HW 03: Difference between revisions

Revision as of 15:48, 12 November 2008

Problem

If $⟨ ϕ_{n} | ϕ_{m} ⟩ = δ_{m n}$ and $ϕ_{n}$ span the space of functions for which $x (t)$ and $y (t)$ are members and $x (t) = \sum_{n} a_{n} ϕ_{n} (t)$ and $y (t) = \sum_{n} b_{n} ϕ_{n} (t)$ , then show

$⟨ x | y ⟩ = \sum_{n} a_{n} b_{n}^{*}$
$⟨ x | x ⟩ = \sum_{n} {| a_{n} |}^{2}$

Notes

$⟨ x | y ⟩ = \int_{- \infty}^{\infty} x (t) y (t)^{*} d t$

This notation is called the Bra $⟨ ϕ |$ Ket $| ψ ⟩$ , or Dirac notation. It denotes the inner product.

Solution

$\int_{- \infty}^{\infty} \sum_{n} a_{n} ϕ_{n} (t) {(\sum_{n} b_{n} ϕ_{n} (t))}^{*} d t$
$\int_{- \infty}^{\infty} \sum_{n} a_{n} ϕ_{n} (t) {(\sum_{n} a_{n} ϕ_{n} (t))}^{*} d t$

@@ Line 1: / Line 1: @@
 ==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 | 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>\
+<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==
+#<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>

HW 03: Difference between revisions

Revision as of 15:48, 12 November 2008

Problem

Notes

Solution

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