- #1
giova7_89
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"Continuous eigenstates" vs "discrete eigenstates"
There's this thing that's bothering me: if I have an Hamiltonian with a discrete and continuous spectrum, every book I read on quantum mechanics says that eigenvectors of discrete eigenvalues are orthogonal in the "Kronecker sense" (their scalar product is a Kronecker delta) and that eigenvectors of continuous eigenvalues are orthogonal in the "Dirac sense" (their scalar product is a Dirac delta). But what about the scalar product between a continuous eigenstate and a discrete eigenstate? Is it 0 or not?
There's this thing that's bothering me: if I have an Hamiltonian with a discrete and continuous spectrum, every book I read on quantum mechanics says that eigenvectors of discrete eigenvalues are orthogonal in the "Kronecker sense" (their scalar product is a Kronecker delta) and that eigenvectors of continuous eigenvalues are orthogonal in the "Dirac sense" (their scalar product is a Dirac delta). But what about the scalar product between a continuous eigenstate and a discrete eigenstate? Is it 0 or not?