In a Hilbert-space whose dimensionality is either finite or countably infinite, we have the discrete resolution of identity(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\sum_n |n\rangle \langle n| = 1

[/tex]

In many cases, for example to obtain the wavefunctions of the discrete states, one employs the continuous form of the resolution of identity, namely

[tex]

\int dx |x\rangle \langle x| = 1

[/tex]

It doesn't seem quite valid to apply a continous operator in a discrete Hilbert space. How can one justify it?

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# Continuous resolution of identity in a discrete Hilbert-space

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