expected to be valid only when $$\nu > \pi c/a_0$$, or equivalently $$k > \pi/a_0$$. It is also valid only for wavelengths longer than the Compton wavelength, or equivalently $$k < mc/\hbar$$. Therefore, one can choose the upper and lower limit of the integral and these limits make the result converge.

 * $$\langle(\delta\vec{r})^2\rangle_{\rm vac}\cong\frac{1}{2\epsilon_0\pi^2}\left(\frac{e^2}{\hbar c}\right)\left(\frac{\hbar}{mc}\right)^2\ln\frac{4\epsilon_0\hbar c}{e^2}$$.

For the atomic orbital and the Coulomb potential,

 * $$\left\langle\nabla^2\left(\frac{-e^2}{4\pi\epsilon_0r}\right)\right\rangle_{\rm at}=\frac{-e^2}{4\pi\epsilon_0}\int d\vec{r}\psi^*(\vec{r})\nabla^2\left(\frac{1}{r}\right)\psi(\vec{r})=\frac{e^2}{\epsilon_0}|\psi(0)|^2,$$

since it is known that

 * $$\nabla^2\left(\frac{1}{r}\right)=-4\pi\delta(\vec{r}).$$

For p orbitals, the nonrelativistic wave function vanishes at the origin (at the nucleus), so there is no energy shift. But for s orbitals there is some finite value at the origin,

 * $$\psi_{2S}(0)=\frac{1}{(8\pi a_0^3)^{1/2}},$$

where the Bohr radius is

 * $$a_0=\frac{4\pi\epsilon_0\hbar^2}{me^2}.$$

Therefore,

 * $$\left\langle\nabla^2\left(\frac{-e^2}{4\pi\epsilon_0r}\right)\right\rangle_{\rm at}=\frac{e^2}{\epsilon_0}|\psi_{2S}(0)|^2=\frac{e^2}{8\pi\epsilon_0a_0^3}$$.

Finally, the difference of the potential energy becomes:

 * $$\langle\Delta V\rangle=\frac{4}{3}\frac{e^2}{4\pi\epsilon_0}\frac{e^2}{4\pi\epsilon_0\hbar c}\left(\frac{\hbar}{mc}\right)^2\frac{1}{8\pi a_0^3}\ln\frac{4\epsilon_0\hbar c}{e^2} = \alpha^5 mc^2 \frac{1}{6\pi} \ln\frac{1}{\pi\alpha},$$

where $$\alpha$$ is the fine-structure constant. This shift is about 500 MHz, within an order of magnitude of the observed shift of 1057 MHz. This is equal to an energy of only 7.00 x 10^-25 J., or 4.37 x 10^-6 eV.

Welton's heuristic derivation of the Lamb shift is similar to, but distinct from, the calculation of the Darwin term using Zitterbewegung, a contribution to the fine structure that is of lower order in $$\alpha$$ than the Lamb shift.

Lamb–Retherford experiment
In 1947 Willis Lamb and Robert Retherford carried out an experiment using microwave techniques to stimulate radio-frequency transitions between 2S1/2 and 2P1/2 levels of hydrogen. By using lower frequencies than for optical transitions the Doppler broadening could be neglected (Doppler broadening is proportional to the frequency). The energy difference Lamb and Retherford found was a rise of about 1000 MHz (0.03 cm−1) of the 2S1/2 level above the 2P1/2 level.

This particular difference is a one-loop effect of quantum electrodynamics, and can be interpreted as the influence of virtual photons that have been emitted and re-absorbed by the atom. In quantum electrodynamics the electromagnetic field is quantized and, like the harmonic oscillator in quantum mechanics, its lowest state is not zero. Thus, there exist small zero-point oscillations that cause the electron to execute rapid oscillatory motions. The electron is "smeared out" and each radius value is changed from r to r + δr (a small but finite perturbation).

The Coulomb potential is therefore perturbed by a small amount and the degeneracy of the two energy levels is removed. The new potential can be approximated (using atomic units) as follows:

 * $$\langle E_\mathrm{pot} \rangle=-\frac{Ze^2}{4\pi\epsilon_0}\left\langle\frac{1}{r+\delta r}\right\rangle.$$

The Lamb shift itself is given by

 * $$\Delta E_\mathrm{Lamb}=\alpha^5 m_e c^2 \frac{k(n,0)}{4n^3}\ \mathrm{for}\ \ell=0\, $$

with k(n, 0) around 13 varying slightly with n, and

 * $$\Delta E_\mathrm{Lamb}=\alpha^5 m_e c^2 \frac{1}{4n^3}\left[k(n,\ell)\pm \frac{1}{\pi(j+\frac{1}{2})(\ell+\frac{1}{2})}\right]\ \mathrm{for}\ \ell\ne 0\ \mathrm{and}\ j=\ell\pm\frac{1}{2},$$

with log(k(n,$\ell$)) a small number (approx. −0.05) making k(n,ℓ) close to unity.

For a derivation of ΔELamb see for example:

In the hydrogen spectrum
In 1947, Hans Bethe was the first to explain the Lamb shift in the hydrogen spectrum, and he thus laid the foundation for the modern development of quantum electrodynamics. Bethe was able to derive the Lamb shift by implementing the idea of mass renormalization, which allowed him to calculate the observed energy shift as the difference between the shift of a bound electron and the shift of a free electron. The Lamb shift currently provides a measurement of the fine-structure constant α to better than one part in a million, allowing a precision test of quantum electrodynamics.

title: Lamb shift


text:
Süreyya Ayhan

Süreyya Ayhan Kop (born September 6, 1978) is a Turkish former female middle distance track runner who specialised in the 1500 metres. In November 2009, she was banned for life by the Court of Arbitration for Sport (CAS) due to her second anti-doping rule violation.

