- #1

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## Homework Statement

For the space L

_{w}

^{2}(a,b), we can write the basis in a discrete fashion as {e

_{n}|n∈ℤ} or in a continuous fashion as |x> (as we would in quantum mechanics for the position representation), such that we may write the identity operator as either

I=∑

_{n}|e

_{n}><e

_{n}|

or

I=∫

_{a}

^{b}dxw(x)|x><x|

Show that these are identity operators, and hence show that δ(x-x

_{0})/√[w(x)w(x

_{0})]=∑

_{n}e

_{n}(x)e

_{n}

^{*}(x

_{0}).

## The Attempt at a Solution

I have done the first bit. For the discrete form,

<f|I|g>=<f|∑

_{n}|e

_{n}><e

_{n}|g>

=<f|∑

_{n}g

_{n}|e

_{n}>

where g

_{n}is the coordinate of |g> with respect to the orthonormal basis vector |e

_{n}>

=<f|g>

For the continuous form,

<f|I|g>=<f|∫

_{a}

^{b}dxw(x)|x><x|g>

=<f|∫

_{a}

^{b}dxg(x)w(x)|x>

as <x|g>=g(x) (note the |g>=∫

_{a}

^{b}g(x)w(x)|x>dx in this representation)

=<f|g>

I'm struggling with the second bit. The RHS makes me think I need to consider <x|x

_{0}> and write

<x|x

_{0}>=<x|I|x

_{0}>

and then use the discrete form of I so that

=∑

_{n}<x|e

_{n}><e

_{n}|x

_{0}>

=∑

_{n}e

_{n}(x)e

_{n}

^{*}(x

_{0})

which sorts out the RHS. This should equal <x|I|x

_{0}> with the continuous identity inserted, but I get

<x|x

_{0}>=<x|∫

_{a}

^{b}dx'w(x')|x'><x'|x

_{0}>

=<x|∫

_{a}

^{b}dx'w(x')|x'>δ(x'-x

_{0})

using the continuous orthonormality condition so <x|x'>=δ(x-x')

=<x|w(x)|x

_{0}>

because of the delta function property, so then I obtain

=w(x)δ(x-x

_{0})

again using orthonormality. So I have

w(x)δ(x-x

_{0})=∑

_{n}e

_{n}(x)e

_{n}

^{*}(x

_{0})

so my LHS has failed somewhere. Can anyone help, thanks :)