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| In [[mathematical physics]] and [[mathematics]], the '''Pauli matrices''' are a set of three 2 × 2 [[complex number|complex]] [[matrix (mathematics)|matrices]] which are [[Hermitian matrix|Hermitian]] and [[Unitary matrix|unitary]].<ref name=planetmath>{{cite web|title=Pauli matrices|url=http://planetmath.org/PauliMatrices|publisher=Planetmath website|accessdate=28 May 2013|date=28 March 2008}}</ref> Usually indicated by the [[Greek (alphabet)|Greek]] letter ''[[sigma]]'' ({{mvar|σ}}), they are occasionally denoted by ''[[tau]]'' ({{mvar|τ}}) when used in connection with [[isospin]] symmetries. They are
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| :<math>
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| \sigma_1 = \sigma_x =
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| \begin{pmatrix}
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| 0&1\\
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| 1&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| \sigma_2 = \sigma_y =
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| \begin{pmatrix}
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| 0&-i\\
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| i&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| \sigma_3 = \sigma_z =
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| \begin{pmatrix}
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| 1&0\\
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| 0&-1
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| \end{pmatrix}
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| </math>
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| which may be compacted into a single expression using the [[Kronecker delta]],
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| :<math>
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| \sigma_j =
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| \begin{pmatrix}
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| \delta_{j3}&\delta_{j1}-i\delta_{j2}\\
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| \delta_{j1}+i\delta_{j2}&-\delta_{j3}
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| \end{pmatrix} ~~.
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| </math> | |
|
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|
| These matrices are named after the physicist [[Wolfgang Pauli]]. In [[quantum mechanics]], they occur in the [[Pauli equation]] which takes into account the interaction of the [[spin (physics)|spin]] of a particle with an external [[electromagnetic field]].
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| Each Pauli matrix is [[Hermitian matrix|Hermitian]], and together with the identity matrix {{mvar|I}} (sometimes considered as the zeroth Pauli matrix {{math|''σ''<sub>0</sub>}}), the Pauli matrices (multiplied by ''real'' coefficients) span the full [[vector space]] of 2 × 2 [[Hermitian matrix|Hermitian matrices]].
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| In the language of quantum mechanics, Hermitian matrices are [[observable]]s, so the Pauli matrices span the space of observables of the 2-dimensional complex [[Hilbert space]]. In the context of Pauli's work, {{math|''σ''<sub>''k''</sub>}} is the observable corresponding to spin along the {{mvar|k}}th coordinate axis in three-dimensional [[Euclidean space]] {{math|ℝ<sup>3</sup>}}.
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| The Pauli matrices (after multiplication by {{mvar|i}} to make them [[skew-Hermitian|anti-Hermitian]]), also generate transformations in the sense of [[Lie algebra]]s: the matrices {{math|''iσ''<sub>1</sub>, ''iσ''<sub>2</sub>, ''iσ''<sub>3</sub>}} form a basis for <math>\mathfrak{su}_2</math>, which [[Exponential_map#Lie_theory|exponentiates]] to the [[spin group]] [[SU(2)#n = 2|SU(2)]], and for the identical Lie algebra <math>\mathfrak{so}_3</math>, which exponentiates to the [[Lie group]] [[Rotation group SO(3)|SO(3)]] of rotations of 3-dimensional space. The [[Algebra over a field|algebra]] generated by the three matrices {{math|''σ''<sub>1</sub>, ''σ''<sub>2</sub>, ''σ''<sub>3</sub>}} is [[isomorphic]] to the [[Clifford algebra]] of {{math|ℝ<sup>3</sup>}}, called the [[algebra of physical space]].
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| == Algebraic properties ==
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| The matrices are [[Involution (mathematics)|involutory]]:
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| :<math>
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| \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I</math>
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| where ''I'' is the [[identity matrix]].
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| *The [[determinant]]s and [[trace of a matrix|trace]]s of the Pauli matrices are:
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| :<math>\det (\sigma_i) = -1,</math>
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| :<math>\operatorname{Tr} (\sigma_i) = 0 .</math>
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| From above we can deduce that the [[eigenvalues]] of each ''σ''<sub>''i''</sub> are ±1.
