HW 03

Problem

If $⟨{\varphi }_n|{\varphi }_m⟩={\delta }_{mn}$ and ${\varphi }_n$ span the space of functions for which $x(t)$ and $y(t)$ are members and $x(t)=\sum _n{a}_n{\varphi }_n(t)$ and $y(t)=\sum _m{b}_m{\varphi }_m(t)$, then show

1. $⟨x|y⟩=\sum _n{a}_n{b}_n^{*}$
2. $⟨x|x⟩=\sum _n{\left|a_n\right|}^2$

Notes

$⟨x|y⟩=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)y(t{)}^{*}dt$

• This notation is called the Bra $⟨\varphi |$ Ket $|\psi ⟩$, or Dirac notation. It denotes the inner product.

Solution

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\sum _n{a}_n{\varphi }_n(t)\sum _m{b}_m{\varphi }_m(t{)}^{*}dt$	$=\sum _n\sum _m{a}_n{b}_m^{*}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\varphi }_n(t){\varphi }_m(t{)}^{*}dt$
	$=\sum _n\sum _m{a}_n{b}_m^{*}⟨{\varphi }_n(t)|{\varphi }_m(t{)}^{*}⟩$
	$=\sum _n\sum _m{a}_n{b}_m^{*}{\delta }_{n{m}^{*}}$
	$=\sum _n{a}_n{b}_n^{*}$

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

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\sum _n{a}_n{\varphi }_n(t)\sum _m{a}_m{\varphi }_m(t{)}^{*}dt$	$=\sum _n\sum _m{a}_n{a}_m^{*}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\varphi }_n(t){\varphi }_m(t{)}^{*}dt$
	$=\sum _n\sum _m{a}_n{a}_m^{*}⟨{\varphi }_n(t)|{\varphi }_n(t{)}^{*}⟩$
	$=\sum _n\sum _m{a}_n{a}_m^{*}{\delta }_{nm}$
	$=\sum _n{\left|a_n\right|}^2$