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I know when you have a system where the energy is discreet (i.e. a bound state), there is an discreet orthonormal base, and you can developpe(?) an arbitrary function in this base:

f(x)=Sum{a[n]*f[n](x)}. And you can find the a[n] by multipling with f[n'] and integrate over x. Then you get a[n']=Int{dx*f(x)*f[n'](x)}.

Now I have a continu(?) base and I can do:

f(x)=Int{dk*a(k)*f[k](x)}.

From now on I begin to doubt:

If I multiply with f[k'](x), the RHS becomes:

Int{dk*a(k)*delta(k-k'))=a(k') and thus:

a(k')=f(x)*f[k'](x)

where did I screw up??

f(x)=Sum{a[n]*f[n](x)}. And you can find the a[n] by multipling with f[n'] and integrate over x. Then you get a[n']=Int{dx*f(x)*f[n'](x)}.

Now I have a continu(?) base and I can do:

f(x)=Int{dk*a(k)*f[k](x)}.

From now on I begin to doubt:

If I multiply with f[k'](x), the RHS becomes:

Int{dk*a(k)*delta(k-k'))=a(k') and thus:

a(k')=f(x)*f[k'](x)

where did I screw up??

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