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| *Together with the 2 × 2 identity matrix ''I'' (sometimes written as ''σ''<sub>0</sub>), the Pauli matrices form an orthogonal basis, in the sense of [[Hilbert–Schmidt operator|Hilbert–Schmidt]], for the real [[Hilbert space]] of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.
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| === Eigenvectors and eigenvalues ===
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| Each of the ([[Hermitian]]) Pauli matrices has two [[eigenvalues]], +1 and −1. The corresponding [[Normalisable wavefunction|normalized]] [[eigenvectors]] are:
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| :<math>
| |
| \begin{array}{lclc}
| |
| \psi_{x+}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{1}\end{pmatrix}, & \psi_{x-}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{-1}\end{pmatrix}, \\
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| \psi_{y+}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{i}\end{pmatrix}, & \psi_{y-}=\displaystyle\frac{1}{\sqrt{2}}\!\!\!\!\! & \begin{pmatrix}{1}\\{-i}\end{pmatrix}, \\
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| \psi_{z+}= & \begin{pmatrix}{1}\\{0}\end{pmatrix}, & \psi_{z-}= & \begin{pmatrix}{0}\\{1}\end{pmatrix}.
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| \end{array}
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| </math>
| |
| | |
| === Pauli vector ===
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| The Pauli vector is defined by
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| :<math>\vec{\sigma} = \sigma_1 \hat{x} + \sigma_2 \hat{y} + \sigma_3 \hat{z} \,</math>
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| and provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows
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| :<math>
| |
| \begin{align}
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| \vec{a} \cdot \vec{\sigma} &= (a_i \hat{x}_i) \cdot (\sigma_j \hat{x}_j ) \\
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| &= a_i \sigma_j \hat{x}_i \cdot \hat{x}_j \\
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| &= a_i \sigma_j \delta_{ij} \\
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| &= a_i \sigma_i
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| \end{align}
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| </math>
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| using the [[Einstein notation|summation convention]]. Further,
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| :<math>
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| \det \vec{a} \cdot \vec{\sigma} = - \vec{a} \cdot \vec{a}= -|\vec{a}|^2.
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| </math>
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|
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| === Commutation relations ===
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| The Pauli matrices obey the following [[commutator|commutation]] relations:
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| :<math>[\sigma_a, \sigma_b] = 2 i \sum_c \varepsilon_{a b c}\,\sigma_c \, , </math>
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| and [[anticommutator|anticommutation]] relations:
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| :<math>\{\sigma_a, \sigma_b\} = 2 \delta_{a b}\,I.</math>
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| where ''ε<sub>abc</sub>'' is the [[Levi-Civita symbol]], ''δ<sub>ab</sub>'' is the [[Kronecker delta]], and ''I'' is the 2 × 2 identity matrix.
