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In pure and
applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
,
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 ...
and
computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
, a tensor operator generalizes the notion of
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
s which are
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
s and
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
s. A special class of these are spherical tensor operators which apply the notion of the
spherical basis In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular ...
and
spherical harmonics In mathematics and 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 table of spherical harmonics co ...
. The spherical basis closely relates to the description 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 ...
in quantum mechanics and spherical harmonic functions. The
coordinate-free A coordinate-free, or component-free, treatment of a scientific theory A scientific theory is an explanation of an aspect of the universe, natural world that can be or that has been reproducibility, repeatedly tested and has corroborating evid ...
generalization of a tensor operator is known as a representation operator.


The general notion of scalar, vector, and tensor operators

In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass M, traveling with a definite center of mass momentum, p , in the z direction. If we rotate the system by 90^ about the y axis, the momentum will change to p , which is in the x direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at p^2/2M. The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are p and p . The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states. In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. Other examples of scalar operators are the total energy operator (more commonly called the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, , and the spin angular momentum, . (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product \cdot of the two vector operators, and , is a scalar operator, which figures prominently in discussions of the
spin–orbit interaction In quantum mechanics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin– ...
. Similarly, the quadrupole moment tensor of our example molecule has the nine components Q_ = \sum_ q_ \left(3 r_r_ - r_^ \delta_\right).Here, the indices i and j can independently take on the values 1, 2, and 3 (or x, y, and z) corresponding to the three Cartesian axes, the index \alpha runs over all particles (electrons and nuclei) in the molecule, q_ is the charge on particle \alpha, and r_ is the i-th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products r_r_ together form a second rank tensor, formed by taking the outer product of the vector operator _ with itself.


Rotations of quantum states


Quantum rotation operator

The rotation operator about the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
n (defining the axis of rotation) through angle ''θ'' is U (\theta, \hat)= \exp\left(-\frac\hat\cdot\mathbf\right) where are the rotation generators (also the angular momentum matrices): J_x = \frac\begin 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end\,\quad J_y = \frac\begin 0 & i & 0 \\ -i & 0 & i \\ 0 & -i & 0 \end\,\quad J_z = \hbar\begin -1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end and let \widehat = \widehat(\theta,\hat) be a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \t ...
. According to the
Rodrigues' rotation formula In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transfo ...
, the rotation operator then amounts to U (\theta, \hat)= 1\!\!1 - \frac\hat\cdot\mathbf -\frac ( \hat\cdot\mathbf )^2. An operator \widehat is invariant under a unitary transformation ''U'' if \widehat = ^\dagger \widehat U ; in this case for the rotation \widehat(R), \widehat = ^\dagger \widehat U(R) = \exp\left(\frac\hat\cdot\mathbf\right) \widehat \exp\left(-\frac\hat\cdot\mathbf\right) .


