Prove. Pauli matrices.

Homework Statement

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

Homework Equations

$$e^{\hat{A}}=\hat{1}+\hat{A}+\frac{\hat{A}^2}{2!}+...$$

The Attempt at a Solution

$$\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\hat{1}+\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x+\frac{1}{2!}(\alpha^2\hat{\sigma}_z^2+\beta^2\hat{\sigma}_x^2+\alpha \beta \hat{\sigma}_x\hat{\sigma}_z+\alpha \beta \hat{\sigma}_z\hat{\sigma}_x)+...$$
from that
$$\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\hat{1}+\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x+\frac{1}{2!}(\alpha^2\hat{1}+\beta^2\hat{1}+\alpha \beta \hat{\sigma}_x\hat{\sigma}_z+\alpha \beta \hat{\sigma}_z\hat{\sigma}_x)+...$$
Is this way to go? I'm not sure?

Related Advanced Physics Homework Help News on Phys.org
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 $\sigma_x \sigma_z + \sigma_z \sigma_x$.

[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.

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?

Hi LagrangeEuler!

The series above simplifies drastically when you figure out what to do with $\sigma_x \sigma_z + \sigma_z \sigma_x$.

[edit: sorry Bloby, posted over you by mistake]
Tnx a lot.

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: