Show that the coefficients satisfy the following condition

Question

Hi I was asked for my QM homework to show that coefficients $c_n$ in superposition

$$ \psi(x) = \sum_{n=1}^\infty c_n\psi_n(x) $$

Satisfy the following condition

$$ \sum_{n=1}^\infty \lvert(c_n)\rvert^2 = 1 $$

Use $$ \int_0^L\lvert\psi(x)\rvert^2dx=1 $$

Can anyone please point me into good direction I tried to substitute $\psi(x)$ for the given sum but I am stuck in a loop. Thank you for any advice.

Answer 1

Assumptions:

• $\langle \psi, \psi \rangle = 1$
• $\langle \psi_m, \psi_n \rangle = \delta_{mn}$

Thus, $$ 1 = \langle \psi, \psi \rangle = \langle \sum_m c_m \psi_m, \sum_n c_n \psi_n \rangle = \sum_m \sum_n c_m^* c_n \langle \psi_m, \psi_n \rangle = \sum_m \sum_n c_m^* c_n \delta_{mn} = \sum_m c_m^* c_m = \sum_m |c_m|^2 . $$