Angular momentum eigenkets

The orthonormal basis set for total angular momentum is , j,m\rangle , where ''j'' is the total angular momentum quantum number and ''m'' is the magnetic angular momentum quantum number, which takes values −''j'', −''j'' + 1, ..., ''j'' − 1, ''j''. A general state within the ''j'' subspace , \psi \rangle = \sum_m c_, j,m\rangle rotates to a new state by: , \bar \rangle = U(R), \psi \rangle = \sum_m c_ U(R), j,m\rangle Using the completeness condition: I = \sum_ , j , m' \rangle \langle j, m' , we have , \bar \rangle = I U(R), \psi \rangle = \sum_ c_ , j , m' \rangle \langle j, m' , U(R) , j , m \rangle Introducing the Wigner D matrix elements: ^_ = \langle j, m' , U(R) , j,m \rangle gives the matrix multiplication: , \bar \rangle = \sum_ c_ D^_ , j , m' \rangle \quad \Rightarrow \quad , \bar \rangle = D^ , \psi \rangle For one basis ket: , \overline \rangle = \sum_ ^_ , j , m' \rangle For the case of orbital angular momentum, the eigenstates , \ell,m\rangle of the orbital
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 ...
L and solutions of
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delt ...
on a 3d sphere 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: Y_\ell^m( \theta , \phi ) = \langle \theta,\phi , \ell , m \rangle = \sqrt \, P_\ell^m ( \cos ) \, e^ where ''P''''m'' is an
associated Legendre polynomial In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalen ...
, is the orbital angular momentum quantum number, and ''m'' is the orbital magnetic
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
which takes the values −, − + 1, ... − 1, The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below. Spherical harmonics are functions of the polar and azimuthal angles, ''ϕ'' and ''θ'' respectively, which can be conveniently collected into a unit vector n(''θ'', ''ϕ'') pointing in the direction of those angles, in the Cartesian basis it is: \hat(\theta,\phi) = \cos\phi \sin\theta \mathbf_x + \sin\phi \sin\theta \mathbf_y + \cos\theta \mathbf_z So a spherical harmonic can also be written Y_^=\langle \mathbf, \ell m\rangle . Spherical harmonic states , m,\ell\rangle rotate according to the inverse rotation matrix U(R^), while , \ell,m\rangle rotates by the initial rotation matrix \widehat(R). , \overline \rangle = \sum_ D_^ (R^), \ell , m' \rangle\,,\quad , \overline \rangle = U(R) , \hat\rangle


Rotation of tensor operators

We define the Rotation of an operator by requiring that the expectation value of the original operator \widehat with respect to the initial state be equal to the expectation value of the rotated operator with respect to the rotated state, \langle \psi' , \widehat , \psi' \rangle = \langle \psi , \widehat , \psi \rangle Now as, , \psi \rangle ~\rightarrow~ , \psi' \rangle = U(R) , \psi \rangle \,, \quad \langle \psi , ~\rightarrow~ \langle \psi' , = \langle \psi , U^\dagger (R) we have, \langle \psi , U^\dagger (R) \widehat' U(R), \psi \rangle = \langle \psi , \widehat , \psi \rangle since, , \psi \rangle is arbitrary, U^\dagger (R) \widehat' U(R) = \widehat


Scalar operators

A scalar operator is invariant under rotations: U(R)^\dagger \widehat U(R) = \widehat This is equivalent to saying a scalar operator commutes with the rotation generators: \left \widehat , \widehat \right= 0 Examples of scalar operators include * the
energy operator In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry. Definition It is given by: \hat = i\hbar\frac It acts on the wave function (the ...
: \widehat \psi = i\hbar\frac \psi *
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
''V'' (in the case of a central potential only) \widehat(r,t) \psi(\mathbf,t) = V(r,t) \psi(\mathbf,t) *
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
''T'':\widehat\psi(\mathbf,t) = -\frac (\nabla^2 \psi)(\mathbf,t) * the spin–orbit coupling: \widehat\cdot\widehat = \widehat_x \widehat_x + \widehat_y \widehat_y + \widehat_z \widehat_z \,.


