HW 03: Difference between revisions

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



{| border="0" cellpadding="0" cellspacing="0"
{| border="0" cellpadding="0" cellspacing="0"
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|-
|<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _n a_n \phi_n (t)^* \,dt</math>
|<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _m a_m \phi_m (t)^* \,dt</math>
|<math>=\sum_n \sum _n a_n a_n^* \int_{-\infty}^{\infty} \phi_n (t) \phi_n (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 a_n a_n^* \left \langle \phi_n (t) | \phi_n (t)^* \right \rangle</math>
|<math>=\sum_n \sum _m a_n a_m^* \left \langle \phi_n (t) | \phi_n (t)^* \right \rangle</math>
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|<math>=\sum_n a_n a_n^* \delta_{nn^*}</math>
|<math>=\sum_n \sum _m a_n a_m^* \delta_{nm}</math>
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Revision as of 16:42, 12 November 2008

Problem

If and span the space of functions for which and are members and and , then show

Notes

  • This notation is called the Bra Ket , or Dirac notation. It denotes the inner product.

Solution

  • Note