Spherical basis




"Spherical tensor" redirects to here. For the concept related to operators see tensor operator.

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 momentum in quantum mechanics and spherical harmonic functions.


While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers.




Contents






  • 1 Spherical basis in three dimensions


    • 1.1 Basis definition


    • 1.2 Commutator definition


    • 1.3 Rotation definition


    • 1.4 Coordinate vectors




  • 2 Properties (three dimensions)


    • 2.1 Orthonormality


    • 2.2 Change of basis matrix


    • 2.3 Cross products


    • 2.4 Inner product in the spherical basis




  • 3 See also


  • 4 References


    • 4.1 General




  • 5 External links





Spherical basis in three dimensions


A vector A in 3D Euclidean space ℝ3 can be expressed in the familiar Cartesian coordinate system in the standard basis ex, ey, ez, and coordinates Ax, Ay, Az:








A=Axex+Ayey+Azez{displaystyle mathbf {A} =A_{x}mathbf {e} _{x}+A_{y}mathbf {e} _{y}+A_{z}mathbf {e} _{z}}{mathbf {A}}=A_{x}{mathbf {e}}_{x}+A_{y}{mathbf {e}}_{y}+A_{z}{mathbf {e}}_{z}









 

 

 

 

(1)


or any other coordinate system with associated basis set of vectors. From this extend the scalars to allow multiplication by complex numbers, so that we are now working in C3{displaystyle mathbb {C} ^{3}}mathbb {C} ^{3} rather than R3{displaystyle mathbb {R} ^{3}}mathbb {R} ^{3}.



Basis definition


In the spherical bases denoted e+, e−, e0, and associated coordinates with respect to this basis, denoted A+, A−, A0, the vector A is:








A=A+e++A−e−+A0e0{displaystyle mathbf {A} =A_{+}mathbf {e} _{+}+A_{-}mathbf {e} _{-}+A_{0}mathbf {e} _{0}}{mathbf {A}}=A_{+}{mathbf {e}}_{{+}}+A_{{-}}{mathbf {e}}_{{-}}+A_{0}{mathbf {e}}_{0}









 

 

 

 

(2)


where the spherical basis vectors can be defined in terms of the Cartesian basis using complex-valued coefficients in the xy plane:[1]








e+=−12ex−i2eye−=+12ex−i2ey⇌e±=∓12(ex±iey){displaystyle {begin{aligned}mathbf {e} _{+}&=-{frac {1}{sqrt {2}}}mathbf {e} _{x}-{frac {i}{sqrt {2}}}mathbf {e} _{y}\mathbf {e} _{-}&=+{frac {1}{sqrt {2}}}mathbf {e} _{x}-{frac {i}{sqrt {2}}}mathbf {e} _{y}\end{aligned}}quad rightleftharpoons quad mathbf {e} _{pm }=mp {frac {1}{sqrt {2}}}left(mathbf {e} _{x}pm imathbf {e} _{y}right),}{begin{aligned}{mathbf {e}}_{+}&=-{frac {1}{{sqrt {2}}}}{mathbf {e}}_{x}-{frac {i}{{sqrt {2}}}}{mathbf {e}}_{y}\{mathbf {e}}_{{-}}&=+{frac {1}{{sqrt {2}}}}{mathbf {e}}_{x}-{frac {i}{{sqrt {2}}}}{mathbf {e}}_{y}\end{aligned}}quad rightleftharpoons quad {mathbf {e}}_{pm }=mp {frac {1}{{sqrt {2}}}}left({mathbf {e}}_{x}pm i{mathbf {e}}_{y}right),









 

 

 

 

(3A)


in which i denotes the imaginary unit, and one normal to the plane in the z direction:


e0=ez{displaystyle mathbf {e} _{0}=mathbf {e} _{z}}{mathbf {e}}_{0}={mathbf {e}}_{z}