Vector operators

Vector operators (as well as
pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ' ...
operators) are a set of 3 operators that can be rotated according to: ^\dagger \widehat_i U(R) = \sum_j R_ \widehat_j Any observable vector quantity of a quantum mechanical system should be invariant of the choice of frame of reference. The transformation of expectation value vector which applies for any wavefunction, ensures the above equality. In Dirac notation:\langle\bar, \widehat_a, \bar\rangle = \langle \psi , ^\dagger \widehat_a U(R) , \psi \rangle = \sum_b R_ \langle \psi , \widehat_b , \psi \rangle where the RHS is due to the rotation transformation acting on the vector formed by expectation values. Since is any quantum state, the same result follows:^\dagger \widehat_a U(R) = \sum_b R_ \widehat_b Note that here, the term "vector" is used two different ways: kets such as are elements of abstract Hilbert spaces, while the vector operator is defined as a quantity whose components transform in a certain way under rotations. From the above relation for infinitesimal rotations and the Baker Hausdorff lemma, by equating coefficients of order \delta\theta, one can derive the commutation relation with the rotation generator: where ''εijk'' is the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers , for some ...
, which all vector operators must satisfy, by construction. The above commutator rule can also be used as an alternative definition for vector operators which can be shown by using the Baker Hausdorff lemma. As the symbol ''εijk'' is a
pseudotensor In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordin ...
, pseudovector operators are invariant
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
a sign: +1 for
proper rotation In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicu ...
s and −1 for
improper rotation In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...
s. Since operators can be shown to form a vector operator by their commutation relation with angular momentum components (which are generators of rotation), its examples include: * 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 ...
: \widehat \psi = \mathbf \psi * the
momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...
: \widehat \psi = -i\hbar \nabla \psi and peusodovector operators include * the orbital
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 ...
: \widehat \psi = -i\hbar \mathbf \times \nabla \psi * as well the
spin operator Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativisti ...
S, and hence the total angular momentum \widehat = \widehat+\widehat\,.


Scalar operators from vector operators

If \vec and \vec are two vector operators, the dot product between the two vector operators can be defined as: \vec \cdot \vec = \sum_^3 \hat \hat Under rotation of coordinates, the newly defined operator transforms as: ^\dagger (\vec\cdot \vec) U(R) = ^\dagger\left(\sum_^3 \hat \hat\right) U(R)= \sum_^3(^\dagger \hat_i U(R))(^\dagger \hat_i U(R))=\sum_^3 \left(\sum_^3 R_ \widehat_j \cdot \sum_^3 R_ \widehat_k \right) Rearranging terms and using transpose of rotation matrix as its inverse property: ^\dagger (\vec\cdot \vec) U(R) =\sum_^3 \sum_^3 \left( \sum_^3 R_^T R_ \right) \widehat_j \widehat_k = \sum_^3 \sum_^3 \delta_ \widehat_j \widehat_k = \sum_^3 \widehat_i \widehat_i Where the RHS is the \vec \cdot \vec operator originally defined. Since the dot product defined is invariant under rotation transformation, it is said to be a scalar operator.


Spherical vector operators

A vector operator in the
spherical basis In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular ...
is where the components are:V_=-\frac(V_x + i V_y)\,\quad V_=\frac(V_x - i V_y)\,,\quad V_0 = V_z \,,
using J_\pm = J_x \pm i J_y \,, the various commutators with the rotation generators and ladder operators are:\begin \left _z, V_\right&= +\hbar V_ \\ ex\left _z, V_\right&= 0 V_ \\ ex\left _z, V_\right&= -\hbar V_ \\ ex\left _+, V_\right&= 0 \\ ex\left _+, V_\right&= \sqrt\hbar V_ \\ ex\left _+, V_\right&= \sqrt\hbar V_ \\ ex\left _-, V_\right&= \sqrt\hbar V_ \\ ex\left _-, V_\right&= \sqrt\hbar V_ \\ ex\left _-, V_\right&= 0 \\ ex\end which are of similar form of\begin J_z , 1,+1 \rangle &= +\hbar , 1,+1 \rangle \\ exJ_z, 1,0 \rangle &= 0 , 1,0 \rangle \\ exJ_z, 1,-1 \rangle &= -\hbar , 1,-1 \rangle\\ exJ_+, 1,+1 \rangle &= 0\\ exJ_+, 1,0 \rangle &= \sqrt\hbar , 1,+1 \rangle \\ exJ_+, 1, -1\rangle &= \sqrt\hbar , 1,0 \rangle\\ exJ_-, 1,+1 \rangle &= \sqrt\hbar , 1,0 \rangle\\ exJ_-, 1,0 \rangle &= \sqrt\hbar , 1,-1 \rangle \\ exJ_-, 1,-1 \rangle &= 0 \\ ex\end In the spherical basis, the generators of rotation are:J_ = \mp \frac J_\pm \,,\quad J_0 = J_z From the transformation of operators and Baker Hausdorff lemma: ^\dagger \widehat_q U(R) = \widehat_q + i\frac\left \hat\cdot \vec, \widehat_q\right\sum_^\infty \frac\widehat_q = exp\left(\right) \widehat_q compared to U(R) , j,k \rangle = , j,k \rangle - i\frac\hat\cdot \vec, j,k \rangle+\sum_^\infty \frac, j,k \rangle= exp\left(\right), j,k \rangle it can be argued that the commutator with operator replaces the action of operator on state for transformations of operators as compared with that of states: U(R) , j,k \rangle = \exp\left(\right), j,k \rangle = \sum_ , j',k' \rangle \langle j',k' , \exp\left(\right), j,k \rangle = \sum_ D_^(R), j,k'\rangle The rotation transformation in the spherical basis (originally written in the Cartesian basis) is then, due to similarity of commutation and operator shown above: ^\dagger \widehat_q U(R) = \sum_ \widehat_ One can generalize the ''vector'' operator concept easily to ''tensorial operators'', shown next.


