Finding Value of Phi*k(x) in Basis Function Method

[ohhhnooo]

Psi(x) = Sumationn CnPhin(x)

in order to find Cn, i have to multiply both side of the above equation by Phi*k(x) and take the integral. The result is Ck if n = k. My question is what is the value of Phi*k(x)?

i know that multiplying both side of the equation by Phi*k(x) would make the function orthonormal when n != k, and normalize when n = k. But i don't know how to find it.

 

[dextercioby]

Is this:$$\psi(x)=\sum_{n}C_{n}\phi_{n}(x)$$(1)...?

Is the basis $$\{\phi_{n} \}_{n=1}^{\infty}$$ orthonormal...?If so,then multiply (1) by $\phi_{k}^{*}(x)$,sum after "k" & integrate after "x"...

Daniel.

 

[ohhhnooo]

what is the condition for $$\{\phi_{n} \}_{n=1}^{\infty}$$ to be orthonormal? thanks

 

[Galileo]

what is the condition for $$\{\phi_{n} \}_{n=1}^{\infty}$$ to be orthonormal? thanks

Per definition, if the following holds, the set $\{\phi_{n} \}_{n=1}^{\infty}$ is orthonormal:

$$\langle \phi_i|\phi_j\rangle = \int \limits_{-\infty}^{+\infty}\phi_i^*(x)\phi_j(x)\; dx=\delta_{ij}$$

 

[ohhhnooo]

can you provide an example?

 

[dextercioby]

Hidrogenoid functions are an interesting example.SHO eigenfunctions are other example.Rigid rotator is another example.Infinite square well and so on,and so forth.

Daniel.