Wigner's D matrix

The Wigner D-matrix is a unitary matrix in an irreducible representation of the three-dimensional rotation group SO (3) or the group SU (2) . It was introduced in 1927 by Eugene Wigner .

The D stands for representation . The Wigner D-matrix is ​​used in the quantum mechanics of the rotating group, so the complex-conjugate D-matrix is ​​an eigenfunction of the Hamilton operator of the spherical and symmetrical rigid rotator. In addition, the D matrix describes the transformation of spin states during rotations. ${\displaystyle |j,m\rangle }$

definition

As usual, let the angular momentum operators be the generators of the Lie algebra of SO (3) and SU (2). They fulfill the commutation relations: ${\displaystyle J_{x},J_{y},J_{z}}$

${\displaystyle [J_{x},J_{y}]=iJ_{z},\quad [J_{z},J_{x}]=iJ_{y},\quad [J_{y},J_{z}]=iJ_{x},}$

where in the angular momentum algebra of quantum mechanics the reduced Planck constant was set equal to 1. The Casimir operator

${\displaystyle J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}$

commutes with the generators and can be diagonalized together with the complete set of basis functions in Bra-Ket notation : ${\displaystyle J_{z}}$

${\displaystyle J^{2}|jm\rangle =j(j+1)|jm\rangle ,\quad J_{z}|jm\rangle =m|jm\rangle ,}$

with j = 0, 1/2, 1, 3/2, 2, ... for SU (2) and j = 0, 1, 2, ... for SO (3) and m = -j, -j +1, ..., j (SU (2) realizes a double superposition of the rotation group SO (3) and its spinor representation describes particles and states with half-integer spin).

In three dimensions, a rotation operator can

${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}},}$

are written with the Euler angles . Right-handed coordinate systems are used and the rotation is positive if it is counterclockwise in the rotation axis when viewed from above. With the Euler angles, the zyz convention is used here and the active interpretation, i.e. rotation of the object - for example a vector - and not of the coordinate system. The rotation of the latter is the passive interpretation, which in this case is obtained from the active one by taking the opposite sign for the angle of rotation. This means that first around the angle around the z-axis, a rotation around the angle around the y-axis and then a rotation around the z-axis. The fixed axes are meant here. The inverse of the rotation operator is . ${\displaystyle \alpha ,\beta ,\gamma }$ ${\displaystyle \gamma }$ ${\displaystyle \beta }$ ${\displaystyle \alpha }$ ${\displaystyle {\mathcal {R}}^{-1}(\alpha ,\beta ,\gamma )={\mathcal {R}}(-\gamma ,-\beta ,-\alpha )}$

The rotation in the angular momentum basis is given by: ${\displaystyle |j,m\rangle }$

${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )\,|j,m\rangle =\sum _{m'=-j}^{+j}D_{m'm}^{j}(\alpha ,\beta ,\gamma )\,|j,m'\rangle }$

with the Wigner D matrix:

${\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )\equiv \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma }}$,

in which

${\displaystyle d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta J_{y}}|jm\rangle =D_{m'm}^{j}(0,\beta ,0)}$

Wigner's little d matrix is.

In this basis, the Wigner D matrix is ​​a unitary square matrix of dimension 2j + 1. The rotated ket vector is eigenvector to but not from but to to the rotated quantization axis . ${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )\,|j,m\rangle }$ ${\displaystyle J^{2}}$ ${\displaystyle J_{z}}$ ${\displaystyle J_{z'}}$ ${\displaystyle {\vec {e}}_{z'}=R(\alpha ,\beta ,\gamma )\,{\vec {e}}_{z}}$

To use instead of active interpretation of the passive interpretation, you have by replacing about, that is complex conjugation and permutation of the indices. ${\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )}$ ${\displaystyle D_{mm'}^{j*}(\alpha ,\beta ,\gamma )}$ ${\displaystyle *}$

Wigner's little d-matrix

Wigner gave the following formula for the small d-matrix (Wigner's formula):

${\displaystyle d_{m'm}^{j}(\beta )=[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\,\sum \limits _{s}\left[{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\cdot \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\right]}$

Here, s is only added over the factorials that are non-negative.

