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cianfa72

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- How to express quantum particle's state in momentum eigenfunctions basis considering the fact that momentum eigenfunctions are not square-integrable

Hi, as discussed in this recent thread, for a particle without spin the quantum state of the particle is described by a "point" in the Hilbert space of the (equivalence classes) of ##L^2## square-integrable functions ##|{\psi} \rangle## defined on ##\mathbb R^3##.

The square-integrable condition for complex-valued function ##f## means that its square ##|f|^2## has finite Lebesgue integral on the measurable space ##(\mathbb R^3, \mathcal A)## where ##\mathcal A## is the sigma-algebra of Lebesgue measurable sets (or perhaps simply the Borel sigma algebra ##\mathcal B(\mathbb R^3)##).

That said, consider the eigenfunctions ##|{\psi_k} \rangle## of momentum operator ##\vec{P}##. Now the Lebesgue integral of each of their equivalence classes is

How do we cope with this ? Thanks.

The square-integrable condition for complex-valued function ##f## means that its square ##|f|^2## has finite Lebesgue integral on the measurable space ##(\mathbb R^3, \mathcal A)## where ##\mathcal A## is the sigma-algebra of Lebesgue measurable sets (or perhaps simply the Borel sigma algebra ##\mathcal B(\mathbb R^3)##).

That said, consider the eigenfunctions ##|{\psi_k} \rangle## of momentum operator ##\vec{P}##. Now the Lebesgue integral of each of their equivalence classes is

*not*finite.How do we cope with this ? Thanks.

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