# PauliMatrix

PauliMatrix[k]

gives the k 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:
•  TargetStructure Automatic the structure of the returned matrix WorkingPrecision Infinity precision at which to create entries
• 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 , a dense matrix is returned.

# Examples

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## Basic Examples(1)

Generate Pauli matrices:

## 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, :

Wolfram Research (2008), PauliMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/PauliMatrix.html (updated 2023).

#### Text

Wolfram Research (2008), PauliMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/PauliMatrix.html (updated 2023).

#### CMS

Wolfram Language. 2008. "PauliMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. 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_2023_paulimatrix, author="Wolfram Research", title="{PauliMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/PauliMatrix.html}", note=[Accessed: 16-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_paulimatrix, organization={Wolfram Research}, title={PauliMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/PauliMatrix.html}, note=[Accessed: 16-April-2024 ]}