Pauli spin matrices: Difference between revisions
imported>Michael Hardy No edit summary |
imported>Meg Taylor No edit summary |
||
(11 intermediate revisions by 5 users not shown) | |||
Line 1: | Line 1: | ||
The '''Pauli spin matrices''' are a set of unitary [[Hermitian matrix|Hermitian matrices]] which form an orthogonal basis (along with the identity matrix) for the real [[Hilbert space]] of 2 × 2 Hermitian matrices and for the complex Hilbert spaces of all | {{subpages}} | ||
<br/> | |||
{{TOC|right}} | |||
The '''Pauli spin matrices''' (named after physicist [[Wolfgang Ernst Pauli]]) are a set of unitary [[Hermitian matrix|Hermitian matrices]] which form an orthogonal basis (along with the [[identity matrix]]) for the real [[Hilbert space]] of 2 × 2 Hermitian matrices and for the complex Hilbert spaces of all 2 × 2 matrices. They are usually denoted:<ref name= Reiher> | |||
{{cite book |title=Relativistic quantum chemistry: the fundamental theory of molecular science |author=Markus Reiher, Alexander Wolf |url=http://books.google.com/books?id=u47v2YmR-P8C&pg=PA141 |pages=p. 141 |isbn=3527312927 |year=2009 |publisher= Wiley-VCH}} | |||
</ref> | |||
: <math>\sigma_x=\begin{pmatrix} | : <math>\sigma_x=\begin{pmatrix} | ||
0 & 1 \\ | 0 & 1 \\ | ||
1 & 0 | 1 & 0 | ||
\end{pmatrix}, | \end{pmatrix}, \quad | ||
\sigma_y=\begin{pmatrix} | \sigma_y=\begin{pmatrix} | ||
0 & -\mathit{i} \\ | 0 & -\mathit{i} \\ | ||
\mathit{i} & 0 | \mathit{i} & 0 | ||
\end{pmatrix}, | \end{pmatrix}, \quad | ||
\sigma_z=\begin{pmatrix} | \sigma_z=\begin{pmatrix} | ||
1 & 0 \\ | 1 & 0 \\ | ||
Line 15: | Line 22: | ||
==Algebraic properties== | ==Algebraic properties== | ||
: <math>\sigma_x^2=\sigma_y^2=\sigma_z^2=I</math> | : <math>\sigma_x^2=\sigma_y^2=\sigma_z^2=I</math> | ||
Line 26: | Line 32: | ||
:<math>\mbox{eigenvalues}=\pm 1\,</math> | :<math>\mbox{eigenvalues}=\pm 1\,</math> | ||
==Commutation relations== | |||
:<math>\sigma_1\sigma_2 = i\sigma_3\,\!</math> | :<math>\sigma_1\sigma_2 = i\sigma_3\,\!</math> | ||
:<math>\sigma_3\sigma_1 = i\sigma_2\,\!</math> | :<math>\sigma_3\sigma_1 = i\sigma_2\,\!</math> | ||
Line 43: | Line 49: | ||
The above two relations can be summarized as: | The above two relations can be summarized as: | ||
:<math>\sigma_i \sigma_j = \delta_{ij} \cdot I + i \varepsilon_{ijk} \sigma_k \,</math>. | :<math>\sigma_i \sigma_j = \delta_{ij} \cdot I + i \varepsilon_{ijk} \sigma_k. \,</math> | ||
==Rotations== | |||
The commutation relations for the Pauli spin matrices can be rearranged as: | |||
:<math> \frac{1}{2}\sigma_{\alpha} \frac{1}{2}\sigma_{\beta} -\frac{1}{2}\sigma_{\beta} \frac{1}{2}\sigma_{\alpha} =i \ \varepsilon_{\alpha \beta \gamma} \frac{1}{2}\sigma_{\gamma} \ , </math> | |||
with αβγ any combination of ''xyz''. | |||
These commutation relations are the same as those satisfied by the generators of infinitesimal rotations in three-dimensional space. If the Pauli matrices are considered to act on a two-dimensional "spin" space, finite rotations in this space can be connected to rotations in three-dimensional space. These spin-space rotations are ''generated'' by the Pauli spin matrices, with a finite rotation in three-space of angle θ about the axis aligned with unit vector '''û''' becoming in spin-space the rotation: | |||
:<math>R_{\hat{\mathbf u}}(\theta) = e^{i\theta {\hat{\mathbf u}}\mathbf{\cdot \boldsymbol{ \sigma}}/2} . </math> | |||
If '''û''' is in the ''z'' direction, for example: | |||
:<math>R_z(\theta)= \begin{pmatrix} | |||
e^{i \theta/2} & 0\\ | |||
0 & e^{-i \theta/2} | |||
\end{pmatrix} \ , </math> | |||
as can be verified using the [[Taylor series]] expansion: | |||
:<math>e^{i\theta \sigma_z/2}= 1 + i\theta {\sigma_z}/2 +\frac{1}{2} \left( i\theta {\sigma_z}/2\right)^2 ... </math> | |||
Given a set of [[Euler angles]] α, β, γ describing orientation of an object in ordinary three-dimensional space, the general spin-space rotation corresponding to these angles is described as:<ref name=Weber> | |||
{{cite book |title=Essential mathematical methods for physicists |author=Hans-Jurgen Weber, George Brown Arfken |url=http://books.google.com/books?id=k046p9v-ZCgC&pg=PA241 |pages=p. 241 |isbn=0120598779 |publisher=Academic Press |year=2004 |edition=5th ed}} | |||
</ref> | |||
:<math>\begin{pmatrix} | |||
e^{i \gamma/2} & 0\\ | |||
0 & e^{-i \gamma/2} | |||
\end{pmatrix} \begin{pmatrix} | |||
\cos (\beta /2) & \sin(\beta /2)\\ | |||