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| For example,
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| :<math>\begin{align}
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| \left[\sigma_1, \sigma_2\right] &= 2i\sigma_3 \,,\\
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| \left[\sigma_2, \sigma_3\right] &= 2i\sigma_1 \,,\\
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| \left[\sigma_2, \sigma_1\right] &= -2i\sigma_3 \,,\\
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| \left[\sigma_1, \sigma_1\right] &= 0\,,\\
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| \left\{\sigma_1, \sigma_1\right\} &= 2I\,,\\
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| \left\{\sigma_1, \sigma_2\right\} &= 0\,.\\
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| \end{align}</math>
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| | |
| ===Relation to dot and cross product===
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| | |
| Adding the commutator to the anticommutator gives:
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| :<math> \begin{align}
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| \left[\sigma_a, \sigma_b\right] + \{\sigma_a, \sigma_b\} & = ( \sigma_a \sigma_b - \sigma_b \sigma_a ) + (\sigma_a \sigma_b + \sigma_b \sigma_a) \\
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| 2i\sum_c\varepsilon_{a b c}\,\sigma_c + 2 \delta_{a b}I & = 2\sigma_a \sigma_b
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| \end{align}</math>
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| and cancelling the factors of 2:
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| :<math> \sigma_a \sigma_b = i\sum_c\varepsilon_{a b c}\,\sigma_c + \delta_{a b}I\,.</math>
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| [[tensor contraction|Contracting]] each side of the equation with components of two 3d vectors ''a<sub>p</sub>'' and ''b<sub>q</sub>'' which commute with the Pauli matrices, i.e. ''a<sub>p</sub>σ<sub>q</sub>'' = ''σ<sub>q</sub>a<sub>p</sub>'' for each matrix ''σ<sub>q</sub>'' and vector component ''a<sub>p</sub>'' (similarly with ''b<sub>q</sub>''), and relabeling indices ''a'', ''b'', ''c'' → ''p'', ''q'', ''r'' to prevent notational conflicts:
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| :<math> \begin{align}
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| a_p b_q \sigma_p \sigma_q & = a_p b_q \left(i\sum_r\varepsilon_{pqr}\,\sigma_r + \delta_{pq}I\right) \\
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| a_p \sigma_p b_q \sigma_q & = i\sum_r\varepsilon_{pqr}\,a_p b_q \sigma_r + a_p b_q \delta_{pq}I
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| \end{align}</math>
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| and translating the index notation for the [[dot product]] and [[cross product#Index notation for tensors|cross product]]:
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| {{NumBlk|:|<math>(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) \, I + i ( \vec{a} \times \vec{b} )\cdot \vec{\sigma}</math>|{{EquationRef|1}}}}
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| ===Exponential of a Pauli vector===
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| For <math>\vec{a} = a \hat{n} </math> and <math> |\hat{n}|=1 </math>, we have, for even powers,
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| :<math>(\hat{n} \cdot \vec{\sigma})^{2n} = I \,</math>
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| which can be shown first for the ''n'' = 1 case using the anticommutation relations.
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| Thus, for odd powers,
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| :<math>(\hat{n} \cdot \vec{\sigma})^{2n+1} = \hat{n} \cdot \vec{\sigma} \, .</math>
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| [[Matrix exponential|Matrix exponentiating]], and using the [[Taylor series#List of Maclaurin series of some common functions|Taylor series for sine and cosine]],
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| :<math>\begin{align}
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| e^{i a(\hat{n} \cdot \vec{\sigma})} & = \sum_{n=0}^\infty{\frac{i^n \left[a (\hat{n} \cdot \vec{\sigma})\right]^n}{n!}} \\
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| & = \sum_{n=0}^\infty{\frac{(-1)^n (a\hat{n}\cdot \vec{\sigma})^{2n}}{(2n)!}} + i\sum_{n=0}^\infty{\frac{(-1)^n (a\hat{n}\cdot \vec{\sigma})^{2n+1}}{(2n+1)!}} \\
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| & = I\sum_{n=0}^\infty{\frac{(-1)^n a^{2n}}{(2n)!}} + i (\hat{n}\cdot \vec{\sigma}) \sum_{n=0}^\infty{\frac{(-1)^n a^{2n+1}}{(2n+1)!}}\\
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| \end{align}</math>
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| and, in the last line, the first sum is the cosine, while the second sum is the sine, so, finally, | |
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| {{NumBlk|:|{{Equation box 1
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| |indent =:
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| |equation = <math>e^{i a(\hat{n} \cdot \vec{\sigma})} = I\cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} \,</math>
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |bgcolor=#F9FFF7}}|{{EquationRef|2}}}}
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| which is analogous to [[Euler's formula]]. Note
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| :<math>\det ( i a(\hat{n} \cdot \vec{\sigma}))= a^2</math>,
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| while the determinant of the exponential itself is just 1, which makes it the generic group element of [[SU(2)]].
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| A more abstract version of this formula, (2), for a general 2×2 matrix can be found in the article on [[Matrix_exponential#via_Laurent_series|matrix exponentials]].
