= Infinite square well
{title2=$E_n=n^2\pi^2\hbar^2/(2mL^2)$}
{wiki=Particle_in_a_box}
= Particle in a box
{synonym}
The <infinite square well> is an ideal confinement model with <potential energy> $V(x)=0$ for $0<x<L$ and infinite walls outside. The walls impose <Dirichlet boundary conditions> $\phi(0)=\phi(L)=0$ and a vanishing exterior <wavefunction>. Inside, the time-independent <Schrödinger equation> is
$$
-\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2}=E\phi,\qquad \phi_n(x)=\sqrt{\frac2L}\sin\frac{n\pi x}{L},\qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2},\quad n=1,2,\ldots.
$$
The first wall removes the <cosine> term; the second requires $\sin(kL)=0$, hence $k=n\pi/L$. The $n=0$ solution is identically zero and cannot satisfy <wavefunction normalization>. Each <energy eigenstate> has $n-1$ interior <nodes of a standing wave>, at $x=jL/n$ for $j=1,\ldots,n-1$. More curvature costs more <kinetic energy>, explaining the $n^2$ growth. The positive <ground-state energy> is a confinement effect: a constant zero-energy <wavefunction> cannot satisfy both walls and remain normalized.
For an <electron> with mass $m_e=9.1093837139\times10^{-31}\,\mathrm{kg}$ in a box of length $L=1\,\mathrm{nm}$, use $h=6.62607015\times10^{-34}\,\mathrm{J\,s}$ and $1\,\mathrm{eV}=1.602176634\times10^{-19}\,\mathrm J$. These constants are from the 2022 CODATA adjustment. <Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>{ref}. Substitution gives
$$
E_1=\frac{h^2}{8m_eL^2}=6.024667\times10^{-20}\,\mathrm J\approx0.376030\,\mathrm{eV},\qquad E_2=4E_1\approx1.504121\,\mathrm{eV}.
$$
Doubling $L$ quarters every <energy eigenvalue>; doubling $m$ halves them. This scaling makes the <particle in a box> useful for estimating confinement energies before attempting a more realistic calculation. A <finite square well> permits exterior tails for <bound states> and has only finitely many bound <energy eigenstates>. For an <electron>, the scalar box calculation assumes fixed <spin> and no spin-dependent interactions.
For the <ground state>, the probability of finding the particle in the middle half of the box is fully computable:
$$
\Pr\left(\frac L4<x<\frac{3L}4\right)=\int_{L/4}^{3L/4}\frac2L\sin^2\frac{\pi x}{L}\,dx=\left[\frac xL-\frac{\sin(2\pi x/L)}{2\pi}\right]_{L/4}^{3L/4}=\frac12+\frac1\pi\approx0.818310.
$$
Thus half the available length contains about $81.8\%$ of the probability. The figure contrasts the signed <wavefunctions> with their nonnegative <probability densities>: a negative lobe is an amplitude with opposite <quantum phase>, not negative probability.
\Image[/quantum-box-eigenstates.png]
{title=Ideal infinite square well: normalized modes, nodes, and analytic position densities}
An equal <quantum superposition> of the first two <energy eigenstates> gives a simple example of moving density:
$$
\psi(x,t)=\frac{\phi_1(x)e^{-iE_1t/\hbar}+\phi_2(x)e^{-iE_2t/\hbar}}{\sqrt2},\qquad |\psi|^2=\frac{\phi_1^2+\phi_2^2}{2}+\phi_1\phi_2\cos\frac{(E_2-E_1)t}{\hbar}.
$$
Its density period is $T=h/(E_2-E_1)\approx3.66608\,\mathrm{fs}$ for the <electron> box above. At $t=0$ the cross term enhances the left side; at $T/4$ it vanishes; at $T/2$ it enhances the right side. More quantitatively,
$$
\Pr(0<x<L/2)=\frac12+\frac4{3\pi}\cos\frac{2\pi t}{T},
$$
which takes the values $0.924413$, $0.5$, and $0.075587$ at those times. <Orthogonality> keeps the total probability one throughout. An <energy measurement> still returns $E_1$ or $E_2$ with equal probabilities: changing position density arises from relative <quantum phase>, not changing populations. A <mixed state> describing a statistical mixture with those populations has the constant <probability density> $(\phi_1^2+\phi_2^2)/2$; it lacks this <matter-wave interference> term.
\Image[/quantum-box-superposition.png]
{title=Ideal box superposition: relative phase changes analytic density while total probability stays one}
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