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How to handle ontinuos spectrum

  • Thread starter jshw
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  • #1
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Homework Statement


In Gasiorowicz's textbook, he provided the schemetic expession of the wave function(95page) ; psi(x) = Sigma[C_n X U_n(x)] + Integral[dE C(E) X u_E(x)]
("X" means multiplication symbol)
In this equation, I think that 1st term(sigma) is the case of the discrete spectrum and 2nd term(integral) is the case of the continuous spectrum.
Textbooks explain many example of bound state(i.e. state having discrete spectrum), but most of them didn't explain about continuos spectrum. I am curious about it.
Would you tell me about refences containing this problem, or tell me some examples of the continuos spectrum?


Homework Equations


psi(x) = Sigma[C_n X U_n(x)] + Integral[dE C(E) X u_E(x)]


The Attempt at a Solution


I tried to find examples of continuos spectrum in many textbooks, but I failed.
 

Answers and Replies

  • #2
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It's indeed the most general expression you can write down.

Take for instance the case of a free particle. In that case the energy eigenfunctions are labeled by the wave number, k, which takes on continous values. You should be able to find this example in pretty much any introductory book... (Griffiths for example)

In systems where there are both bound and unbound states, it are these unbound states which correspond to free particles - and hence the continous spectrum.

Take for example the hydrogen atom. The lowest energy state has a (relative) energy of -13.6 eV. There are infinite number of bound states with an energy <0 eV. But there are also unbound states, namely those with an energy larger than 0 eV and these states form a continous spectrum. So formally, these should be taken into account since all eigenstates together form a complete basis of the Hilbert space.
 

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