Orthogonality of momentum space wavefunctions

[ehrenfest]

Page 152 Robinett:

Consider the (non-normalized) even momentum space wavefunctions for the symmetric well:,

[tex]\phi_n^+(p) = 2sin(w-m)/(w-m)+sin(w+m)/(w+m)[/tex] where
w = sin((n-1/2)pi) and
m = ap/hbar.

Show that

[tex] \int_{-\infty}^{\infty}\phi_n^+(p)^*\cdot \phi_n^+(p) dp = \delta_{n,m} [/tex]

The hint is to use partial fractions to rewrite the product found in the denominators and then use an integral table.

So, there are there are terms in the expansion of that integrand. Do I need to rewrite all of them in terms of partial fractions?

The first is 2sin(w-m)sin(w+m)/(w-m)(w+m), which I am having trouble with partial fractions. I get A=B=0 for the numerators?

 

[olgranpappy]

Page 152 Robinett:

Consider the (non-normalized) even momentum space wavefunctions for the symmetric well:,

[tex]\phi_n^+(p) = 2sin(w-m)/(w-m)+sin(w+m)/(w+m)[/tex] where
w = sin((n-1/2)pi) and
m = ap/hbar.

Show that

[tex] \int_{-\infty}^{\infty}\phi_n^+(p)^*\cdot \phi_n^+(p) dp = \delta_{n,m} [/tex]

The hint is to use partial fractions to rewrite the product found in the denominators and then use an integral table.

So, there are there are terms in the expansion of that integrand. Do I need to rewrite all of them in terms of partial fractions?

The first is 2sin(w-m)sin(w+m)/(w-m)(w+m)...

no, it's 2sin(w-m)(w+m)/(w-m)(w+m).

 

[ehrenfest]

So the integrand is

[tex] \frac{a}{2\pi\hbar} \left(sin^2(w-m)/(w-m) + 2 sin (m-w) sin (w+m)/(w-m)(w+m) + sin^2(w+n)/w+m\right) dp[/tex]

I think I can integrate the squared terms, but I am not sure how to do partial decomposition on the middle term to derive something useful from it.

When I try partial fraction decomposition on that term I get

2 sin(w-m) sin(w+m)/(2n- pi) (1/(w-m) +1/(w+m) )Sorry. The statement of the problem is wrong. (n-1/2)pi not the sine of that.

 

[ehrenfest]

Sorry. The statement of the problem is wrong. (n-1/2)pi not the sine of that.