Tensor operators

In general, a tensor operator is one that transforms according to a tensor: U(R)^\dagger \widehat_^ U(R) = R_R_R_\cdots \widehat_^ R_^R_^R_^ \cdots where the basis are transformed by R^ or the vector components transform by R . In the subsequent discussion surrounding tensor operators, the index notation regarding covariant/contravariant behavior is ignored entirely. Instead, contravariant components is implied by context. Hence for an n times contravariant tensor: U(R)^\dagger \widehat_ U(R) = R_R_R_\cdots \widehat_


Examples of tensor operators

* The
Quadrupole A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure re ...
moment operator, Q_ = \sum_q_ (3 r_r_ - r_^ \delta_) * Components of two tensor vector operators can be multiplied to give another Tensor operator. T_ = V_ W_ In general, n number of tensor operators will also give another tensor operator T_ = V_^ V_^ V_^\cdots V_^ or, T_ = V_ W_ ''Note:'' In general, a tensor operator cannot be written as the tensor product of other tensor operators as given in the above example.


Tensor operator from vector operators

If \vec and \vec are two three dimensional vector operators, then a rank 2 Cartesian dyadic tensors can be formed from nine operators of form \hat_ = \hat \hat, ^\dagger \hat_ U(R) = ^\dagger(\hat \hat) U(R)= (^\dagger \hat_i U(R))(^\dagger \hat_j U(R))= \left(\sum_^3 R_ \hat_l \cdot \sum_^3 R_ \hat_k \right) Rearranging terms, we get: ^\dagger \hat_ U(R) = \sum_^3 \sum_^3 \left( R_ R_ \hat_ \right) The RHS of the equation is change of basis equation for twice contravariant tensors where the basis are transformed by R^ or the vector components transform by R which matches transformation of vector operator components. Hence the operator tensor described forms a rank 2 tensor, in tensor representation,\hat = \vec \otimes \vec = (\hat_i \hat_j) (\mathbf_i \otimes \mathbf_j) Similarly, an n-times contravariant tensor operator can be formed similarly by n vector operators. We observe that the subspace spanned by linear combinations of the rank two tensor components form an invariant subspace, ie. the subspace does not change under rotation since the transformed components itself is a linear combination of the tensor components. However, this subspace is not irreducible ie. it can be further divided into invariant subspaces under rotation. Otherwise, the subspace is called reducible. In other words, there exists specific sets of different linear combinations of the components such that they transforms into a linear combination of the same set under rotation. In the above example, we will show that the 9 independent tensor components can be divided into a set of 1, 3 and 5 combination of operators that each form irreducible invariant subspaces.