-\sin(\beta /2) & \cos (\beta /2) | |||
\end{pmatrix} \begin{pmatrix} | |||
e^{i \alpha/2} & 0\\ | |||
0 & e^{-i \alpha/2} | |||
\end{pmatrix}</math> | |||
::::::<math> =\begin{pmatrix} | |||
e^{i (\alpha +\gamma)/2}\cos (\beta /2) & e^{-i (\alpha -\gamma)/2} \sin(\beta /2)\\ | |||
-e^{i (\alpha -\gamma)/2} \sin(\beta /2) & e^{-i (\alpha +\gamma)/2}\cos (\beta /2) | |||
\end{pmatrix} \ . </math> | |||
The two-dimensional matrices describing all rotations in "spin space" form a representation of the special unitary group of transformations of two complex variables, usually denoted as [[SU(2)]], formally described as the [[group theory|group of transformations]] of two-dimensional complex vectors leaving their [[inner product]] fixed, and hence also the norm of a complex vector.<ref name= Mirman> | |||
For example, see {{cite book |title=Group Theory: An Intuitive Approach |author=R. Mirman |chapter=§X.5 The unitary, unimodular group SU(2) |pages=pp. 284 ''ff'' |url=http://books.google.com/books?id=LFakv848R2oC&pg=PA284 |isbn=9810233655 |year=1995 |publisher=World Scientific}} | |||
</ref> | |||
===Extensions=== | |||
These commutation relations can be viewed as applying in general, and the question opened as to what general mathematical objects might satisfy these rules. A set of symbols with a defined sum and a product taken as a commutator of the symbols is called a [[Lie algebra]].<ref name=LieAlgebra> For a mathematical discussion see {{cite book |title=Group Theory: An Intuitive Approach |author=R. Mirman |chapter=§X.7 Angular momentum operators and their algebra |isbn=9810233655 |year=1997 |pages =pp. 292 ''ff'' |url=http://books.google.com/books?id=LFakv848R2oC&pg=PA292 |publisher=World Scientific Publishing Company}} Matrices satisfying the commutation rules are called a ''matrix representation'' of the Lie algebra. See {{cite book |title=Dynamical groups and spectrum generating algebras, vol. 1 |author=BG Adams, J Cizek, J Paldus |editor=Arno Böhm ''et al.'' |edition=Reprint of [http://books.google.com/books?hl=en&lr=&id=sGsR18jJujYC&oi=fnd&pg=PA1 article] in ''Advances in Quantum Chemistry'', vol. 19, Academic Press, 1987 |url=http://books.google.com/books?id=KANTJADFKG4C&pg=PA114 |pages=pp. 114 ''ff'' |chapter= §2.2 Matrix representation of a Lie algebra |isbn=9971501473 |year=1987 |publisher=World Scientific}} | |||
</ref> | |||
One can construct sets of square matrices of various dimensions that satisfy these commutation rules; each set is a so-called ''representation'' of the rules. One finds that there are many such sets, but they can be sorted into two kinds: irreducible and reducible. The reducible sets of matrices can be shown to be equivalent to matrices with smaller irreducible matrices down the diagonal, so that the rules are satisfied within these smaller constituent matrices, and the entire matrix is not essential. The irreducible sets cannot be arranged this way.<ref name=representations> | |||
For a discussion see {{cite book |title=The theory of groups and quantum mechanics |author=Hermann Weyl |isbn=0486602699 |year=1950 |publisher=Courier Dover Publications |edition=Reprint of 1932 ed |url=http://books.google.com/books?id=jQbEcDDqGb8C&pg=PA185 |pages=pp. 185 ''ff'' |chapter=Chapter IV A §1 ''The representation induced in system space by the rotation group''}}, or {{cite book |title =Rotational spectroscopy of diatomic molecules |author=John M. Brown, Alan Carrington |url=http://books.google.com/books?id=TU4eA7MoDrQC&pg=PA143 |chapter=§5.2.4 Representations of the rotation group |pages=pp. 143 ''ff'' |isbn=0521530784 |publisher=Cambridge University Press |year=2003}} | |||
</ref> The Pauli matrices are the basis for an irreducible two-dimensional representation of this Lie algebra. | |||
==Notes== | |||
{{reflist}} | |||
Latest revision as of 05:49, 17 October 2013
![](http://s9.addthis.com/button1-share.gif)
The Pauli spin matrices (named after physicist Wolfgang Ernst Pauli) are a set of unitary Hermitian matrices which form an orthogonal basis (along with the identity matrix) for the real Hilbert space of 2 × 2 Hermitian matrices and for the complex Hilbert spaces of all 2 × 2 matrices. They are usually denoted:[1]
Algebraic properties
For i = 1, 2, 3:
Commutation relations
The Pauli matrices obey the following commutation and anticommutation relations:
- where is the Levi-Civita symbol, is the Kronecker delta, and I is the identity matrix.