The inverse relations are:








ex=−12e++12e−ey=+i2e++i2e−ez=e0{displaystyle {begin{aligned}mathbf {e} _{x}&=-{frac {1}{sqrt {2}}}mathbf {e} _{+}+{frac {1}{sqrt {2}}}mathbf {e} _{-}\mathbf {e} _{y}&=+{frac {i}{sqrt {2}}}mathbf {e} _{+}+{frac {i}{sqrt {2}}}mathbf {e} _{-}\mathbf {e} _{z}&=mathbf {e} _{0}end{aligned}}}{begin{aligned}{mathbf {e}}_{x}&=-{frac {1}{{sqrt {2}}}}{mathbf {e}}_{+}+{frac {1}{{sqrt {2}}}}{mathbf {e}}_{{-}}\{mathbf {e}}_{y}&=+{frac {i}{{sqrt {2}}}}{mathbf {e}}_{+}+{frac {i}{{sqrt {2}}}}{mathbf {e}}_{{-}}\{mathbf {e}}_{z}&={mathbf {e}}_{0}end{aligned}}









 

 

 

 

(3B)



Commutator definition


While giving a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank k{displaystyle k}k is 1. For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator Tq(k){displaystyle T_{q}^{(k)}}T_{q}^{{(k)}} that satisfies the following relations is a spherical tensor :



[J±,Tq(k)]=ℏ(k∓q)(k±q+1)Tq±1(k){displaystyle [J_{pm },T_{q}^{(k)}]=hbar {sqrt {(kmp q)(kpm q+1)}}T_{qpm 1}^{(k)}}[J_{{pm }},T_{q}^{{(k)}}]=hbar {sqrt {(kmp q)(kpm q+1)}}T_{{qpm 1}}^{{(k)}}



[Jz,Tq(k)]=ℏqTq(k){displaystyle [J_{z},T_{q}^{(k)}]=hbar qT_{q}^{(k)}}[J_{{z}},T_{q}^{{(k)}}]=hbar qT_{q}^{{(k)}}





Rotation definition


Analogously to how the spherical harmonics transform under a rotation, a general spherical tensor transforms as follows, when the states transform under the unitary Wigner D-matrix D(R){displaystyle {mathcal {D}}(R)}{mathcal {D}}(R), where R is a (3×3 rotation) group element in SO(3). That is, these matrices represent the rotation group elements. With the help of its Lie algebra, one can show these two definitions are equivalent.


D(R)Tq(k)D†(R)=∑q′=−kkTq′(k)Dq′q(k){displaystyle {mathcal {D}}(R)T_{q}^{(k)}{mathcal {D}}^{dagger }(R)=sum _{q^{prime }=-k}^{k}T_{q^{prime }}^{(k)}{mathcal {D}}_{q^{prime }q}^{(k)}}{displaystyle {mathcal {D}}(R)T_{q}^{(k)}{mathcal {D}}^{dagger }(R)=sum _{q^{prime }=-k}^{k}T_{q^{prime }}^{(k)}{mathcal {D}}_{q^{prime }q}^{(k)}}

See also: Wigner D-matrix


Coordinate vectors


Main article: Coordinate vector

For the spherical basis, the coordinates are complex-valued numbers A+, A0, A−, and can be found by substitution of (3B) into (1), or directly calculated from the inner product ⟨, ⟩ (5):








