PauliMatrix [k]
gives the k^(th) Pauli spin matrix .
PauliMatrix
PauliMatrix [k]
gives the k^(th) Pauli spin matrix .
Details and Options
- PauliMatrix gives 2×2 constant matrices with the property .
- PauliMatrix [0] and PauliMatrix [4] give the identity matrix.
- The following options can be given:
-
- Possible settings for TargetStructure include:
-
Automatic automatically choose the representation returned"Dense" represent the matrix as a dense matrix"Hermitian" represent the matrix as a Hermitian matrix"Sparse" represent the matrix as a sparse array"Unitary" represent the matrix as a unitary matrix
- With the setting TargetStructure Automatic , a dense matrix is returned.
Examples
open all close allBasic Examples (1)
Generate Pauli matrices:
Scope (1)
PauliMatrix threads element-wise over lists:
Options (6)
TargetStructure (4)
Return the Pauli matrix as a dense matrix:
Return the Pauli matrix as a sparse array:
Return the Pauli matrix as a Hermitian matrix:
Return the Pauli matrix as a unitary matrix:
WorkingPrecision (2)
Create a machine-precision Pauli matrix:
Create an arbitrary-precision Pauli matrix:
Applications (4)
Pauli's differential equation:
Pauli matrices' algebra:
Build a unitary matrix representing the rotation of the spinor around the axis through angle :
Rotation by 360° changes the spinor's direction:
In quantum mechanics, systems with finitely many states are represented by unit vectors and physical quantities by matrices that act on them. Consider a spin-1/2 particle such as an electron in the following state:
The operator for the component of angular momentum is given by the following matrix:
Compute the expected angular momentum in this state as :
The uncertainty in the angular momentum is :
The uncertainty in the component of angular momentum is computed analogously:
The uncertainty principle gives a lower bound on the product of uncertainties, :
Related Guides
Text
Wolfram Research (2008), PauliMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/PauliMatrix.html (updated 2024).
CMS
Wolfram Language. 2008. "PauliMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/PauliMatrix.html.
APA
Wolfram Language. (2008). PauliMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PauliMatrix.html
BibTeX
@misc{reference.wolfram_2025_paulimatrix, author="Wolfram Research", title="{PauliMatrix}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/PauliMatrix.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_paulimatrix, organization={Wolfram Research}, title={PauliMatrix}, year={2024}, url={https://reference.wolfram.com/language/ref/PauliMatrix.html}, note=[Accessed: 16-November-2025]}