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The Wigner–Eckart theorem is a
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
of
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
and
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. It states that
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
elements of spherical tensor operators in the basis of
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
eigenstate In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch–Gordan coefficient. The name derives from physicists
Eugene Wigner Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
and Carl Eckart, who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.Eckart Biography
– The National Academies Press.
Mathematically, the Wigner–Eckart theorem is generally stated in the following way. Given a tensor operator T^ and two states of angular momenta j and j', there exists a constant \langle j \, T^ \, j' \rangle such that for all m, m', and q, the following equation is satisfied: : \langle j \, m , T^_q , j' \, m'\rangle = \langle j' \, m' \, k \, q , j \, m \rangle \langle j \, T^ \, j'\rangle, where *T^_q is the -th component of the spherical tensor operator T^ of rank , * , j m\rangle denotes an eigenstate of total angular momentum and its ''z'' component , * \langle j' m' k q , j m\rangle is the Clebsch–Gordan coefficient for coupling with to get , * \langle j \, T^ \, j' \rangle denotes some value that does not depend on , , nor and is referred to as the reduced matrix element. The Wigner–Eckart theorem states indeed that operating with a spherical tensor operator of rank on an angular momentum eigenstate is like adding a state with angular momentum ''k'' to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch–Gordan coefficient, which arises when considering adding two angular momenta. When stated another way, one can say that the Wigner–Eckart theorem is a theorem that tells how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator. This definition is given in the book ''Quantum Mechanics'' by Cohen–Tannoudji, Diu and Laloe.


Background and overview


Motivating example: position operator matrix elements for 4d → 2p transition

Let's say we want to calculate
transition dipole moment The transition dipole moment or transition moment, usually denoted \mathbf_ for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. In general the t ...
s for an electron transition from a 4d to a 2p orbital of a hydrogen atom, i.e. the matrix elements of the form \langle 2p,m_1 , r_i , 4d,m_2 \rangle, where ''r''''i'' is either the ''x'', ''y'', or ''z'' component of the
position operator In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues ...
, and ''m''1, ''m''2 are the
magnetic quantum number In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The orbital magnetic quantum number ( or ) disting ...
s that distinguish different orbitals within the 2p or 4d subshell. If we do this directly, it involves calculating 45 different integrals: there are 3 possibilities for ''m''1 (−1, 0, 1), 5 possibilities for ''m''2 (−2, −1, 0, 1, 2), and 3 possibilities for ''i'', so the total is 3 × 5 × 3 = 45. The Wigner–Eckart theorem allows one to obtain the same information after evaluating just ''one'' of those 45 integrals (''any'' of them can be used, as long as it is nonzero). Then the other 44 integrals can be inferred from that first one—without the need to write down any wavefunctions or evaluate any integrals—with the help of Clebsch–Gordan coefficients, which can be easily looked up in a table or computed by hand or computer.


Qualitative summary of proof

The Wigner–Eckart theorem works because all 45 of these different calculations are related to each other by rotations. If an electron is in one of the 2p orbitals, rotating the system will generally move it into a ''different'' 2p orbital (usually it will wind up in a
quantum superposition Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödi ...
of all three basis states, ''m'' = +1, 0, −1). Similarly, if an electron is in one of the 4d orbitals, rotating the system will move it into a different 4d orbital. Finally, an analogous statement is true for the position operator: when the system is rotated, the three different components of the position operator are effectively interchanged or mixed. If we start by knowing just one of the 45 values (say, we know that \langle 2p,m_1 , r_i , 4d,m_2 \rangle = K) and then we rotate the system, we can infer that ''K'' is also the matrix element between the rotated version of \langle 2p,m_1 , , the rotated version of r_i, and the rotated version of , 4d,m_2 \rangle. This gives an algebraic relation involving ''K'' and some or all of the 44 unknown matrix elements. Different rotations of the system lead to different algebraic relations, and it turns out that there is enough information to figure out all of the matrix elements in this way. (In practice, when working through this math, we usually apply
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum pro ...
s to the states, rather than rotating the states. But this is fundamentally the same thing, because of the close mathematical relation between rotations and angular momentum operators.)


In terms of representation theory

To state these observations more precisely and to prove them, it helps to invoke the mathematics of
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
. For example, the set of all possible 4d orbitals (i.e., the 5 states ''m'' = −2, −1, 0, 1, 2 and their
quantum superposition Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödi ...
s) form a 5-dimensional abstract
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
. Rotating the system transforms these states into each other, so this is an example of a "group representation", in this case, the 5-dimensional
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
("irrep") of the rotation group SU(2) or SO(3), also called the "spin-2 representation". Similarly, the 2p quantum states form a 3-dimensional irrep (called "spin-1"), and the components of the position operator also form the 3-dimensional "spin-1" irrep. Now consider the matrix elements \langle 2p,m_1 , r_i , 4d,m_2 \rangle. It turns out that these are transformed by rotations according to the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of those three representations, i.e. the spin-1 representation of the 2p orbitals, the spin-1 representation of the components of r, and the spin-2 representation of the 4d orbitals. This direct product, a 45-dimensional representation of SU(2), is ''not'' an
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
, instead it is the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of a spin-4 representation, two spin-3 representations, three spin-2 representations, two spin-1 representations, and a spin-0 (i.e. trivial) representation. The nonzero matrix elements can only come from the spin-0 subspace. The Wigner–Eckart theorem works because the direct product decomposition contains one and only one spin-0 subspace, which implies that all the matrix elements are determined by a single scale factor. Apart from the overall scale factor, calculating the matrix element \langle 2p,m_1 , r_i , 4d,m_2 \rangle is equivalent to calculating the
projection Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...
of the corresponding abstract vector (in 45-dimensional space) onto the spin-0 subspace. The results of this calculation are the Clebsch–Gordan coefficients. The key qualitative aspect of the Clebsch–Gordan decomposition that makes the argument work is that in the decomposition of the tensor product of two irreducible representations, each irreducible representation occurs only once. This allows
Schur's lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a gro ...
to be used.


