Matrix with fractions for indices?

Tags:
1. May 20, 2015

Shawnyboy

Hi PF Peeps!

Something came up while I was studying for my QM1 class. Basically we want to represent operators as matrices and in one case the matrix element is defined by the formula :

$<m'|m> = \frac{h}{2\pi}\sqrt{\frac{15}{4} - m(m+1)} \delta_{m',m+1}$

But the thing is we know m takes on the fractional values -3/2, -1/2, 1/2, 3/2. So basically my question is simply put: can you have a matrix with fractions for indeces?

Thanks,
Shawn

2. May 20, 2015

sk1105

Yes you can. A matrix is essentially just a compact way of specifying what action to perform on each component of a vector to make another vector, and can contain whatever you want. Is there something you've seen or heard that made you think fractional indices weren't possible?

3. May 22, 2015

jk22

Maybe it is because your matrix represents a continuous operator hence having continuous indices ?

4. May 23, 2015

theodoros.mihos

I think your equation is not complete. If this is an eigenvalues equation then the diagonal matrix must have elements like:
$$\sqrt{\frac{15}{4}-i(i+1)}$$where i the correspond to matrix column/row element by the same index on m values.