Commuting Pauli Matrices: A Tricky Homework Challenge

[unscientific]

Homework Statement

Express the product

where σ^y and σ^z are the other two Pauli matrices defined above.

Homework Equations

The Attempt at a Solution

I'm not sure if this is a trick question, because right away both exponentials combine to give 1, where the result is simply σ^x

 

[kevinferreira]

That's the whole point of the exercise, to see that the exponentials do not combine to give 1. To do such a thing, you would have to pass [itex]exp(i\alpha\sigma^z)[/itex] to the other side of [itex]\sigma^x[/itex]. But [itex]\left[\sigma^z,\sigma^x\right]\neq0[/itex] so that you can't simply commute them.

Hint: express the exponential as a series.

 

[unscientific]

That's the whole point of the exercise, to see that the exponentials do not combine to give 1. To do such a thing, you would have to pass [itex]exp(i\alpha\sigma^z)[/itex] to the other side of [itex]\sigma^x[/itex]. But [itex]\left[\sigma^z,\sigma^x\right]\neq0[/itex] so that you can't simply commute them.

Hint: express the exponential as a series.

Ah, I see what you mean, as the sum is a matrix and not a number. Silly me.