Numerical Homework $1$

a$.$ Consider a string of identical point masses $m$ along the $x-$axis, each two masses

g connected by an internal spring with spring constant $k$ to model a diatomic

molecule. Let $\frac{N}{2}$ be the total number of these two$-$atom molecules that are then

connected by external springs having spring constant $k_{0}d()()n=1\mathrm{.}2$dots. for $k_0=1$ then plot ${}_n=1{}_n-\frac{{}_2}{a}s$ a function $of\frac{N}{2}$ for

$n=10,20,N.$ What happen $to$ these differences $asN$ becomes very large. Estimate

the energy gap, $,$ the region where there $isno$ eigenfrequency mode for large $N.$

Find the corresponding "density $of$ eigenvalues" defined $by$:

$()=\frac{d}{d}$

numerically through

for $k_0=0$ and

for $k_0=1$ then plot ${}_n=1{}_n-\frac{{}_2}{a}s$ a function $of\frac{N}{2}$ for

$n=10,20,N.$ What happen $to$ these differences $asN$ becomes very large. Estimate

the energy gap, $,$ the region where there $isno$ eigenfrequency mode for large $N.$

Find the corresponding "density $of$ eigenvalues" defined $by$:

$()=\frac{d}{d}$

numerically through

$()=\frac{(+d)-()}{d}$

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