HW 03 - Revision history

Fonggr: /* Solution */

2008-11-13T01:27:23Z

Solution

← Older revision		Revision as of 18:27, 12 November 2008
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	\|-		\|-
	\|		\|
	\|<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^* \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 \sum _m a_n b_m^* \delta_{nm}</math>
	\|-		\|-
	\|		\|
	\|<math>=\sum_n a_n b_n^*</math>		\|<math>=\sum_n a_n b_n^*</math>
	\|}		\|}
	Note <math>\delta_{nm^}=\delta_{nm}</math>

	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
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	\|-		\|-
	\|		\|
	\|<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^* \left \langle \phi_n (t) \| \phi_m (t) \right \rangle</math>
	\|-		\|-
	\|		\|

Fonggr: /* Solution */

2008-11-12T23:49:42Z

Solution

← Older revision		Revision as of 16:49, 12 November 2008
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	\|-		\|-
	\|		\|
	\|<math>=\sum_n \sum _m a_n a_m^* \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_m (t)^* \right \rangle</math>
	\|-		\|-
	\|		\|

Fonggr at 23:42, 12 November 2008

2008-11-12T23:42:33Z

← Older revision		Revision as of 16:42, 12 November 2008
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	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-
	\|<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>
	\|}		\|}
			Note <math>\delta_{nm^}=\delta_{nm}</math>

	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-
	\|<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>
	\|-		\|-
	\|		\|
	\|<math>=\sum_n a_n ~~a_n~~^* \delta_{nn^*}</math>		\|<math>=\sum_n \sum _m a_n a_m^* \delta_{nm}</math>
	\|-		\|-
	\|		\|
	\|<math>=\sum_n \left \| a_n \right \|^2</math>		\|<math>=\sum_n \left \| a_n \right \|^2</math>
	\|}		\|}

Fonggr at 23:31, 12 November 2008

2008-11-12T23:31:54Z

← Older revision		Revision as of 16:31, 12 November 2008
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	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-
	\|<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 _n a_n \phi_n (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 _n a_n a_n^* \int_{-\infty}^{\infty} \phi_n (t) \phi_n (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 a_n a_n^* \left \langle \phi_n (t) \| \phi_n (t)^* \right \rangle</math>
	\|-		\|-
	\|		\|
	\|<math>=\sum_n ~~\sum _m~~ a_n ~~b_m~~^* \delta_{nm^*}</math>		\|<math>=\sum_n a_n a_n^* \delta_{nn^*}</math>
	\|-		\|-
	\|		\|
	\|<math>=\sum_n a_n ~~b_n~~^*</math>		\|<math>=\sum_n \left \| a_n \right \|^2</math>
	\|}		\|}

Fonggr: /* Solution */

2008-11-12T23:26:41Z

Solution

← Older revision		Revision as of 16:26, 12 November 2008
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	<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _n a_n \phi_n (t)^* \,dt</math>		{\| border="0" cellpadding="0" cellspacing="0"
			\|-
			\|<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _m b_n \phi_n (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>
			\|}

Fonggr at 23:26, 12 November 2008

2008-11-12T23:26:10Z

← Older revision		Revision as of 16:26, 12 November 2008
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	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		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>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>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 \| y \right \rangle = \sum_n a_n b_n^*</math>
Line 14:		Line 14:
	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-
	\|<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _n b_n \phi_n (t)^* \,dt</math>		\|<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \sum _m b_n \phi_n (t)^* \,dt</math>
	\|<math>=\sum_n a_n ~~b_n~~ \int_{-\infty}^{\infty} \phi_n (t) \~~phi_n~~ (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 a_n ~~b_n~~ \left \langle \phi_n (t) \| \~~phi_n~~ (t)^* \right \rangle</math>		\|<math>=\sum_n \sum _m a_n b_m^* \left \langle \phi_n (t) \| \phi_m (t)^* \right \rangle</math>
	\|-		\|-
	\|		\|
	\|<math>=\sum_n a_n ~~b_n~~ \delta_{nn^*}</math>		\|<math>=\sum_n \sum _m a_n b_m^* \delta_{nm^*}</math>
			\|-
			\|
			\|<math>=\sum_n a_n b_n^*</math>
	\|}		\|}


	#<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 _n a_n \phi_n (t)^* \,dt</math>

Fonggr at 23:16, 12 November 2008

2008-11-12T23:16:53Z

@@ Line 12: / Line 12: @@
 ==Solution==
-#<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \left ( \sum _n b_n \phi_n (t) \right )^*\,dt</math>
+{| border="0" cellpadding="0" cellspacing="0"
-#<math>\int_{-\infty}^{\infty} \sum _n a_n \phi_n (t) \left ( \sum _n a_n \phi_n (t) \right )^*\,dt</math>
+|-

Fonggr at 22:48, 12 November 2008

2008-11-12T22:48:09Z

← Older revision		Revision as of 15:48, 12 November 2008
Line 1:		Line 1:
	==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>

Fonggr: /* Notes */

2008-11-12T22:42:38Z

Notes

← Older revision		Revision as of 15:42, 12 November 2008
Line 5:		Line 5:

	==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.

Fonggr: New page: ==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 $...$

2008-11-12T22:35:19Z

New page: ==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>...

New page

==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
#<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>

HW 03 - Revision history

Fonggr: /* Solution */

Fonggr: /* Solution */

Fonggr at 23:42, 12 November 2008

Fonggr at 23:31, 12 November 2008

Fonggr: /* Solution */

Fonggr at 23:26, 12 November 2008

Fonggr at 23:16, 12 November 2008

Fonggr at 22:48, 12 November 2008

Fonggr: /* Notes */

Fonggr: New page: ==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 ...

Fonggr: New page: ==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 $...$