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# Lect3_Basis - Representations in Hilbert Space A...

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Representations in Hilbert Space A representation is a choice of orthonormal basis in Hilbert space analogous to choosing a coordinate system in 3D space A basis is a complete set of unit vectors that spans the state space Basis sets come in two flavors: ‘discrete’ and ‘continuous’ A discrete basis is what we have been considering so far. The unit vectors can be labeled by integers, e.g. {|1 ! , |2 ! ,…, | M ! } , where M can be either finite or infinite The number of basis vectors is either finite or ‘countable infinity’. A continuous basis is a generalization whereby the unit vectors are labeled by real numbers, e.g. {| x ! }; x min < x < x max , where the upper and lower bounds can be either finite or infinite The number of basis vectors is `uncountable infinity’. Properties of basis vectors Matrix element operator expansion projector component/ wavefunction state expansion normalization orthogonality continuous discrete property jk k j ! = ) ( x x x x !

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Lect3_Basis - Representations in Hilbert Space A...

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