Since the Laguerre polynomial is deeply connected to the hydrogen-like electron orbital. The radial solution is

From the normalization condition, we can get the coefficient A

Beside of calculating the normalization factor, the general expectation value also need to evaluation of the integration of the Laguerre polynomial.

In here, we will show the calculation for 3 integrals:

Using the Rodrigues formula

For ,

It is obvious that due to the . Since , then at . Thus,

Using integration by path,

For the same reason, the first term is zero for . Repeat the integration by path times,

When , can go to , then,

When , can only go to . then,

For the similar result as , the result is zero.

To summarize, the following formula suitable for all

To evaluate the integral,

We need the know that the Laguerre polynomial can be expressed as,

Thus, we can evaluate

Then combine everything,

put inside the series expression,

In general, .

For , the sum only has 1 term for

Using the formula, for , we have

To evaluate the expectation value of radial function

We have to calculate the integral,

For , we get the normalization constant,

In general,

For the integral, for

When , when , the integral is zero.

When

The sum only non-zero when or

This shows that the Laguerre polynomial is orthogonal with weighting .

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