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| A straightforward application of this formula allows directly solving for {{mvar|c}} in
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| :<math>e^{i a(\hat{n} \cdot \vec{\sigma})} e^{i b(\hat{n}' \cdot \vec{\sigma})} = I\cos{c} + i (\hat{n}'' \cdot \vec{\sigma}) \sin{c} = e^{i c(\hat{n}'' \cdot \vec{\sigma})},</math>
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| that is, specification of generic group multiplication in SU(2), where <math>~~\cos c = \cos a \cos b - \hat{n} \cdot\hat{n}'~ \sin a \sin b</math> .
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| The fact that any 2 × 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the [[Bloch sphere]] representation of 2 × 2 [[mixed state (physics)|mixed state]]s' density matrix, (2 × 2 positive semidefinite matrices with trace 1). This can be seen by simply first writing an arbitrary Hermitian matrix as a real linear combination of {''σ''<sub>0</sub>, ''σ''<sub>1</sub>, ''σ''<sub>2</sub>, ''σ''<sub>3</sub>} as above, and then imposing the positive-semidefinite and [[Trace (linear algebra)|trace]] 1 conditions.
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| === Completeness relation ===
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| An alternative notation that is commonly used for the Pauli matrices is to write the vector index {{mvar|i}} in the superscript, and the matrix indices as subscripts, so that the element in row {{mvar|α}} and column {{mvar|β}} of the {{mvar|i}}-th Pauli matrix is {{math|''σ'' <sup>''i''</sup><sub>''αβ''</sub>}}.
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| In this notation, the '''completeness relation''' for the Pauli matrices can be written
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| :<math>\vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv \sum_{i=1}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma} - \delta_{\alpha\beta}\delta_{\gamma\delta}.\,</math>
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| <div style="clear:both;width:65%;" class="NavFrame">
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| <div class="NavHead" style="background:#ccf; text-align:left; font-size:larger;">Proof</div>
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| <div class="NavContent" style="text-align:left;">
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| The fact that the Pauli matrices, along with the identity matrix ''I'', form an orthogonal basis for the complex Hilbert space of all 2 × 2 matrices mean that we can express any matrix ''M'' as
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| :<math>M = c \mathbf{I} + \sum_i a_i \sigma^i</math>
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| where ''c'' is a complex number, and ''a'' is a 3-component complex vector. It is straightforward to show, using the properties listed above, that
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| :<math>\mathrm{tr}\, \sigma^i\sigma^j = 2\delta_{ij}</math>
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| where "tr" denotes the [[Trace (linear algebra)|trace]], and hence that <math>c=\frac{1}{2}\mathrm{tr}\,M</math> and <math>a_i = \frac{1}{2}\mathrm{tr}\,\sigma^i M</math>.
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| This therefore gives
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| :<math>2M = I \mathrm{tr}\, M + \sum_i \sigma^i \mathrm{tr}\, \sigma^i M</math>
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| which can be rewritten in terms of matrix indices as
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| :<math>2M_{\alpha\beta} = \delta_{\alpha\beta} M_{\gamma\gamma} + \sum_i \sigma^i_{\alpha\beta} \sigma^i_{\gamma\delta} M_{\delta\gamma}</math>
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| where [[Einstein notation|summation is implied]] over the repeated indices ''γ'' and ''δ''. Since this is true for any choice of the matrix ''M'', the completeness relation follows as stated above.
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| </div>
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| </div>
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| As noted above, it is common to denote the 2 × 2 unit matrix by ''σ''<sub>0</sub>, so ''σ''<sup>0</sup><sub>''αβ''</sub> = ''δ''<sub>''αβ''</sub>. The completeness relation can therefore alternatively be expressed as
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| :<math>\sum_{i=0}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma}\,</math>.
