Problem with the Laguerre polynomials

[Logarythmic]

My task is to explicitly write down the first three Laguerre polynomials by using a power series ansatz.
What should this ansatz look like? Should it be the Rodrigues representation

$$L_n (x) = \frac{e^x}{n!} \frac{d^n}{dx^n} x^n e^{-x}$$

?

 

[dicerandom]

Sounds to me like they either want you to solve the DE using the method of series solutions (see, e.g., Chapter 12 of Boas' Mathematical Methods for Physicists) or that they want you to generate the polynomials from a power series of the generating function (see (19) & (20) here).

 

[Logarythmic]

I guess it should be the method with the generating function

$$g(x,z) = \frac{e^{- \frac{xz}{1-z}}}{1-z} = \sum_{n \geq 0} a_n L_n (x) z^n$$

but how do I get rid of the z?

 

[dicerandom]

Taylor expand the generating function in powers of $z$ and the coefficients of the $z^n$ are the Laguerre polynomials. The additional factor of $a_n$ is present because Wolfram is using a definition of the Laguerre polynomials that does not include the $1/n!$ term in your definition, so you should absorb that $a_n$ term into the $L_n$ for your purposes.

 

[Logarythmic]

Yes, I got the correct answer. The second part of the exercise tells me to use the normalization

$$<y_n|y_m>_{e^{-x}} = \int_0^\infty e^{-x}y_n(x)y_m(x)dx = \delta_{nm}$$

to determine the overall constant.

What constant?

 

[dicerandom]

The $\delta_{nm}$ is 1 if n=m and 0 otherwise. So if you perform that integral with n=m the result you get should be equal to one. Using this condition will allow you to determine the normalization constant for the polynomials (i.e. what you have to multiply $L_n$ by in order to satisfy the equation).

This Kronecker delta property is extremely important when using the Laguerre polynomials as an eigenbasis for quantum mechanical states. It allows us to think of the set of Laguerre polynomials as an orthonomal vector space which we can then use as an eigenbasis to represent wavefunctions. The integral you're computing is the dot product between the basis vectors in the Laguerre polynomial space.

 

[Logarythmic]

Yes I'm aware of all this, but if I solve the integral I get:

For n = m = 0 and n = m = 1, I = 1.
For n = m = 2, I = 2.

Where is the overall constant in this case?

 

[dicerandom]

You should do the integral for arbitrary n and see what you get. Alternatively you can try doing it for a few more specific values of n and see if you notice a pattern.

 

[Logarythmic]

Thanks for your help!