Irreducible tensor operators

The subspace spanned by \ can be divided two subspaces; three independent antisymmetric components \ and six independent symmetric component \ , defined as \hat_ = \frac (\hat_-\hat_) and \hat_ = \frac (\hat_+\hat_) . Using the \ transformation under rotation formula, it can be shown that both \ and \ are transformed into a linear combination of members of its own sets. Although \ is irreducible, the same cannot be said about \ . The six independent symmetric component set can be divided into five independent traceless symmetric component and the invariant trace can be its own subspace. Hence, the invariant subspaces of \ are formed respectively by: # One invariant trace of the tensor, \hat = \sum_^3\hat_ # Three linearly independent antisymmetric components from: \hat_ = \frac (\hat_-\hat_) # Five linearly independent traceless symmetric components from \hat_ = \frac (\hat_+\hat_) - \frac\hat\delta_ If \hat_ = \hat \hat, the invariant subspaces of \ formed are represented by: # One invariant scalar operator \vec \cdot \vec # Three linearly independent components from \frac (\hat_\hat_-\hat_\hat_) # Five linearly independent components from \frac (\hat_\hat_+\hat_\hat_) - \frac(\vec \cdot \vec)\delta_ From the above examples, the nine component \ are split into subspaces formed by one, three and five components. These numbers add up to the number of components of the original tensor in a manner similar to the dimension of vector subspaces adding to the dimension of the space that is a direct sum of these subspaces. Similarly, every element of \ can be expressed in terms of a linear combination of components from its invariant subspaces: \hat_ = \frac\hat\delta_+\hat_+\hat_ or \hat_ = \frac(\vec \cdot \vec)\delta_+\left( \frac (\hat_\hat_-\hat_\hat_) \right) +\left( \frac (\hat_\hat_+\hat_\hat_) - \frac(\vec \cdot \vec)\delta_ \right) = \mathbf^ + \mathbf^ + \mathbf^ where:\widehat^_ = \frac\delta_ \widehat^_ = \frac \left widehat_i \widehat_j - \widehat_j \widehat_i\right= \widehat_ \widehat_ \widehat^_ = \tfrac \left(\widehat_i \widehat_j + \widehat_j \widehat_i\right) - \tfrac \widehat_k \widehat_k \delta_ = \widehat_ \widehat_ - T^_ In general cartesian tensors of rank greater than 1 are reducible. In quantum mechanics, this particular example bears resemblance to the addition of two spin one particles where both are 3 dimensional, hence the total space being 9 dimensional, can be formed by spin 0, spin 1 and spin 2 systems each having 1 dimensional, 3 dimensional and 5 dimensional space respectively. These three terms are irreducible, which means they cannot be decomposed further and still be tensors satisfying the defining transformation laws under which they must be invariant. Each of the irreducible representations T(0), T(1), T(2) ... transform like angular momentum eigenstates according to the number of independent components. It is possible that a given tensor may have one or more of these components vanish. For example, the quadrupole moment tensor is already symmetric and traceless, and hence has only 5 independent components to begin with.