The elements of the d-matrix are real for the Euler angles in this convention, which is why it is used. In the zxz convention of the Euler angles, the factor in the above formula has to be replaced by Wigner , which makes half of the functions imaginary. ${\displaystyle (-1)^{m'-m+s}}$ ${\displaystyle (-1)^{s}\,i^{m-m'}}$

The elements of the d matrix are related to Jacobi polynomials with non-negative a, b. Be ${\displaystyle P_{k}^{(a,b)}(\cos \beta )}$

${\displaystyle k=\min(j+m,\,j-m,\,j+m',\,j-m').}$

${\displaystyle {\hbox{Falls}}\quad k={\begin{cases}j+m:&\quad a=m'-m;\quad \lambda =m'-m\\j-m:&\quad a=m-m';\quad \lambda =0\\j+m':&\quad a=m-m';\quad \lambda =0\\j-m':&\quad a=m'-m;\quad \lambda =m'-m\\\end{cases}}}$

Then with (it applies ): ${\displaystyle b=2j-2k-a\,}$ ${\displaystyle a,b\geq 0}$

${\displaystyle d_{m'm}^{j}(\beta )=(-1)^{\lambda }{\binom {2j-k}{k+a}}^{1/2}{\binom {k+b}{b}}^{-1/2}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),}$

Properties of the Wigner D-matrix

The complex conjugate D matrix satisfies a number of differential equations. For this purpose, the following differential operators are also defined , which in quantum mechanics are rotational operators for the spatially fixed system of the rigid rotator: ${\displaystyle (x,\,y,\,z)=(1,\,2,\,3)}$

${\displaystyle {\begin{array}{lcl}{\hat {\mathcal {J}}}_{1}&=&i\left(\cos \alpha \cot \beta \,{\partial \over \partial \alpha }\,+\sin \alpha \,{\partial \over \partial \beta }\,-{\cos \alpha \over \sin \beta }\,{\partial \over \partial \gamma }\,\right)\\{\hat {\mathcal {J}}}_{2}&=&i\left(\sin \alpha \cot \beta \,{\partial \over \partial \alpha }\,-\cos \alpha \;{\partial \over \partial \beta }\,-{\sin \alpha \over \sin \beta }\,{\partial \over \partial \gamma }\,\right)\\{\hat {\mathcal {J}}}_{3}&=&-i\;{\partial \over \partial \alpha },\end{array}}}$

Furthermore, one has the rotation operators of the body-fixed system of the rigid rotator in quantum mechanics:

${\displaystyle {\begin{array}{lcl}{\hat {\mathcal {P}}}_{1}&=&\,i\left({\cos \gamma \over \sin \beta }{\partial \over \partial \alpha }-\sin \gamma {\partial \over \partial \beta }-\cot \beta \cos \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{2}&=&\,i\left(-{\sin \gamma \over \sin \beta }{\partial \over \partial \alpha }-\cos \gamma {\partial \over \partial \beta }+\cot \beta \sin \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{3}&=&-i{\partial \over \partial \gamma },\\\end{array}}}$

They fulfill the commutator relations

${\displaystyle \left[{\mathcal {J}}_{1},\,{\mathcal {J}}_{2}\right]=i{\mathcal {J}}_{3},\qquad {\hbox{und}}\qquad \left[{\mathcal {P}}_{1},\,{\mathcal {P}}_{2}\right]=-i{\mathcal {P}}_{3}}$

and accordingly with cyclic permutation of the indices.

In contrast to the anomalous commutator relations, they have a minus sign on the right-hand side. ${\displaystyle {\mathcal {P}}_{i}}$ ${\displaystyle {\mathcal {J}}_{i}}$

The two types of operators commute

${\displaystyle \left[{\mathcal {P}}_{i},\,{\mathcal {J}}_{j}\right]=0,\quad i,\,j=1,\,2,\,3,}$

and their squares are equal:

${\displaystyle {\mathcal {J}}^{2}\equiv {\mathcal {J}}_{1}^{2}+{\mathcal {J}}_{2}^{2}+{\mathcal {J}}_{3}^{2}={\mathcal {P}}^{2}\equiv {\mathcal {P}}_{1}^{2}+{\mathcal {P}}_{2}^{2}+{\mathcal {P}}_{3}^{2}.}$

One has explicitly:

${\displaystyle {\mathcal {J}}^{2}={\mathcal {P}}^{2}=-{\frac {1}{\sin ^{2}\beta }}\left({\frac {\partial ^{2}}{\partial \alpha ^{2}}}+{\frac {\partial ^{2}}{\partial \gamma ^{2}}}-2\cos \beta {\frac {\partial ^{2}}{\partial \alpha \partial \gamma }}\right)-{\frac {\partial ^{2}}{\partial \beta ^{2}}}-\cot \beta {\frac {\partial }{\partial \beta }}.}$

The operators act on the first index of the D matrix (the row index): ${\displaystyle {\mathcal {J}}_{i}}$

