View source for HW 03

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

#<math> \left \langle x | y \right \rangle = \sum_n a_n b_n^*</math>
#<math> \left \langle x | x \right \rangle = \sum_n \left | a_n \right |^2</math>

==Notes==
<math> \left \langle x | y \right \rangle = \int_{-\infty}^{\infty}x(t)y(t)^*\,dt</math>
*This notation is called the Bra <math> \langle\phi| </math> Ket <math>|\psi\rangle</math>, or Dirac notation. It denotes the inner product.

==Solution==
{| border="0" cellpadding="0" cellspacing="0"
|-
|<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _m b_m \phi_m (t)^* \,dt</math>
|<math>=\sum_n \sum _m a_n b_m^* \int_{-\infty}^{\infty} \phi_n (t) \phi_m (t)^* \,dt</math>
|-
|
|<math>=\sum_n \sum _m a_n b_m^* \left \langle \phi_n (t) | \phi_m (t) \right \rangle</math>
|-
|
|<math>=\sum_n \sum _m a_n b_m^* \delta_{nm}</math>
|-
|
|<math>=\sum_n a_n b_n^*</math>
|}

{| border="0" cellpadding="0" cellspacing="0"
|-
|<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _m a_m \phi_m (t)^* \,dt</math>
|<math>=\sum_n \sum _m a_n a_m^* \int_{-\infty}^{\infty} \phi_n (t) \phi_m (t)^* \,dt</math>
|-
|
|<math>=\sum_n \sum _m a_n a_m^* \left \langle \phi_n (t) | \phi_m (t) \right \rangle</math>
|-
|
|<math>=\sum_n \sum _m a_n a_m^* \delta_{nm}</math>
|-
|
|<math>=\sum_n \left | a_n \right |^2</math>
|}