Spherical tensor operators

Spherical tensor operators are generally defined as operators with the following transformation rule, under rotation of coordinate system: \widehat_m^ \rightarrow U(R)^\dagger \widehat_m^ U(R) =\sum_ D_^(R^)\widehat_^ The commutation relations can be found by expanding LHS and RHS as: U(R)^\dagger \widehat_m^ U(R) = \left(1+\frac +\mathcal(\epsilon ^2)\right)\widehat_m^\left(1-\frac + \mathcal(\epsilon ^2)\right) =\sum_ \langle j,m' , \left( 1+\frac +\mathcal(\epsilon ^2) \right), j,m\rangle \widehat_^ Simplifying and applying limits to select only first order terms, we get: ,\widehat_m^] = \sum_ \widehat_^ \langle j, m', \vec\cdot\hat, j,m\rangle For choices of \hat=\hat\pm i \hat or \hat=\hat, we get:\begin \left J_\pm , \widehat^_ \right&= \hbar \sqrt \widehat^_ \\ ex\left J_z , \widehat^_ \right&= \hbar m \widehat^_ \end Note the similarity of the above to:\begin J_\pm , j,m \rangle &= \hbar \sqrt , j, m \pm 1 \rangle \\ exJ_z, j,m \rangle &= \hbar m , j,m \rangle \end Since J_x and J_y are linear combinations of J_\pm, they share the same similarity due to linearity. If, only the commutation relations hold, using the following relation, , j,m \rangle \rightarrow U(R) , j,m \rangle = exp\left(-\right), j,m \rangle=\sum_ D_^(R), j,m' \rangle we find due to similarity of actions of J on wavefunction , j,m \rangle and the commutation relations on \widehat_m^, that: \widehat_m^ \rightarrow U(R)^\dagger \widehat_m^ U(R) = exp\left(\right)\widehat_m^ =\sum_ D_^(R^)\widehat_^ where the exponential form is given by Baker–Hausdorff lemma. Hence, the above commutation relations and the transformation property are equivalent definitions of spherical tensor operators. It can also be shown that \ transform like a vector due to their commutation relation. In the following section, construction of spherical tensors will be discussed. For example, since example of spherical vector operators is shown, it can be used to construct higher order spherical tensor operators. In general, spherical tensor operators can be constructed from two perspectives. One way is to specify how spherical tensors transform under a physical rotation - a group theoretical definition. A rotated angular momentum eigenstate can be decomposed into a linear combination of the initial eigenstates: the coefficients in the linear combination consist of Wigner rotation matrix entries. Or by continuing the previous example of the second order dyadic tensor T = a ⊗ b, casting each of a and b into the spherical basis and substituting into T gives the spherical tensor operators of the second order.


Construction using Clebsch–Gordan coefficients

Combination of two spherical tensors A_^ and B_^in the following manner involving the
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 m ...
can be proved to give another spherical tensor of the form:T_^ = \sum_ \langle k_1 , k_2 ; q_1 , q_2 , k_1 , k_2; k , q \rangle A_^ B_^ This equation can be used to construct higher order spherical tensor operators, for example, second order spherical tensor operators using two first order spherical tensor operators, say A and B, discussed previously: \begin \widehat^_ &= \widehat_ \widehat_ \\ ex\widehat^_ &= \tfrac\left( \widehat_ \widehat_0 + \widehat_0 \widehat_ \right) \\ ex\widehat^_ &= \tfrac\left( \widehat_ \widehat_ + \widehat_ \widehat_ + 2 \widehat_0 \widehat_0 \right) \end Using the infinitesimal rotation operator and its Hermitian conjugate, one can derive the commutation relation in the spherical basis:\left _a, \widehat^_ \right= \sum_ ^_ \widehat_^ = \sum_ \langle j 2, m q , J_a , j 2, m q' \rangle \widehat_^ and the finite rotation transformation in the spherical basis can be verified:^\dagger \widehat^_q U(R) = \sum_ ^* \widehat_^