A+=⟨A,e+⟩=−Ax2+iAy2A−=⟨A,e−⟩=+Ax2+iAy2⇌A±=⟨e±,A⟩=12(∓Ax+iAy){displaystyle {begin{aligned}A_{+}&=leftlangle mathbf {A} ,mathbf {e} _{+}rightrangle =-{frac {A_{x}}{sqrt {2}}}+{frac {iA_{y}}{sqrt {2}}}\A_{-}&=leftlangle mathbf {A} ,mathbf {e} _{-}rightrangle =+{frac {A_{x}}{sqrt {2}}}+{frac {iA_{y}}{sqrt {2}}}\end{aligned}}quad rightleftharpoons quad A_{pm }=leftlangle mathbf {e} _{pm },mathbf {A} rightrangle ={frac {1}{sqrt {2}}}left(mp A_{x}+iA_{y}right)}{displaystyle {begin{aligned}A_{+}&=leftlangle mathbf {A} ,mathbf {e} _{+}rightrangle =-{frac {A_{x}}{sqrt {2}}}+{frac {iA_{y}}{sqrt {2}}}\A_{-}&=leftlangle mathbf {A} ,mathbf {e} _{-}rightrangle =+{frac {A_{x}}{sqrt {2}}}+{frac {iA_{y}}{sqrt {2}}}\end{aligned}}quad rightleftharpoons quad A_{pm }=leftlangle mathbf {e} _{pm },mathbf {A} rightrangle ={frac {1}{sqrt {2}}}left(mp A_{x}+iA_{y}right)}









 

 

 

 

(4A)


A0=⟨e0,A⟩=⟨ez,A⟩=Az{displaystyle A_{0}=leftlangle mathbf {e} _{0},mathbf {A} rightrangle =leftlangle mathbf {e} _{z},mathbf {A} rightrangle =A_{z}}A_{0}=leftlangle {mathbf {e}}_{0},{mathbf {A}}rightrangle =leftlangle {mathbf {e}}_{z},{mathbf {A}}rightrangle =A_{z}

with inverse relations:








Ax=−12A++12A−Ay=−i2A+−i2A−Az=A0{displaystyle {begin{aligned}A_{x}&=-{frac {1}{sqrt {2}}}A_{+}+{frac {1}{sqrt {2}}}A_{-}\A_{y}&=-{frac {i}{sqrt {2}}}A_{+}-{frac {i}{sqrt {2}}}A_{-}\A_{z}&=A_{0}end{aligned}}}{begin{aligned}A_{x}&=-{frac {1}{{sqrt {2}}}}A_{+}+{frac {1}{{sqrt {2}}}}A_{{-}}\A_{y}&=-{frac {i}{{sqrt {2}}}}A_{+}-{frac {i}{{sqrt {2}}}}A_{{-}}\A_{z}&=A_{0}end{aligned}}









 

 

 

 

(4B)


In general, for two vectors with complex coefficients in the same real-valued orthonormal basis ei, with the property ei·ej = δij, the inner product is:








⟨a,b⟩=a⋅b⋆=∑jajbj⋆{displaystyle leftlangle mathbf {a} ,mathbf {b} rightrangle =mathbf {a} cdot mathbf {b} ^{star }=sum _{j}a_{j}b_{j}^{star }}leftlangle {mathbf {a}},{mathbf {b}}rightrangle ={mathbf {a}}cdot {mathbf {b}}^{star }=sum _{j}a_{j}b_{j}^{star }









 

 

 

 

(5)


where · is the usual dot product and the complex conjugate * must be used to keep the magnitude (or "norm") of the vector positive definite.



Properties (three dimensions)



Orthonormality


The spherical basis is an orthonormal basis, since the inner product ⟨, ⟩ (5) of every pair vanishes meaning the basis vectors are all mutually orthogonal:


⟨e+,e−⟩=⟨e−,e0⟩=⟨e0,e+⟩=0{displaystyle leftlangle mathbf {e} _{+},mathbf {e} _{-}rightrangle =leftlangle mathbf {e} _{-},mathbf {e} _{0}rightrangle =leftlangle mathbf {e} _{0},mathbf {e} _{+}rightrangle =0}leftlangle {mathbf {e}}_{+},{mathbf {e}}_{{-}}rightrangle =leftlangle {mathbf {e}}_{{-}},{mathbf {e}}_{0}rightrangle =leftlangle {mathbf {e}}_{0},{mathbf {e}}_{+}rightrangle =0

and each basis vector is a unit vector:


⟨e+,e+⟩=⟨e−,e−⟩=⟨e0,e0⟩=1{displaystyle leftlangle mathbf {e} _{+},mathbf {e} _{+}rightrangle =leftlangle mathbf {e} _{-},mathbf {e} _{-}rightrangle =leftlangle mathbf {e} _{0},mathbf {e} _{0}rightrangle =1}leftlangle {mathbf {e}}_{+},{mathbf {e}}_{{+}}rightrangle =leftlangle {mathbf {e}}_{{-}},{mathbf {e}}_{{-}}rightrangle =leftlangle {mathbf {e}}_{0},{mathbf {e}}_{0}rightrangle =1

hence the need for the normalizing factors of 1/√2.