${\displaystyle {\mathcal {J}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=m'\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}$

and

${\displaystyle ({\mathcal {J}}_{1}\pm i{\mathcal {J}}_{2})\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m'(m'\pm 1)}}\,D_{m'\pm 1,m}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

The operators act on the second index (column index) of the D matrix: ${\displaystyle {\mathcal {P}}_{i}}$

${\displaystyle {\mathcal {P}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=m\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}$

Because of the anomalous commutator relations, the associated ladder operators are defined with a different sign:

${\displaystyle ({\mathcal {P}}_{1}\mp i{\mathcal {P}}_{2})\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m(m\pm 1)}}\,D_{m',m\pm 1}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

${\displaystyle {\mathcal {J}}^{2}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\mathcal {P}}^{2}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=j(j+1)D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

The rows and columns of the complex-conjugated D-matrix form an irreducible representation of the isomorphic and generated Lie algebras. ${\displaystyle \{{\mathcal {J}}_{i}\}}$ ${\displaystyle \{-{\mathcal {P}}_{i}\}}$

From the commutator of it the time-reverse operator follows: ${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )}$ ${\displaystyle T\,}$

${\displaystyle \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =\langle jm'|T^{\,\dagger }{\mathcal {R}}(\alpha ,\beta ,\gamma )T|jm\rangle =(-1)^{m'-m}\langle j,-m'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|j,-m\rangle ^{*},}$

or

${\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )=(-1)^{m'-m}D_{-m',-m}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

The anti-unitarity of , and was used . ${\displaystyle T\,}$ ${\displaystyle T|jm\rangle =(-1)^{j-m}|j,-m\rangle }$ ${\displaystyle (-1)^{2j-m'-m}=(-1)^{m'-m}}$

Orthogonality relation

The D-matrices for the Euler angles , and satisfy the orthogonality relations: ${\displaystyle D_{mk}^{j}(\alpha ,\beta ,\gamma )}$ ${\displaystyle \alpha }$ ${\displaystyle \beta ,}$ ${\displaystyle \gamma }$

${\displaystyle \int _{0}^{2\pi }d\alpha \int _{0}^{\pi }\sin \beta d\beta \int _{0}^{2\pi }d\gamma \,\,D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )^{\ast }D_{mk}^{j}(\alpha ,\beta ,\gamma )={\frac {8\pi ^{2}}{2j+1}}\delta _{m'm}\delta _{k'k}\delta _{j'j}.}$

According to Peter-Weyl's theorem , the orthogonal basis they form is complete.

The group characters of SU (2) depend only on the angle and are class functions: ${\displaystyle \beta }$

${\displaystyle \chi ^{j}(\beta )\equiv \sum _{m}D_{mm}^{j}(\beta )=\sum _{m}d_{mm}^{j}(\beta )={\frac {\sin((2j+1)\beta /2)}{\sin(\beta /2)}},}$

For them, simpler orthogonality relations apply using the hair measure of the group:

${\displaystyle {\frac {1}{\pi }}\int _{0}^{2\pi }d\beta ~\sin ^{2}(\beta /2)~\chi ^{j}(\beta )\chi ^{j'}(\beta )=\delta _{j'j}.}$

The completeness relation is:

${\displaystyle \sum _{j}\chi ^{j}(\beta )\chi ^{j}(\beta ')=\delta (\beta -\beta '),}$

The following applies to: ${\displaystyle \beta '=0}$

${\displaystyle \sum _{j}\chi ^{j}(\beta )(2j+1)=\delta (\beta )~.}$

Kronecker product from D-Matrices and Clebsch-Gordan series

The Kronecker products from Matrizen

${\displaystyle \mathbf {D} ^{j}(\alpha ,\beta ,\gamma )\otimes \mathbf {D} ^{j'}(\alpha ,\beta ,\gamma )}$

provide a reducible representation of the groups SO (3) or SU (2) and reduction into irreducible components results in the Clebsch-Gordan series:

${\displaystyle D_{mk}^{j}(\alpha ,\beta ,\gamma )D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )=\sum _{J=|j-j'|}^{j+j'}\langle jmj'm'|J\left(m+m'\right)\rangle \langle jkj'k'|J\left(k+k'\right)\rangle D_{\left(m+m'\right)\left(k+k'\right)}^{J}(\alpha ,\beta ,\gamma )}$

there is a Clebsch-Gordan coefficient . ${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle }$