Using Spherical Harmonics

Define an operator by its spectrum: \Upsilon^m_l, r\rangle= r^l Y^m_l(\theta,\phi), r\rangle=\Upsilon^m_l(\vec), r\rangleSince for spherical harmonics under rotation: Y_^ (\mathbf)= \langle \mathbf, k,q\rangle \rightarrow U(R)^\dagger Y_^ (\mathbf) U(R) = Y_^ (R\mathbf) = \langle \mathbf, D(R)^\dagger, k,q\rangle = \sum_ D^_(R^)Y_^ (\mathbf)It can also been shown that: \Upsilon^m_l(\vec) \rightarrow U(R)^\dagger \Upsilon^m_l(\vec) U(R) = \sum_ D^_(R^)\Upsilon_^ (\vec) Then \Upsilon^m_l(\vec), where \vec is a vector operator, also transforms in the same manner ie, is a spherical tensor operator. The process involves expressing \Upsilon^m_l(\vec)=r^l Y^m_l(\theta,\phi)=\Upsilon^m_l(x,y,z) in terms of x, y and z and replacing x, y and z with operators Vx Vy and Vz which from vector operator. The resultant operator is hence a spherical tensor operator \hat_m^. This may include constant due to normalization from spherical harmonics which is meaningless in context of operators. 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 ...
of a spherical tensor may be defined as(T^\dagger)_^ = (-1)^ (T^_)^\dagger.There is some arbitrariness in the choice of the phase factor: any factor containing will satisfy the commutation relations. The above choice of phase has the advantages of being real and that the tensor product of two commuting
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...
operators is still Hermitian. Some authors define it with a different sign on , without the , or use only the
floor A floor is the bottom surface of a room or vehicle. Floors vary from wikt:hovel, simple dirt in a cave to many layered surfaces made with modern technology. Floors may be stone, wood, bamboo, metal or any other material that can support the ex ...
of .


Angular momentum and spherical harmonics


Orbital angular momentum and spherical harmonics

Orbital angular momentum operators have the
ladder operator In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
s: L_\pm = L_x \pm i L_y which raise or lower the orbital magnetic quantum number ''m'' by one unit. This has almost exactly the same form as the spherical basis, aside from constant multiplicative factors.


Spherical tensor operators and quantum spin

Spherical tensors can also be formed from algebraic combinations of the spin operators ''Sx'', ''Sy'', ''Sz'', as matrices, for a spin system with total quantum number ''j'' = + ''s'' (and = 0). Spin operators have the ladder operators: S_\pm = S_x \pm i S_y which raise or lower the spin magnetic quantum number ''ms'' by one unit.


Applications

Spherical bases have broad applications in pure and applied mathematics and physical sciences where spherical geometries occur.


Dipole radiative transitions in a single-electron atom (alkali)

The transition amplitude is proportional to matrix elements of the dipole operator between the initial and final states. We use an electrostatic, spinless model for the atom and we consider the transition from the initial energy level Enℓ to final level En′ℓ′. These levels are degenerate, since the energy does not depend on the magnetic quantum number m or m′. The wave functions have the form, \psi_(r,\theta,\phi) = R_(r) Y_(\theta,\phi) The dipole operator is proportional to the position operator of the electron, so we must evaluate matrix elements of the form, \langle n'\ell'm', \mathbf , n\ell m \rangle where, the initial state is on the right and the final one on the left. The position operator r has three components, and the initial and final levels consist of 2ℓ + 1 and 2ℓ′ + 1 degenerate states, respectively. Therefore if we wish to evaluate the intensity of a spectral line as it would be observed, we really have to evaluate 3(2ℓ′+ 1)(2ℓ+ 1) matrix elements, for example, 3×3×5 = 45 in a 3d → 2p transition. This is actually an exaggeration, as we shall see, because many of the matrix elements vanish, but there are still many non-vanishing matrix elements to be calculated. A great simplification can be achieved by expressing the components of r, not with respect to the Cartesian basis, but with respect to the spherical basis. First we define, r_ = \hat_\cdot \mathbf Next, by inspecting a table of the ''Yℓm''′s, we find that for ℓ = 1 we have, \begin r Y_(\theta,\phi) &=&&-r \sqrt\sin(\theta) e^ &=& \sqrt\left(-\frac\right) \\ r Y_(\theta,\phi) &=&& r \sqrt\cos(\theta) &=& \sqrtz \\ r Y_(\theta,\phi) &=&& r \sqrt\sin(\theta) e^ &=& \sqrt\left(\frac\right) \end where, we have multiplied each ''Y''1''m'' by the radius ''r''. On the right hand side we see the spherical components ''rq'' of the position vector r. The results can be summarized by, r Y_(\theta,\phi) = \sqrt r_ for q = 1, 0, −1, where q appears explicitly as a magnetic quantum number. This equation reveals a relationship between vector operators and the angular momentum value ℓ = 1, something we will have more to say about presently. Now the matrix elements become a product of a radial integral times an angular integral, \langle n'\ell'm', r_, n\ell m \rangle = \left(\int_0^ r^2 dr R_^* (r) r R_(r)\right) \left(\sqrt\int \sind\Omega Y_^*(\theta,\phi) Y_(\theta,\phi) Y_(\theta,\phi)\right) We see that all the dependence on the three magnetic quantum numbers (m′,q,m) is contained in the angular part of the integral. Moreover, the angular integral can be evaluated by the three-''Yℓm'' formula, whereupon it becomes proportional to the Clebsch-Gordan coefficient, \langle \ell'm', \ell1mq\rangle The radial integral is independent of the three magnetic quantum numbers (''m''′, ''q'', ''m''), and the trick we have just used does not help us to evaluate it. But it is only one integral, and after it has been done, all the other integrals can be evaluated just by computing or looking up Clebsch–Gordan coefficients. The selection rule ''m''′ = ''q'' + ''m'' in the Clebsch–Gordan coefficient means that many of the integrals vanish, so we have exaggerated the total number of integrals that need to be done. But had we worked with the Cartesian components ''r''i of r, this selection rule might not have been obvious. In any case, even with the selection rule, there may still be many nonzero integrals to be done (nine, in the case 3d → 2p). The example we have just given of simplifying the calculation of matrix elements for a dipole transition is really an application of the Wigner–Eckart theorem, which we take up later in these notes.


