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v_pino
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Homework Statement
1. Mixed Spectrum
The finite square well has a mixed spectrum or a mixed set of basis functions. The set of
eigenfunctions that corresponds to the bound states are discrete (call this set {ψ_i(x)}) and
the set that corresponds to the scattering states are continuous (call this set {ψ_k(x)}). Thus
the complete set of basis functions are {ψ_i(x), ψ_k(x)}. Write down for this set of basis
functions
a) the orthonormal condition;
b) the completeness condition;
c) the expansion for an arbitrary wavefunction ψ(x) in terms of the basis functions; and
d) the expressions for the expansion coefficients in part c).
Homework Equations
An orthonormal basis may be formed from a linear combination of basis vectors.
We can write vectors as expansions of orthonormal basis so that
[tex] |a>=\sum_{n}^{i=1}a_i|i> [/tex]
[tex] <a|b>=\sum_{n}^{i,j=1}a_i^*b_j<i|j>=\sum_{n}{i}a_i^*b_i [/tex]
The Attempt at a Solution
I am having trouble starting this problem. In (a), do I simply normalize the wavefunctions given?