Position and Momentum operator in Legendre base

[Mengahna]

First of all, I don't really know if this problem corresponds to this section, but anyway I have this as a probem in my matemathical physics class.

The problem is stated something like this:

Find the matrix elements of the position and momentum operators in the legendre base (on the interval [-1, 1]).

Ok, I know how to proceed...but I'm having a hard time integrating. Can someone help me with this?... or at least give me the correct answer just to know if I'm doing right.

Sorry for the bad english, I'm not a native speaker. Also I'm new in this so I don't know if I'm posting in the right forum...please tell me.

 

[BigJDubs]

The position and momentum operators can be written as, Position operator: $\hat{x}=\int_{-1}^{1}x|x\rangle\langle x|dx$Momentum operator: $\hat{p}=-i\hbar\int_{-1}^{1}\frac{\partial}{\partial x}|x\rangle\langle x|dx$The matrix elements of the operators in the Legendre base are given by, Position operator: $\langle n|\hat{x}|m\rangle=\int_{-1}^{1}xP_n(x)P_m(x)dx$Momentum operator: $\langle n|\hat{p}|m\rangle=-i\hbar\int_{-1}^{1}\frac{\partial}{\partial x}P_n(x)P_m(x)dx$The integrals can be evaluated using the following recursive relation, $\int_{-1}^{1} xP_n(x)P_m(x)dx = \frac{2}{2n+1}\left[ (n+m+1)\int_{-1}^{1} P_{n+1}(x)P_m(x)dx + (n-m)\int_{-1}^{1} P_{n-1}(x)P_m(x)dx \right]$In a similar way, the integral for the momentum operator can be evaluated using the following recursive relation, $\int_{-1}^{1}\frac{\partial}{\partial x}P_n(x)P_m(x)dx = \frac{2}{2n+1}\left[ (n+m+1)\int_{-1}^{1} \frac{\partial}{\partial x}P_{n+1}(x)P_m(x)dx - (n-m)\int_{-1}^{1} \frac{\partial}{\partial x}P_{n-1}(x)P_m(x)