Magnetic resonance

The spherical tensor formalism provides a common platform for treating coherence and relaxation in
nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are disturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
. In
NMR Nuclear magnetic resonance (NMR) is a physical phenomenon in which atomic nucleus, nuclei in a strong constant magnetic field are disturbed by a weak oscillating magnetic field (in the near and far field, near field) and respond by producing ...
and EPR, spherical tensor operators are employed to express the quantum dynamics of
particle spin Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic qu ...
, by means of an equation of motion for the
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...
entries, or to formulate dynamics in terms of an equation of motion in Liouville space. The Liouville space equation of motion governs the observable averages of spin variables. When relaxation is formulated using a spherical tensor basis in Liouville space, insight is gained because the relaxation matrix exhibits the cross-relaxation of spin observables directly.


Image processing and computer graphics


See also

*
Wigner–Eckart theorem The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be expressed as the product of two factors, one ...
*
Structure tensor In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix (mathematics), matrix derived from the gradient of a function (mathematics), function. It describes the distribution of the gradient in a specified ne ...
* Clebsch–Gordan coefficients for SU(3)


References


Notes


Sources

* * * * * * * * *


Further reading


Spherical harmonics

* * * * * *


Angular momentum and spin

* *


Condensed matter physics

* * * * * *


Magnetic resonance

* * *


Image processing

* * * *


External links


(2012) ''Clebsch-Gordon (sic) coefficients and the tensor spherical harmonics''''The tensor spherical harmonics''(2010) ''Irreducible Tensor Operators and the Wigner-Eckart Theorem''

''Tensor_Operators''(2009) ''Tensor Operators and the Wigner Eckart Theorem''''The Wigner-Eckart theorem''(2004) ''Rotational Transformations and Spherical Tensor Operators''''Tensor operators''''Evaluation of the matrix elements for radiative transitions''D.K. Ghosh, (2013) ''Angular Momentum - III : Wigner- Eckart Theorem''B. Baragiola (2002) ''Tensor Operators''
{{Tensors Image processing Quantum mechanics Condensed matter physics Linear algebra Tensors Spherical geometry