in order to fine out the matrix elements, we just need to know 2 equations
The calculation is straight forward, but be careful of the sign.
i define the coefficient :
and the matrix coefficients are:
the Kronecker Delta indicated that only 1st upper and lower diagonal elements are non-zero.
δμ(ν-1) means the 1st upper diagonal elements. since ν = μ+1 to make it non-zero.
To compute , we just need to multiply the upper diagonal with and the lower diagonal with .
The coefficient is every interesting. if we only look at the first upper diagonal. and take the square of each element.
and we use this to form a bigger matrix
if we read from the top right hand corner, and take the diagonal element. we can see that they fit the 1st upper diagonal element of , if we take square root of each one.
and the pattern are just the multiplication table! How nice~
so, i don’t have to compute for j = 5/2.
but the physical reason for this trick , i don’t know. for the Pascal triangle, we understand the reason for each element – it is the multiplicity.