# HW 03

## Problem

If $\left \langle \phi_n | \phi_m \right \rangle = \delta_{mn}$ and $\phi_n \,\!$ span the space of functions for which $x(t)\,\!$ and $y(t)\,\!$ are members and $x(t)= \sum _n a_n \phi_n (t)\,\!$ and $y(t)= \sum _m b_m \phi_m (t)\,\!$, then show

1. $\left \langle x | y \right \rangle = \sum_n a_n b_n^*$
2. $\left \langle x | x \right \rangle = \sum_n \left | a_n \right |^2$

## Notes

$\left \langle x | y \right \rangle = \int_{-\infty}^{\infty}x(t)y(t)^*\,dt$

• This notation is called the Bra $\langle\phi|$ Ket $|\psi\rangle$, or Dirac notation. It denotes the inner product.

## Solution

 $\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _m b_m \phi_m (t)^* \,dt$ $=\sum_n \sum _m a_n b_m^* \int_{-\infty}^{\infty} \phi_n (t) \phi_m (t)^* \,dt$ $=\sum_n \sum _m a_n b_m^* \left \langle \phi_n (t) | \phi_m (t) \right \rangle$ $=\sum_n \sum _m a_n b_m^* \delta_{nm}$ $=\sum_n a_n b_n^*$
 $\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _m a_m \phi_m (t)^* \,dt$ $=\sum_n \sum _m a_n a_m^* \int_{-\infty}^{\infty} \phi_n (t) \phi_m (t)^* \,dt$ $=\sum_n \sum _m a_n a_m^* \left \langle \phi_n (t) | \phi_m (t) \right \rangle$ $=\sum_n \sum _m a_n a_m^* \delta_{nm}$ $=\sum_n \left | a_n \right |^2$