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GR9277 #52
Problem
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\prob{52}
A particle of mass m is confined to an infinitely deep square-well potential:
$\begin{eqnarray} V(x)&=&\infty,\;\;x\leq0,x\geq a \\ V(x)&=&0,\;\;0\lt x \lt a\end{eqnarray}$

The normalized eigenfunctions, labeled by the quantum number n, are $\psi_n=\sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}$

The eigenfunctions satisfy the condition $\int_0^a \psi_n^*(x)\psi_l(x)dx=\delta_{nl},\delta_{nl}=1$ if $n=l$, otherwise $\delta_{nl}=0$. This is a statement that the eigenfunctions are

1. solutions to the Schrodinger equation
2. orthonormal
3. bounded
4. linearly dependent
5. symmetric

Quantum Mechanics$\Rightarrow$}Orthonormality

$\langle \psi_m | \psi_n \rangle = \delta_{nm}$

This is the definition of orthonormality, i.e., something that is both orthogonal (self dot others = 0) and normal (self dot self = 1).

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