How can I exponentiate spin matrices on a 4x1 matrix?

[FunkyDwarf]

Hey guys,

I know you can represent exponentiated matrices as a series if the matrix is nilpotent and it collapses or for other special properties, but say i want to act exp[-i sigmax/hbar] on a 4x1 matrix, err how do i do it? =D I know that sigmax ^n where n is an even number its I but still the series goes on forever doesn't it?

Cheers
-G

 

[George Jones]

What is sigma (or sigmax)? Is it a spin matrix? If it is, why is it 4x1?

 

[FunkyDwarf]

Yeah sorry it is the first spin matrix and it is acting to the right on a column vector with 4 rows although theyve been written as a 2x1 matrix with two equal size matrices in side so i think muiltiplication still works

 

[borgwal]

If you're talking about spin-1/2 particles, then sigma_x is a 2x2 matrix; for spin-1, sigma_x is a 4x4 matrix...

In any case, the series is infinitely long, but that doesn't mean the series doesn't converge! You can write down the series expansion for the number exp(ix) in a similar way and see there are infinitely many terms that, obviously, do converge.

The expansion for exp(i\sigma_x) will look very similar, and converge very similarly.

 

[turin]

spin-1 is usually represented by 3x3 matrices. I don't even think it is possible to represent spin-1 with 4x4 matrices, unless you just put zeros for the extra elements of the generators and a one in the extra diagonal element of the rotation matrix.

funky:
If you're asking what I think you're asking, then consider that you can find recursion relations for the Pauli matrices: sigma^2=1 => sigma^3=sigma. Then, you only have two kinds of terms in the sum: those propotional to the identity matrix, and those proportional to sigma, and they alternate. This is a standard trick; you can find it in most introductions to spin.

EDIT: After reading your post again, maybe you are familiar with this? Then, I'm not sure what your confusion is. You basically end up with two summations, and they converge to cosine and sine.