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| === Relation with the permutation operator ===
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| Let {{math|''P<sub>ij</sub>''}} be the [[transposition (mathematics)|transposition]] (also known as a permutation) between two spins {{math|''σ''<sub>''i''</sub>}} and {{math|''σ''<sub>''j''</sub>}} living in the [[tensor product]] space {{math|ℂ<sup>2</sup> ⊗ ℂ<sup>2</sup>}},
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| :<math> P_{ij}|\sigma_i \sigma_j\rangle = |\sigma_j \sigma_i\rangle \,. </math>
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| This operator can also be written more explicitly as [[Exchange_interaction#Inclusion_of_spin|Dirac's spin exchange operator]],
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| :<math> P_{ij} = \tfrac{1}{2}(\vec{\sigma}_i\cdot\vec{\sigma}_j+1)\,.</math>
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| Its eigenvalues are 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.
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| == SU(2) ==
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| The group [[SU(2)]] is the [[Lie group]] of [[unitary matrix|unitary]] 2×2 matrices with unit determinant; its [[Lie algebra]] is the set of all 2×2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the [[Lie algebra]] '''su'''(2) is the 3-dimensional real algebra [[Linear span|spanned]] by the set { {{math|''iσ<sub>j</sub>''}} }. In compact notation,
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| :<math> \mathfrak{su}(2) = \operatorname{span} \{ i \sigma_1, i \sigma_2 , i \sigma_3 \}.</math>
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| As a result, each {{math|''iσ<sub>j</sub>''}} can be seen as an [[Lie group#The Lie algebra associated with a Lie group|infinitesimal generator]] of SU(2). The elements of SU(2) are
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| exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector.
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| As SU(2) is a compact group, its [[Cartan decomposition]] is trivial.
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| === SO(3) ===
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| The Lie algebra '''su'''(2) is [[isomorphism|isomorphic]] to the Lie algebra '''so'''(3), which corresponds to the Lie group [[Rotation group SO(3)|SO(3)]], the [[group (mathematics)|group]] of [[rotation]]s in three-dimensional space. In other words, one can say that the ''iσ<sub>j</sub>'' are a realization (and, in fact, the lowest-dimensional realization) of ''infinitesimal'' rotations in three-dimensional space. However, even though '''su'''(2) and '''so'''(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a [[Double covering group|double cover]] of SO(3), meaning that there is a two-to-one group homomorphism from SU(2) to SO(3).
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| === Quaternions ===
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| The real linear span of {''I'', ''iσ''<sub>1</sub>, ''iσ''<sub>2</sub>, ''iσ''<sub>3</sub>} is isomorphic to the real algebra of [[quaternions]] '''H'''. The isomorphism from '''H''' to this set is given by the following map (notice the reversed signs for the Pauli matrices):
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| :<math>
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| 1 \mapsto I, \quad
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| i \mapsto - i \sigma_1, \quad
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| j \mapsto - i \sigma_2, \quad
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| k \mapsto - i \sigma_3.
| |
| </math> | |
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| Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,<ref>{{citation | first=Mikio |last=Nakahara | title=Geometry, topology, and physics | edition=2nd | publisher=CRC Press | isbn=978-0-7503-0606-5 | year=2003}}, [http://books.google.com/books?id=cH-XQB0Ex5wC&pg=PR22 pp. xxii].</ref>
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| :<math>
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| 1 \mapsto I, \quad
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| i \mapsto i \sigma_3, \quad
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| j \mapsto i \sigma_2, \quad
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| k \mapsto i \sigma_1.
| |
| </math>
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| | |
| As the quaternions of unit norm is group-isomorphic to SU(2), this gives yet another way of describing SU(2) via the Pauli matrices. The two-to-one homomorphism from SU(2) to SO(3) can also be explicitly given in terms of the Pauli matrices in this formulation.
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| Quaternions form a [[division algebra]]—every non-zero element has an inverse—whereas Pauli matrices do not. For a quaternionic version of the algebra generated by Pauli matrices see [[biquaternion]]s, which is a venerable algebra of eight real dimensions.
