Question: Is there a way to get the correct solution to a commutator involving powers of a Pauli Matrix?

 

Hello Community,

Wondering if anyone has run into the problem outlined in my code below. I am looking to evalute a commutator involving Pauli spin matrices. One of the matrices is squared. Such a commutator should evaluate to zero. However, the way I implemented it below doesn't give such a simplified result. See Result 7 below. Any suggestions as to how to get Maple to evaluate the exponential before the Commutator? Or to otherwise get to a final, simplified result?

Many thanks.

NULL

with(Physics):

Setup(noncommutativeprefix = {S}):

Setup(dimens = 3, signature = `+`, spacetimeindices = lowercaselatin):

`* Partial match of  'dimens' against keyword 'dimension'`

(1)

Setup(alg = {%Commutator(S[x], S[y]) = Physics:-`*`(I, S[z])})

`* Partial match of  'alg' against keyword 'algebrarules'`

 

[algebrarules = {%Commutator(S[x], S[y]) = I*S[z], %Commutator(S[y], S[z]) = I*S[x], %Commutator(S[z], S[x]) = -I*S[y], %AntiCommutator(S[x], S[x]) = 2, %AntiCommutator(S[y], S[y]) = 2, %AntiCommutator(S[z], S[z]) = 2}]

(2)

Setup(alg = {%Commutator(S[z], S[x]) = -Physics:-`*`(I, S[y])})

`* Partial match of  'alg' against keyword 'algebrarules'`

 

[algebrarules = {%Commutator(S[x], S[y]) = I*S[z], %Commutator(S[y], S[z]) = I*S[x], %Commutator(S[z], S[x]) = -I*S[y], %AntiCommutator(S[x], S[x]) = 2, %AntiCommutator(S[y], S[y]) = 2, %AntiCommutator(S[z], S[z]) = 2}]

(3)

Setup(alg = {%Commutator(S[y], S[z]) = Physics:-`*`(I, S[x])})

`* Partial match of  'alg' against keyword 'algebrarules'`

 

[algebrarules = {%Commutator(S[x], S[y]) = I*S[z], %Commutator(S[y], S[z]) = I*S[x], %Commutator(S[z], S[x]) = -I*S[y], %AntiCommutator(S[x], S[x]) = 2, %AntiCommutator(S[y], S[y]) = 2, %AntiCommutator(S[z], S[z]) = 2}]

(4)

Setup(%AntiCommutator(S[x], S[x]) = 2)

[algebrarules = {%Commutator(S[x], S[y]) = I*S[z], %Commutator(S[y], S[z]) = I*S[x], %Commutator(S[z], S[x]) = -I*S[y], %AntiCommutator(S[x], S[x]) = 2, %AntiCommutator(S[y], S[y]) = 2, %AntiCommutator(S[z], S[z]) = 2}]

(5)

Setup(%AntiCommutator(S[y], S[y]) = 2, %AntiCommutator(S[z], S[z]) = 2)

[algebrarules = {%Commutator(S[x], S[y]) = I*S[z], %Commutator(S[y], S[z]) = I*S[x], %Commutator(S[z], S[x]) = -I*S[y], %AntiCommutator(S[x], S[x]) = 2, %AntiCommutator(S[y], S[y]) = 2, %AntiCommutator(S[z], S[z]) = 2}]

(6)

Commutator(Physics:-`^`(S[x], 2), S[y])

I*(-I*S[y]+2*Physics:-`*`(S[x], S[z]))

(7)

Commutator(2, S[y])

0

(8)

Commutator(Physics:-`^`(S[y], 2), Physics:-`^`(S[x], 2))

-I*(-I*AntiCommutator(S[y], S[y])+2*AntiCommutator(S[y], Physics:-`*`(S[x], S[z])))

(9)

Commutator(S[x], S[y])

I*S[z]

(10)

``

NULL

 

 

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