for discrete eigenstates $|\psi _{i}\rangle$ forming a discrete basis, so any state is a
$|\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle$ where c_i are complex numbers such that |c_i|² = c_i^*c_i is the probability of measuring the state $|\phi _{i}\rangle$ , and the corresponding set of eigenvalues a_i is also discrete - either finite or countably infinite. In this case, the inner product of two eigenstates is given by $\langle \phi _{i}\vert \phi _{j}\rangle =\delta _{ij}$ , where $\delta _{mn}$ denotes the Kronecker Delta. However,

sum

for a continuum of eigenstates $|\psi _{i}\rangle$ forming a continuous basis, any state is an
$|\psi \rangle =\int c(\phi )\,d\phi |\phi \rangle$ where c(φ) is a complex function such that |c(φ)|² = c(φ)^*c(φ) is the probability of measuring the state $|\phi \rangle$ , and there is an uncountably infinite set of eigenvalues a. In this case, the inner product of two eigenstates is defined as $\langle \phi '\vert \phi \rangle =\delta (\phi -\phi ')$ , where here $\delta (x-y)$ denotes the Dirac Delta.

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integral