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| == Physics ==
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| === Quantum mechanics ===
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| In [[quantum mechanics]], each Pauli matrix is related to an [[operator (physics)|operator]] that corresponds to an [[observable]] describing the [[spin (physics)|spin]] of a [[spin-½|spin ½]] particle, in each of the three spatial directions. Also, as an immediate consequence of the Cartan decomposition mentioned above, {{math|''iσ<sub>j</sub>''}} are the generators of rotation acting on [[theory of relativity|non-relativistic]] particles with spin ½. The [[mathematical formulation of quantum mechanics|state]] of the particles are represented as two-component [[spinor]]s. An interesting property of spin ½ particles is that they must be rotated by an angle of 4{{mvar|π}} in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the [[2-sphere]] {{mvar|S}} <sup>2</sup>, they are actually represented by [[orthogonal]] vectors in the two dimensional complex [[Hilbert space]].
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| For a spin ½ particle, the spin operator is given by {{math|'''''J'''''{{=}}{{sfrac|''ħ''|2}}'''''σ'''''}}. It is possible to construct the [[spin operator]]s for higher spin systems in three spatial dimensions. For arbitrarily large ''j'', the spin matrices can be calculated using the [[spin operator]] and [[Ladder operator#Angular momentum|ladder operators]]. For example, the spin matrices for spin 1 and spin {{sfrac|3|2}} are
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| <math>j=1</math>:
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| :<math>
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| J_x = \frac{\hbar}{\sqrt{2}}
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| \begin{pmatrix}
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| 0&1&0\\
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| 1&0&1\\
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| 0&1&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| J_y = \frac{\hbar}{\sqrt{2}}
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| \begin{pmatrix}
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| 0&-i&0\\
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| i&0&-i\\
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| 0&i&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| J_z = \hbar
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| \begin{pmatrix}
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| 1&0&0\\
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| 0&0&0\\
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| 0&0&-1
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| \end{pmatrix}
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| </math>
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| <math>j=\textstyle\frac{3}{2}</math>:
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| :<math>
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| J_x = \frac\hbar2
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| \begin{pmatrix}
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| 0&\sqrt{3}&0&0\\
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| \sqrt{3}&0&2&0\\
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| 0&2&0&\sqrt{3}\\
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| 0&0&\sqrt{3}&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| J_y = \frac\hbar2
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| \begin{pmatrix}
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| 0&-i\sqrt{3}&0&0\\
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| i\sqrt{3}&0&-2i&0\\
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| 0&2i&0&-i\sqrt{3}\\
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| 0&0&i\sqrt{3}&0
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| \end{pmatrix}
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| </math>
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| :<math>
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| J_z = \frac\hbar2
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| \begin{pmatrix}
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| 3&0&0&0\\
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| 0&1&0&0\\
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| 0&0&-1&0\\
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| 0&0&0&-3
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| \end{pmatrix}.
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| </math>
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| Also useful in the [[quantum mechanics]] of multiparticle systems, the general [[Pauli group]] {{math|''G<sub>n</sub>''}} is defined to consist of all {{mvar|n}}-fold [[tensor]] products of Pauli matrices.
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| === Quantum information ===
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| *In [[quantum information]], single-[[qubit]] [[quantum gate]]s are ''2'' × ''2'' unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the ''Z–Y decomposition of a single-qubit gate''. Choosing a different Cartan pair gives a similar ''X–Y decomposition of a single-qubit gate''.
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| == See also ==
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| * [[Angular momentum]]
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| * [[Gell-Mann matrices]]
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| * [[Poincaré group]]
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| * [[Generalizations of the Pauli matrices]]
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| * [[Bloch sphere]]
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| == Notes ==
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| <references/>
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| == References ==
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| *{{cite book | author=[[Liboff, Richard L.]] | title=Introductory Quantum Mechanics | publisher=Addison-Wesley | year=2002 | isbn=0-8053-8714-5}}
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| *{{cite book | author=Schiff, Leonard I. | title=Quantum Mechanics | publisher=McGraw-Hill | year=1968 | id=ISBN 007-Y85643-5}}
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| *{{cite book | author=Leonhardt, Ulf | title=Essential Quantum Optics | publisher=Cambridge University Press | year=2010 | isbn=0-521-14505-8}}
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| [[Category:Lie groups]]
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| [[Category:Matrices]]
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| [[Category:Rotational symmetry]]
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| [[Category:Articles containing proofs]]
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