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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?