= One-dimensional Anderson chain
{c}
{title2=$H=-t\sum_n(\lvert n\rangle\langle n+1\rvert+\lvert n+1\rangle\langle n\rvert)+\sum_n\epsilon_n\lvert n\rangle\langle n\rvert$}
Consider a noninteracting <Electron> in a finite <tight-binding model> with open ends:
$$
H=-t\sum_{n=1}^{N-1}\bigl(\lvert n\rangle\langle n+1\rvert+\lvert n+1\rangle\langle n\rvert\bigr)+\sum_{n=1}^N\epsilon_n\lvert n\rangle\langle n\rvert,\qquad t>0.
$$
The <vectors> $\lvert n\rangle$ form an <orthonormal basis> of site states. The hopping <energy> is $t$; it is not the time coordinate. Draw each $\epsilon_n$ independently from the <continuous uniform distribution> on $[-W/2,W/2]$ and then hold it fixed. The disorder strength $W/t$ is dimensionless. Open ends mean no hopping bond connects sites $N$ and $1$; equivalently set $\psi_0=\psi_{N+1}=0$. There are no <Electron> interactions, <phonons> or <wave absorption> in this model.
The <eigenvalue equation> in the interior is
$$
-t(\psi_{n-1}+\psi_{n+1})+\epsilon_n\psi_n=E\psi_n.
$$
As a simple local calculation, if a state's peak is at $n_0$, a neighbouring site with $|E-\epsilon_n|\gg t$ receives a small leading contribution $\psi_n\simeq-t\psi_{n_0}/(E-\epsilon_n)$. Several successive large mismatches multiply small factors. This explains why large disorder can restrict spreading, but is not a proof of <Anderson localization>: near-resonant denominators and alternative <interference> paths prevent treating this expansion as uniformly valid.
For the first original calculation, take $N=80$, $t=1$, $W/t=6$ and the fixed random seed $20261015$. Diagonalizing the resulting real <symmetric matrix> with `numpy.linalg.eigh` and selecting the <eigenvalue> closest to zero gives $E/t\simeq-0.05962$, a probability peak at site $29$, and $P\simeq6.0031$. The selected <eigenvector> has total probability one. This is one chosen realization, not a disorder average or a measured specimen.
\Image[/anderson-chain-eigenstate.png]
{title=One random potential and a concentrated <energy eigenstate> in a <One-dimensional Anderson chain>}
{id=anderson-chain-eigenstate}
{description=Original calculation for $80$ sites and $W/t=6$. The top panel shows all site <energies> and the selected <eigenvalue>. The lower panels show the same probability on linear and logarithmic scales, compared with a clean central <standing wave>. No fitted <localization length> is claimed.}
The clean comparison needs care. With $W=0$, the open chain has <standing waves>
$$
\psi_n^{(m)}=\sqrt{\frac{2}{N+1}}\sin\frac{m\pi n}{N+1},\qquad E_m=-2t\cos\frac{m\pi}{N+1},\qquad m=1,\ldots,N.
$$
For even $N$, the identity $\sin^4x=(3-4\cos2x+\cos4x)/8$ and finite trigonometric sums give $\sum_n|\psi_n^{(m)}|^4=3/[2(N+1)]$, hence $P=2(N+1)/3$. Thus the clean $80$-site comparison has $P=54$, although it extends across the chain. Extended does not mean uniform probability. For odd $N$, the special central mode $m=(N+1)/2$ has zeros on alternating sites and instead $P=(N+1)/2$; the even-size assumption avoids that exception.
The second original calculation compares $N=40$ and $80$ over $W/t=0,0.5,1,2,3,4,6,8$. For each size, draw $24$ independent site patterns using seed $20261016$, with the size included in the random-seed construction. Scale each fixed pattern across disorder strengths. At every point, average $P$ over the four <energy eigenstates> closest to $E=0$. The solid line is the median across these sample averages; shading spans their $10$th–$90$th percentiles. The selected states can change as $W/t$ varies: lines connect one sample's selections rather than tracking a single mode.
\Image[/anderson-participation-disorder.png]
{title=<Participation ratio> varies across disorder realizations and finite chain sizes}
{id=anderson-participation-disorder}
{description=Original <One-dimensional Anderson chain> calculations. Faint lines show individual sample averages, solid lines their medians and bands their $10$th–$90$th percentiles. The left panel plots $P$; the right divides by site count $N$. These finite samples illustrate concentration and variability, without a fitted critical disorder.}
At $W/t=0$ the two clean values are $27\frac13$ and $54$. At $W/t=8$, the computed median sample averages are approximately $3.17$ and $3.01$. In the latter regime, doubling $N$ hardly changes the typical occupied site count, so $P/N$ falls. At weak disorder the larger chain permits more spatial extent. Neither a single curve nor the overlap of finite-size bands proves the infinite-system statement under <mobility edge>; a long <localization length> can hide confinement in either size.
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