Relationship to spherical functions and Legendre functions

For integer and second index equal to zero, the D matrix elements are proportional to spherical surface functions and assigned Legendre polynomials . With normalization to 1 and phase convention according to Condon and Shortley: ${\displaystyle l}$

${\displaystyle D_{m0}^{\ell }(\alpha ,\beta ,\gamma )={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m*}(\beta ,\alpha )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta })\,e^{-im\alpha }.}$

From this it follows for the d matrix:

${\displaystyle d_{m0}^{\ell }(\beta )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta }).}$

If both indices are set to zero, the elements of the D matrix are given by Legendre polynomials :

${\displaystyle D_{0,0}^{\ell }(\alpha ,\beta ,\gamma )=d_{0,0}^{\ell }(\beta )=P_{\ell }(\cos \beta ).}$

It follows from the behavior of the D matrix at time reversal

${\displaystyle \left(Y_{\ell }^{m}\right)^{*}=(-1)^{m}Y_{\ell }^{-m}.}$

The following applies to spin-weighted spherical surface functions:

${\displaystyle D_{-ms}^{\ell }(\alpha ,\beta ,-\gamma )=(-1)^{m}{\sqrt {\frac {4\pi }{2{\ell }+1}}}{}_{s}Y_{{\ell }m}(\beta ,\alpha )e^{is\gamma }.}$

The rotational behavior of the spherical functions can be expressed with Wigner's D-matrices. The Euler angles parameterize the rotation of the coordinate system (x, y, z) in (X, Y, Z). Let be the polar angle of a unit vector in the system (x, y, z) and in the system (X, Y, Z). Then the spherical function can be understood as a Bra-Ket vector with the transformation: ${\displaystyle \omega =(\phi ,\theta )}$ ${\displaystyle \Omega =(\Phi ,\Theta )}$ ${\displaystyle Y_{l}^{m}(\omega )}$ ${\displaystyle |l,m\rangle }$

${\displaystyle Y_{l}^{m}(\Omega )=\sum _{m'=-l}^{+l}Y_{l}^{m'}(\omega )D_{m'm}^{l}(\alpha ,\beta ,\gamma )}$

Relationship to Bessel functions

For one has the Bessel function and finite . ${\displaystyle \ell \gg m,m^{\prime }}$ ${\displaystyle D_{mm^{\prime }}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -im^{\prime }\gamma }J_{m-m^{\prime }}(\ell \beta )}$ ${\displaystyle J_{m-m^{\prime }}(\ell \beta )}$ ${\displaystyle \ell \beta }$

List of elements of the d matrix

The d-matrices are given in Wigner's sign convention.

For j = 1/2

• ${\displaystyle d_{1/2,1/2}^{1/2}=\cos(\beta /2)}$
• ${\displaystyle d_{1/2,-1/2}^{1/2}=-\sin(\beta /2)}$

For j = 1

• ${\displaystyle d_{1,1}^{1}={\frac {1+\cos \beta }{2}}}$
• ${\displaystyle d_{1,0}^{1}={\frac {-\sin \beta }{\sqrt {2}}}}$
• ${\displaystyle d_{1,-1}^{1}={\frac {1-\cos \beta }{2}}}$
• ${\displaystyle d_{0,0}^{1}=\cos \beta }$

For j = 3/2

• ${\displaystyle d_{3/2,3/2}^{3/2}={\frac {1+\cos \beta }{2}}\cos {\frac {\beta }{2}}}$
• ${\displaystyle d_{3/2,1/2}^{3/2}=-{\sqrt {3}}{\frac {1+\cos \beta }{2}}\sin {\frac {\beta }{2}}}$
• ${\displaystyle d_{3/2,-1/2}^{3/2}={\sqrt {3}}{\frac {1-\cos \beta }{2}}\cos {\frac {\beta }{2}}}$
• ${\displaystyle d_{3/2,-3/2}^{3/2}=-{\frac {1-\cos \beta }{2}}\sin {\frac {\beta }{2}}}$
• ${\displaystyle d_{1/2,1/2}^{3/2}={\frac {3\cos \beta -1}{2}}\cos {\frac {\beta }{2}}}$
• ${\displaystyle d_{1/2,-1/2}^{3/2}=-{\frac {3\cos \beta +1}{2}}\sin {\frac {\beta }{2}}}$

