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Physics
Quantum Physics
Eigenkets belonging to a range of eigenvalues
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[QUOTE="stevendaryl, post: 6018281, member: 372855"] I looked over the thread about Rigged Hilbert Spaces, and I'm not sure that it completely explained the relationship between continuous and discrete bases. In a non-rigorous way, you can think of the continuous basis as a limiting case of the discrete basis. However, in going from discrete to continuous, the normalization convention for basis elements changes. Let me illustrate. Suppose you have an operator ##\Lambda## with discrete eigenvalues ##\lambda_j##. I think in order for the continuum limit to make sense int the most straightforward way, you need to assume that ##\lambda_{j+1} > \lambda_j##, and that there are infinitely many ##\lambda_j##, and that the corresponding eigenstates ##|n\rangle## form a complete orthonormal basis. That means that [LIST=1] [*]If ##n \neq m##, then ##\langle n|m\rangle = 0## [*]##\langle n|n\rangle = 1## [*]If ##|\psi\rangle## is a properly normalized state, then ##|\psi\rangle = \sum_n \langle n|\psi\rangle |n\rangle## [*]##\sum_n |\langle n|\psi\rangle|^2 = 1## [/LIST] Now, if the coefficients ##\langle n|\psi\rangle## change slowly with ##n## (and maybe we also have to assume that ##(\Delta \lambda)_n \equiv \lambda_{n+1} - \lambda_n## remains bounded? I'm not sure...) then we can define a new ket with a different normalization: ##|\lambda_n\rangle \equiv \frac{1}{\sqrt{(\Delta \lambda)_n}} |n\rangle## In terms of the ##|\lambda_n\rangle##, we have: ##|\psi\rangle = \sum_n (\Delta \lambda)_n \langle \lambda_n |\psi\rangle |\lambda_n\rangle## The kets ##|\lambda_n\rangle## have a different normalization: [LIST] [*]##\langle \lambda_n | \lambda_m \rangle = 0## (if ##m \neq n##) [*]##\langle \lambda_n | \lambda_n \rangle = \frac{1}{(\Delta \lambda)_n}## [/LIST] If the states ##|\lambda_n\rangle## change smoothly with ##n##, then this can be approximated by an integral: ##|\psi\rangle = \int d\lambda \langle \lambda |\psi\rangle |\lambda\rangle## [/QUOTE]
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Eigenkets belonging to a range of eigenvalues
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