HW 03: Difference between revisions

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==Problem==
==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 _n b_n \phi_n (t)\,\!</math>, then show
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 _n b_n \phi_n (t)\,\!</math>, then show

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


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

==Solution==
#<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \left ( \sum _n b_n \phi_n (t) \right )^*\,dt</math>
#<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \left ( \sum _n a_n \phi_n (t) \right )^*\,dt</math>

Revision as of 14:48, 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