Change of basis matrix


See also: change of basis

The defining relations (3A) can be summarized by a transformation matrix U:


(e+e−e0)=U(exeyez),U=(−12−i20+12−i20001),{displaystyle {begin{pmatrix}mathbf {e} _{+}\mathbf {e} _{-}\mathbf {e} _{0}end{pmatrix}}=mathbf {U} {begin{pmatrix}mathbf {e} _{x}\mathbf {e} _{y}\mathbf {e} _{z}end{pmatrix}},,quad mathbf {U} ={begin{pmatrix}-{frac {1}{sqrt {2}}}&-{frac {i}{sqrt {2}}}&0\+{frac {1}{sqrt {2}}}&-{frac {i}{sqrt {2}}}&0\0&0&1end{pmatrix}},,}{begin{pmatrix}{mathbf {e}}_{+}\{mathbf {e}}_{{-}}\{mathbf {e}}_{0}end{pmatrix}}={mathbf {U}}{begin{pmatrix}{mathbf {e}}_{x}\{mathbf {e}}_{y}\{mathbf {e}}_{z}end{pmatrix}},,quad {mathbf {U}}={begin{pmatrix}-{frac {1}{{sqrt {2}}}}&-{frac {i}{{sqrt {2}}}}&0\+{frac {1}{{sqrt {2}}}}&-{frac {i}{{sqrt {2}}}}&0\0&0&1end{pmatrix}},,

with inverse:


(exeyez)=U−1(e+e−e0),U−1=(−12+120+i2+i20001).{displaystyle {begin{pmatrix}mathbf {e} _{x}\mathbf {e} _{y}\mathbf {e} _{z}end{pmatrix}}=mathbf {U} ^{-1}{begin{pmatrix}mathbf {e} _{+}\mathbf {e} _{-}\mathbf {e} _{0}end{pmatrix}},,quad mathbf {U} ^{-1}={begin{pmatrix}-{frac {1}{sqrt {2}}}&+{frac {1}{sqrt {2}}}&0\+{frac {i}{sqrt {2}}}&+{frac {i}{sqrt {2}}}&0\0&0&1end{pmatrix}},.}{begin{pmatrix}{mathbf {e}}_{x}\{mathbf {e}}_{y}\{mathbf {e}}_{z}end{pmatrix}}={mathbf {U}}^{{-1}}{begin{pmatrix}{mathbf {e}}_{+}\{mathbf {e}}_{{-}}\{mathbf {e}}_{0}end{pmatrix}},,quad {mathbf {U}}^{{-1}}={begin{pmatrix}-{frac {1}{{sqrt {2}}}}&+{frac {1}{{sqrt {2}}}}&0\+{frac {i}{{sqrt {2}}}}&+{frac {i}{{sqrt {2}}}}&0\0&0&1end{pmatrix}},.

It can be seen that U is a unitary matrix, in other words its Hermitian conjugate U† (complex conjugate and matrix transpose) is also the inverse matrix U−1.


