Matrix with fractions for indices?

AI Thread Summary
Matrices can indeed have fractional indices, as they are simply a representation of operations on vector components. The discussion highlights that the presence of fractional values like -3/2, -1/2, 1/2, and 3/2 does not invalidate the use of matrices. Concerns about fractional indices may stem from the perception that matrices represent discrete operators, while they can also represent continuous ones. The equation provided may need clarification, particularly regarding its completeness and the nature of the eigenvalue representation. Overall, fractional indices in matrices are valid and can be utilized effectively in quantum mechanics contexts.
Shawnyboy
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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
 
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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?
 
Maybe it is because your matrix represents a continuous operator hence having continuous indices ?
 
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.
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