HW 03

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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