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:

where

the Kronecker Delta indicated that only 1st upper and lower diagonal elements are non-zero.

δ_{μ(ν-1) }means the 1st upper diagonal elements. since ν = μ+1 to make it non-zero.

For example:

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.

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binarias

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