Proof

Starting with the definition of a spherical tensor operator, we have : _\pm, T^_q= \hbar \sqrtT_^, which we use to then calculate : \begin &\langle j \, m , _\pm, T^_q, j' \, m' \rangle = \hbar \sqrt \, \langle j \, m , T^_ , j' \, m' \rangle. \end If we expand the commutator on the LHS by calculating the action of the on the bra and ket, then we get : \begin \langle j \, m , _\pm, T^_q, j' \, m' \rangle = &\hbar\sqrt \, \langle j \, (m \mp 1) , T^_q , j' \, m' \rangle \\ &-\hbar\sqrt \, \langle j \, m , T^_q , j' \, (m' \pm 1) \rangle. \end We may combine these two results to get : \begin \sqrt \langle j \, (m \mp 1) , T^_q , j' \, m' \rangle = &\sqrt \, \langle j \, m , T^_q , j' \, (m' \pm 1) \rangle \\ &+\sqrt \, \langle j \, m , T^_ , j' \, m' \rangle. \end This recursion relation for the matrix elements closely resembles that of the Clebsch–Gordan coefficient. In fact, both are of the form . We therefore have two sets of linear homogeneous equations: : \begin \sum_c a_ x_c &= 0, & \sum_c a_ y_c &= 0. \end one for the Clebsch–Gordan coefficients () and one for the matrix elements (). It is not possible to exactly solve for . We can only say that the ratios are equal, that is :\frac = \frac or that , where the coefficient of proportionality is independent of the indices. Hence, by comparing recursion relations, we can identify the Clebsch–Gordan coefficient with the matrix element , then we may write : \langle j' \, m' , T^_ , j \, m\rangle \propto \langle j \, m \, k \, (q \pm 1) , j' \, m' \rangle.


Alternative conventions

There are different conventions for the reduced matrix elements. One convention, used by Racah and Wigner, includes an additional phase and normalization factor, : \langle j \, m , T^_q , j' \, m'\rangle = \frac = (-1)^ \begin j & k & j' \\ -m & q & m' \end \langle j \, T^ \, j'\rangle_. where the array denotes the 3-j symbol. (Since in practice is often an integer, the factor is sometimes omitted in literature.) With this choice of normalization, the reduced matrix element satisfies the relation: :\langle j \, T^ \, j'\rangle_ = (-1)^ \langle j' \, T^ \, j\rangle_^*, where the
Hermitian adjoint In mathematics, specifically in operator theory, each linear operator A on an inner product space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \l ...
is defined with the convention. Although this relation is not affected by the presence or absence of the phase factor in the definition of the reduced matrix element, it is affected by the phase convention for the Hermitian adjoint. Another convention for reduced matrix elements is that of Sakurai's '' Modern Quantum Mechanics'': : \langle j \, m , T^_q , j' \, m'\rangle = \frac.


Example

Consider the position expectation value . This matrix element is the expectation value of a Cartesian operator in a spherically symmetric hydrogen-atom-eigenstate basis, which is a nontrivial problem. However, the Wigner–Eckart theorem simplifies the problem. (In fact, we could obtain the solution quickly using parity, although a slightly longer route will be taken.) We know that is one component of , which is a vector. Since vectors are rank-1 spherical tensor operators, it follows that must be some linear combination of a rank-1 spherical tensor with . In fact, it can be shown that :x = \frac, where we define the spherical tensors asJ. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley). :T^_ = \sqrt r Y_1^q and are
spherical harmonic In mathematics and Outline of physical science, physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The tabl ...
s, which themselves are also spherical tensors of rank . Additionally, , and :T^_ = \mp \frac. Therefore, : \begin \langle n \, j \, m , x , n' \, j' \, m' \rangle & = \left\langle n \, j \, m \left, \frac \ n' \, j' \, m' \right\rangle \\ & = \frac \langle n \, j \, T^ \, n' \, j'\rangle \, \big(\langle j' \, m' \, 1 \, (-1) , j \, m \rangle - \langle j' \, m' \, 1 \, 1 , j \, m \rangle\big). \end The above expression gives us the matrix element for in the basis. To find the expectation value, we set , , and . The selection rule for and is for the spherical tensors. As we have , this makes the Clebsch–Gordan Coefficients zero, leading to the expectation value to be equal to zero.


See also

* Tensor operator *
Landé g-factor In physics, the Landé ''g''-factor is a particular example of a ''g''-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921. In atomic physics, the Landé '' ...


References


General

*


External links

*J. J. Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, . *
Wigner–Eckart theorem
{{DEFAULTSORT:Wigner-Eckart theorem Representation theory of Lie groups Theorems in quantum mechanics Theorems in representation theory