HW 03

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

$\int _{-\infty }^{\infty }\sum _{n}a_{n}\phi _{n}(t)\sum _{m}b_{m}\phi _{m}(t)^{*}\,dt$	$=\sum _{n}\sum _{m}a_{n}b_{m}^{*}\int _{-\infty }^{\infty }\phi _{n}(t)\phi _{m}(t)^{*}\,dt$
	$=\sum _{n}\sum _{m}a_{n}b_{m}^{*}\left\langle \phi _{n}(t)|\phi _{m}(t)^{*}\right\rangle$
	$=\sum _{n}\sum _{m}a_{n}b_{m}^{*}\delta _{nm^{*}}$
	$=\sum _{n}a_{n}b_{n}^{*}$

• Note $\delta _{nm^{*}}=\delta _{nm}$

$\int _{-\infty }^{\infty }\sum _{n}a_{n}\phi _{n}(t)\sum _{m}a_{m}\phi _{m}(t)^{*}\,dt$	$=\sum _{n}\sum _{m}a_{n}a_{m}^{*}\int _{-\infty }^{\infty }\phi _{n}(t)\phi _{m}(t)^{*}\,dt$
	$=\sum _{n}\sum _{m}a_{n}a_{m}^{*}\left\langle \phi _{n}(t)|\phi _{n}(t)^{*}\right\rangle$
	$=\sum _{n}\sum _{m}a_{n}a_{m}^{*}\delta _{nm}$
	$=\sum _{n}\left|a_{n}\right|^{2}$