Ayhan ran for the sports clubs MTA Ankara and Fenerbahçe Istanbul. She then joined the Gaziantep Metropolitan Municipality Sports Club. She holds Turkish records in 800 m (2:00.64) and 1500 m (3:55.33). She became the first Turkish woman ever to reach an Olympic semi-final during her participation in the 2000 Summer Olympics in Sydney, Australia. The next year, she became the first Turkish woman to reach a World Championship final. She was the best European woman athlete running 1500 m in two consecutive years 2002 and 2003.

Early career
Ayhan's father is a former amateur athlete, a local cross-country runner. He was a role model and supporter for Süreyya when she started athletics in the junior high school. In 1992, she started running competitively. She graduated from the Kahramanmaraş Sütçü İmam Üniversitesi in sport and physical education.

Born in Korgun, Çankırı, she established herself as Turkey's top 1500 m runner with a win at the 1999 national championships, setting a Turkish record. The following year she became the country's first female semi-finalist at the Olympic Games. She reached the final of the 2001 World Championships in Athletics (another first for Turkish female athletes), became the Universiade 1500 m champion and won a silver at the 2001 Mediterranean Games.

World-class breakthrough
In 2002, she won Turkey's first gold medal in a European Championships by out-sprinting World and Olympics champion Gabriela Szabo from Romania for the 1500 m title 2 seconds ahead with 3:58.79, leading from the gun to the finish. She improved further in 2003, taking two seconds off her personal best winning a silver medal in the 1500 m final of the 2003 World Championships.

Doping ban
Ayhan, one of Turkey's best hopes for a gold medal at the 2004 Athens Olympics, withdrew from the games due to an injured tendon during a training in Germany.

Allegations that Ayhan may have attempted to cheat on a pre-Olympic doping test surfaced in August after testers reportedly complained of being obstructed from carrying out their work. She was cleared of doping allegations by the IAAF, but she violated rules while taking her test. IAAF ruled that Ayhan had not taken performance-enhancing drugs, but said the athlete had broken testing rules, and she was banned for two years.

She returned to competition but failed another test while she was training in the United States in November 2007, this time after her sample tested positive for the steroids stanozolol and methandienone metabolites. This second offence meant she received a lifetime ban from the sport, but she contested the ruling, taking the case to the Court of Arbitration for Sport (CAS). The Turkish Court of Arbitration for Sport reduced her ban to four years. Ayhan took the banning to the CAS in Lausanne, however the international court reversed the decision in November 2009, upholding the lifetime ban.

Dedications
On May 26, 2003, The Turkish Mint issued a 925 silver commemoration coin worth of 10 US dollars in honor of her.

National titles

 * Turkish Athletics Championships
 * 1500 m: 1999
 * Turkish Clubs Athletics Championships
 * 800 m: 2000

Circuit wins

 * Turkish Stars Indoor Championship 800 m: 1993 (2:18 )
 * Israeli Championships 1500 m: 1996, 1997
 * Memorial Van Damme 1500 m: 2002 (3:57:75 ), 2003 (3:55:33 )
 * ISTAF 1500 m: 2002, 2003
 * Weltklasse Zürich 1500 m: 2003 (3:55:60 )

title: Süreyya Ayhan


text: Category:Iranian judges

title: Category:Iranian judges


text: Category:Prime ministers of Iran

title: Category:Prime ministers of Iran


text:
Camden Society

The Camden Society was a text publication society founded in London in 1838 to publish early historical and literary materials, both unpublished manuscripts and new editions of rare printed books. It was named after the 16th-century antiquary and historian William Camden. In 1897 it merged with the Royal Historical Society, which continues to publish texts in what are now known as the Camden Series.

History
The formation of the Camden Society in 1838 was the initiative of Thomas Wright, John Gough Nichols, and John Bruce. It was modelled on the Surtees Society (founded in 1834 to publish materials relating to the history of northern England), of which Nichols had been the first treasurer. Other founder members included Thomas Amyot, Thomas Crofton Croker, Sir Frederic Madden, John Payne Collier, and Rev. Joseph Hunter. Lord Francis Egerton served as the first President.

For an annual subscription (originally £1 per year), members received the Society's publications, roughly two volumes per year. The original membership was 500. In 1838, 20% of members were clergymen, 9% held legal qualifications, and 36% were Fellows of the Society of Antiquaries of London. Membership reached a peak of 1250 in 1845, and began to decline thereafter. By the 1880s, the Camden Society was experiencing financial problems, brought about in part by a project to create a general index to its first 100 volumes. On 2 May 1897 it was amalgamated with (or effectively absorbed into) the Royal Historical Society.

The Society published some works jointly with the Early English Text Society.

Publications
Between 1838 and 1872 the Society published 105 volumes in its first series (now sometimes known as the Old Series); followed by a further 56 volumes in its second series (known as the New Series), which appeared between 1871 and 1896. Following the merger with the Royal Historical Society, the joint society issued another 6 volumes in the New Series, of which the last, volume 62, appeared in 1901. A fresh series was then begun, published under the Royal Historical Society imprint, but named the Camden Third Series, in which 94 volumes appeared between 1900 and 1963. A further 44 volumes were published between 1969 and 1993 in the Camden Fourth Series; and the Camden Fifth Series, which began publication in 1993, had reached volume 56 by 2019, and remains in progress. The majority of volumes have contained book-length texts (occasionally running into multiple volumes), or thematic groups of shorter texts; but they have also from time to time – both in the two original Camden Society series and in the three RHS Camden Series – been made up of two or more disparate shorter texts, under the series title Camden Miscellany. A total of 36 Miscellany volumes had appeared by 2015.

Volumes are now published for the RHS by Cambridge University Press.

title: Camden Society