For j = 2

• ${\displaystyle d_{2,2}^{2}={\frac {1}{4}}\left(1+\cos \beta \right)^{2}}$
• ${\displaystyle d_{2,1}^{2}=-{\frac {1}{2}}\sin \beta \left(1+\cos \beta \right)}$
• ${\displaystyle d_{2,0}^{2}={\sqrt {\frac {3}{8}}}\sin ^{2}\beta }$
• ${\displaystyle d_{2,-1}^{2}=-{\frac {1}{2}}\sin \beta \left(1-\cos \beta \right)}$
• ${\displaystyle d_{2,-2}^{2}={\frac {1}{4}}\left(1-\cos \beta \right)^{2}}$
• ${\displaystyle d_{1,1}^{2}={\frac {1}{2}}\left(2\cos ^{2}\beta +\cos \beta -1\right)}$
• ${\displaystyle d_{1,0}^{2}=-{\sqrt {\frac {3}{8}}}\sin 2\beta }$
• ${\displaystyle d_{1,-1}^{2}={\frac {1}{2}}\left(-2\cos ^{2}\beta +\cos \beta +1\right)}$
• ${\displaystyle d_{0,0}^{2}={\frac {1}{2}}\left(3\cos ^{2}\beta -1\right)}$

Elements with swapped lower indices can be obtained from:

${\displaystyle d_{m',m}^{j}=(-1)^{m-m'}d_{m,m'}^{j}=d_{-m,-m'}^{j}}$.

Examples of D-matrices

For one has (from left to right and top to bottom indices in order ): ${\displaystyle j={\frac {1}{2}}}$ ${\displaystyle m={\frac {1}{2}},-{\frac {1}{2}}}$

${\displaystyle D^{\frac {1}{2}}(\alpha ,\beta ,\gamma )={\begin{pmatrix}e^{-{\frac {1}{2}}i\alpha }\cos({\frac {\beta }{2}})e^{-{\frac {1}{2}}i\gamma }&-e^{-{\frac {1}{2}}i\alpha }\sin({\frac {\beta }{2}})e^{{\frac {1}{2}}i\gamma }\\e^{{\frac {1}{2}}i\alpha }\sin({\frac {\beta }{2}})e^{-{\frac {1}{2}}i\gamma }&e^{{\frac {1}{2}}i\alpha }\cos({\frac {\beta }{2}})e^{{\frac {1}{2}}i\gamma }\end{pmatrix}}}$

and thus

${\displaystyle d^{\frac {1}{2}}(\beta )={\begin{pmatrix}\cos({\frac {\beta }{2}})&-\sin({\frac {\beta }{2}})\\\sin({\frac {\beta }{2}})&\cos({\frac {\beta }{2}})\end{pmatrix}}}$

For one has (quantum numbers or indices in sequence ): ${\displaystyle j=1}$ ${\displaystyle m=1,0,-1}$

${\displaystyle d^{1}(\beta )={\begin{pmatrix}{\frac {1}{2}}(1+\cos(\beta ))&-{\frac {1}{2}}{\sqrt {2}}\sin(\beta )&{\frac {1}{2}}(1-\cos(\beta ))\\{\frac {1}{2}}{\sqrt {2}}\sin(\beta )&\cos(\beta )&-{\frac {1}{2}}{\sqrt {2}}\sin(\beta )\\{\frac {1}{2}}(1-\cos(\beta ))&{\frac {1}{2}}{\sqrt {2}}\sin(\beta )&{\frac {1}{2}}(1+\cos(\beta ))\end{pmatrix}}}$

Web links

• PDG table of Clebsch-Gordan coefficients, spherical functions and d-functions
• Morrison, Parker, A guide to rotations in quantum mechanics , Australian Journal of Physics, Volume 40, 1987, pp. 465-497

Individual evidence

1. ↑ Sometimes the Euler angles are rotated around the rotated axes, then rotated first around and last around . For example, M. Morrison, G. Parker, A guide to rotations in quantum mechanics, Australian J. Phys., Vol. 40, 1987, pp. 495-4997 ${\displaystyle \alpha }$ ${\displaystyle \gamma }$
2. ↑ Morrison, Parker, Australian J. Phys., Vol. 40, 1987, p. 478
3. ↑ Messiah, Quantum Mechanics, Volume 2, p. 533
4. ↑ Biedenharn, Louck, Angular Momentum in Quantum Physics, Addison-Wesley 1981
5. ^ Julian Schwinger , On angular momentum, DOE 1952
6. ^ ME Rose: Elementary Theory of Angular Momentum, Wiley 1957, p. 58
7. ^ Morrison, Parker, A guide to rotations in quantum mechanics, Australian Journal of Physics, Volume 40, 1987, p. 487, online
8. ↑ Messiah, Quantum Mechanics, Volume 2, p. 535
9. ↑ Messiah, Quantum Mechanics, Volume 2, p. 534