For the coordinates:


(A+A−A0)=U∗(AxAyAz),U∗=(−12+i20+12+i20001),{displaystyle {begin{pmatrix}A_{+}\A_{-}\A_{0}end{pmatrix}}=mathbf {U} ^{mathrm {*} }{begin{pmatrix}A_{x}\A_{y}\A_{z}end{pmatrix}},,quad mathbf {U} ^{mathrm {*} }={begin{pmatrix}-{frac {1}{sqrt {2}}}&+{frac {i}{sqrt {2}}}&0\+{frac {1}{sqrt {2}}}&+{frac {i}{sqrt {2}}}&0\0&0&1end{pmatrix}},,}{begin{pmatrix}A_{+}\A_{{-}}\A_{0}end{pmatrix}}={mathbf {U}}^{{mathrm {*}}}{begin{pmatrix}A_{x}\A_{y}\A_{z}end{pmatrix}},,quad {mathbf {U}}^{{mathrm {*}}}={begin{pmatrix}-{frac {1}{{sqrt {2}}}}&+{frac {i}{{sqrt {2}}}}&0\+{frac {1}{{sqrt {2}}}}&+{frac {i}{{sqrt {2}}}}&0\0&0&1end{pmatrix}},,

and inverse:


(AxAyAz)=(U∗)−1(A+A−A0),(U∗)−1=(−12+120−i2−i20001).{displaystyle {begin{pmatrix}A_{x}\A_{y}\A_{z}end{pmatrix}}=(mathbf {U} ^{mathrm {*} })^{-1}{begin{pmatrix}A_{+}\A_{-}\A_{0}end{pmatrix}},,quad (mathbf {U} ^{mathrm {*} })^{-1}={begin{pmatrix}-{frac {1}{sqrt {2}}}&+{frac {1}{sqrt {2}}}&0\-{frac {i}{sqrt {2}}}&-{frac {i}{sqrt {2}}}&0\0&0&1end{pmatrix}},.}{begin{pmatrix}A_{x}\A_{y}\A_{z}end{pmatrix}}=({mathbf {U}}^{{mathrm {*}}})^{{-1}}{begin{pmatrix}A_{+}\A_{{-}}\A_{0}end{pmatrix}},,quad ({mathbf {U}}^{{mathrm {*}}})^{{-1}}={begin{pmatrix}-{frac {1}{{sqrt {2}}}}&+{frac {1}{{sqrt {2}}}}&0\-{frac {i}{{sqrt {2}}}}&-{frac {i}{{sqrt {2}}}}&0\0&0&1end{pmatrix}},.


Cross products


Taking cross products of the spherical basis vectors, we find an obvious relation:


eq×eq=0{displaystyle mathbf {e} _{q}times mathbf {e} _{q}={boldsymbol {0}}}{mathbf {e}}_{{q}}times {mathbf {e}}_{{q}}={boldsymbol {0}}

where q is a placeholder for +, −, 0, and two less obvious relations:



e±×e∓=±ie0{displaystyle mathbf {e} _{pm }times mathbf {e} _{mp }=pm imathbf {e} _{0}}{mathbf {e}}_{{pm }}times {mathbf {e}}_{{mp }}=pm i{mathbf {e}}_{0}

e±×e0=±ie±{displaystyle mathbf {e} _{pm }times mathbf {e} _{0}=pm imathbf {e} _{pm }}{mathbf {e}}_{{pm }}times {mathbf {e}}_{{0}}=pm i{mathbf {e}}_{{pm }}



Inner product in the spherical basis


The inner product between two vectors A and B in the spherical basis follows from the above definition of the inner product:


⟨A,B⟩=A+B+⋆+A−B−⋆+A0B0⋆{displaystyle leftlangle mathbf {A} ,mathbf {B} rightrangle =A_{+}B_{+}^{star }+A_{-}B_{-}^{star }+A_{0}B_{0}^{star }}leftlangle {mathbf {A}},{mathbf {B}}rightrangle =A_{+}B_{+}^{star }+A_{{-}}B_{{-}}^{star }+A_{0}B_{0}^{star }


See also



  • Wigner–Eckart theorem

  • Wigner D matrix



References





  1. ^ W.J. Thompson (2008). Angular Momentum. John Wiley & Sons. p. 311. ISBN 9783527617838..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}




General



  • S. S. M. Wong (2008). Introductory Nuclear Physics (2nd ed.). John Wiley & Sons. ISBN 978-35-276-179-13.


External links









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