The above two relations can be summarized as:
Rotations
The commutation relations for the Pauli spin matrices can be rearranged as:
with αβγ any combination of xyz.
These commutation relations are the same as those satisfied by the generators of infinitesimal rotations in three-dimensional space. If the Pauli matrices are considered to act on a two-dimensional "spin" space, finite rotations in this space can be connected to rotations in three-dimensional space. These spin-space rotations are generated by the Pauli spin matrices, with a finite rotation in three-space of angle θ about the axis aligned with unit vector û becoming in spin-space the rotation:
If û is in the z direction, for example:
as can be verified using the Taylor series expansion:
Given a set of Euler angles α, β, γ describing orientation of an object in ordinary three-dimensional space, the general spin-space rotation corresponding to these angles is described as:[2]
The two-dimensional matrices describing all rotations in "spin space" form a representation of the special unitary group of transformations of two complex variables, usually denoted as SU(2), formally described as the group of transformations of two-dimensional complex vectors leaving their inner product fixed, and hence also the norm of a complex vector.[3]
Extensions
These commutation relations can be viewed as applying in general, and the question opened as to what general mathematical objects might satisfy these rules. A set of symbols with a defined sum and a product taken as a commutator of the symbols is called a Lie algebra.[4]
One can construct sets of square matrices of various dimensions that satisfy these commutation rules; each set is a so-called representation of the rules. One finds that there are many such sets, but they can be sorted into two kinds: irreducible and reducible. The reducible sets of matrices can be shown to be equivalent to matrices with smaller irreducible matrices down the diagonal, so that the rules are satisfied within these smaller constituent matrices, and the entire matrix is not essential. The irreducible sets cannot be arranged this way.[5] The Pauli matrices are the basis for an irreducible two-dimensional representation of this Lie algebra.
Notes
- ↑ Markus Reiher, Alexander Wolf (2009). Relativistic quantum chemistry: the fundamental theory of molecular science. Wiley-VCH, p. 141. ISBN 3527312927.
- ↑ Hans-Jurgen Weber, George Brown Arfken (2004). Essential mathematical methods for physicists, 5th ed. Academic Press, p. 241. ISBN 0120598779.
- ↑ For example, see R. Mirman (1995). “§X.5 The unitary, unimodular group SU(2)”, Group Theory: An Intuitive Approach. World Scientific, pp. 284 ff. ISBN 9810233655.
- ↑ For a mathematical discussion see R. Mirman (1997). “§X.7 Angular momentum operators and their algebra”, Group Theory: An Intuitive Approach. World Scientific Publishing Company, pp. 292 ff. ISBN 9810233655. Matrices satisfying the commutation rules are called a matrix representation of the Lie algebra. See BG Adams, J Cizek, J Paldus (1987). “§2.2 Matrix representation of a Lie algebra”, Arno Böhm et al.: Dynamical groups and spectrum generating algebras, vol. 1, Reprint of article in Advances in Quantum Chemistry, vol. 19, Academic Press, 1987. World Scientific, pp. 114 ff. ISBN 9971501473.
- ↑ For a discussion see Hermann Weyl (1950). “Chapter IV A §1 The representation induced in system space by the rotation group”, The theory of groups and quantum mechanics, Reprint of 1932 ed. Courier Dover Publications, pp. 185 ff. ISBN 0486602699. , or John M. Brown, Alan Carrington (2003). “§5.2.4 Representations of the rotation group”, Rotational spectroscopy of diatomic molecules. Cambridge University Press, pp. 143 ff. ISBN 0521530784.