the Pauli's 2 x 2 sigma matrices - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Physics : Physics/Psigma

Physics[Psigma] - the Pauli's 2 x 2 sigma matrices

Calling Sequence

Psigma[n]

Parameters

n

-

an integer between 0 and 4, or an algebraic expression representing it, identifying a Pauli matrix

Description

• 

The Psigma[n] command represents the three Pauli matrices; that is, the set of Hermitian and unitary matrices:

Psigma1=0110,Psigma2=0−II0,Psigma3=100−1

  

where I is the imaginary unit (to represent it with a lowercase i, see interface,imaginaryunit). Together with Psigma4=Psigma0, representing the 2 x 2 identity matrix, the Pauli matrices form an orthogonal basis. The Psigman matrices are displayed as σn.

• 

When multiplied by the imaginary unit, these matrices are a realization of the Lie algebra of the SU(2) group, which is isomorphic to the Lie algebra of SO(3). So, the Iσi are also a matrix realization of infinitesimal rotations in 3D space, hence serving as representation for the 3D angular momentum operator in Physics.

• 

The Pauli matrices satisfy the commutation relations σi,σj=2Iεijkσk, where εijk is the Levi-Civita symbol, and i,j,k range from 1 to 3. The σi also satisfy the anticommutation relations σi,σj=2δij, where δij is the Kronecker delta. Those two relations can be written as σiσj=Iεijkσk+δij.

• 

For i from 1 to 3, the Pauli matrices satisfy Detσi=−1,Traceσi=0, and σi2=1 (the 2 x 2 identity matrix), where Det represents the determinant, and Trace represents/computes the trace. In the context of the Physics package (see conventions), you can also use the index 0, as in Psigma0, and it will be automatically mapped into Psigma4.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Psigma1

σ1

(2)

Library:-RewriteInMatrixForm

0110

(3)

Psigma1Psigma2+Psigma2Psigma1

σ1σ2+σ2σ1

(4)

Library:-RewriteInMatrixForm

`.`0110,0II0+`.`0II0,0110

(5)

Library:-PerformMatrixOperations

0000

(6)

Psigma1Psigma1

σ12

(7)

Trace

2

(8)

Psigma1Psigma2

σ1σ2

(9)

Trace

0

(10)

See Also

conventions, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics[*], Physics[Library], Trace

Compatibility

• 

The Physics[Psigma] command was updated in Maple 2019.