Prove Pauli Matrices: 65-Character Title

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Homework Help Overview

The discussion revolves around proving an expression involving the exponential of a linear combination of Pauli matrices, specifically \(\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)\). The context is within quantum mechanics and linear algebra, focusing on the properties of these matrices.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the series expansion of the exponential function and its simplification using the properties of Pauli matrices. There are inquiries about the expression for powers of the dot product of a unit vector with the Pauli matrices, and some participants express uncertainty about the relevance of certain connections to the main problem.

Discussion Status

There is ongoing exploration of the simplification techniques related to the Pauli matrices. Some participants have offered insights into the anticommutation properties of the matrices, which may lead to cancellations in the series expansion. However, there is no explicit consensus on the best approach yet.

Contextual Notes

Participants are navigating the complexities of the problem, including the need for a deeper understanding of the algebraic properties of the Pauli matrices and their implications for the given expression. Some participants mention potential connections to other mathematical expressions, but these connections are not universally accepted as relevant.

LagrangeEuler
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Homework Statement


Prove
[tex]\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\cosh \sqrt{\alpha^2+\beta^2}+\frac{\sinh \sqrt{\alpha^2+\beta^2}}{\sqrt{\alpha^2+\beta^2}}(\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)[/tex]



Homework Equations


[tex]e^{\hat{A}}=\hat{1}+\hat{A}+\frac{\hat{A}^2}{2!}+...[/tex]



The Attempt at a Solution


[tex]\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\hat{1}+\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x+\frac{1}{2!}(\alpha^2\hat{\sigma}_z^2+\beta^2\hat{\sigma}_x^2+\alpha \beta \hat{\sigma}_x\hat{\sigma}_z+\alpha \beta \hat{\sigma}_z\hat{\sigma}_x)+...[/tex]
from that
[tex]\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\hat{1}+\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x+\frac{1}{2!}(\alpha^2\hat{1}+\beta^2\hat{1}+\alpha \beta \hat{\sigma}_x\hat{\sigma}_z+\alpha \beta \hat{\sigma}_z\hat{\sigma}_x)+...[/tex]
Is this way to go? I'm not sure?
 
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Have you seen an expression for ##(\vec{n}\cdot \hat{\vec{\sigma}})^k##, with ##\vec{n}## a unit vector?

[edit: no problem]
 
Last edited:
Hi LagrangeEuler!

The series above simplifies drastically when you figure out what to do with [itex]\sigma_x \sigma_z + \sigma_z \sigma_x[/itex].

:smile:

[edit: sorry Bloby, posted over you by mistake]
 
Check out the anticommuntation properties {sig_x,sig_z} = 0, so there is a lot of cancelations.
 
bloby said:
Have you seen an expression for ##(\vec{n}\cdot \hat{\vec{\sigma}})^k##, with ##\vec{n}## a unit vector?

[edit: no problem]

I don't see connection with this problem?
 
Oxvillian said:
Hi LagrangeEuler!

The series above simplifies drastically when you figure out what to do with [itex]\sigma_x \sigma_z + \sigma_z \sigma_x[/itex].

:smile:

[edit: sorry Bloby, posted over you by mistake]

Tnx a lot.
 
LagrangeEuler said:
I don't see connection with this problem?

With ##\sqrt{\alpha^2+\beta^2}=a##, ##exp((\beta , 0 , \alpha)\cdot (\hat{\sigma}_x , \hat{\sigma}_y , \hat{\sigma}_z))=exp(\sqrt{\beta^2+\alpha^2}(\hat{n}\cdot \hat{\vec{\sigma}}))=I+a(\hat{n}\cdot\hat{\vec{\sigma}})+\frac{1}{2!}a^2(\quad)^2+...##
 
Last edited:

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