Particle in a box

Intuition [Beginner]

Imagine a marble rolling inside a pipe. The pipe has hard walls at both ends. The marble bounces back and forth, never escaping. In classical mechanics the marble has a definite speed and position at every instant, and it can have any kinetic energy at all — slow, fast, anything in between.

In quantum mechanics this same setup — a particle trapped between two impenetrable walls — behaves differently. The particle is described by a wave function $\psi (x)$ spread across the inside of the box, and the boundary conditions at the walls force the wave function to be zero at both ends. Only certain wave shapes fit: those that go to zero at the walls and have the right number of half-wavelengths in between. Each allowed wave shape corresponds to a specific energy, and those energies form a discrete ladder — numbered $n=1,2,3,\dots$

The key word is discrete. A particle in a box cannot have arbitrary energy. It can only sit on one of the rungs of a ladder, and the rungs are spaced according to $E_n=n^2{\pi }^2{\hslash }^2/(2m{L}^2)$ where $m$ is the particle's mass and $L$ is the box length. The lowest rung, $n=1$, is called the ground state. It has a nonzero energy — the particle is never at rest inside the box. This minimum energy is called zero-point energy.

The wave function for the $n$-th rung looks like a standing wave with $n$ half-wavelengths fitting inside the box: ${\psi }_n(x)=\sqrt{2/L}\sin (n\pi x/L)$. The squareroot factor ensures the total probability sums to one. The probability of finding the particle near position $x$ is $|{\psi }_n(x){|}^2=(2/L){\sin }^2(n\pi x/L)$, which is not uniform — it has peaks and nodes (flat spots where the probability is zero).

For an electron confined to a box of length $L=1$ nanometre, the ground-state energy works out to about 0.376 eV. The first excited state ($n=2$) has four times this energy — about 1.504 eV. The spacing between adjacent levels grows as $n$ increases, which is the signature of the $n^2$ dependence.

The particle in a box is the simplest quantum system that shows quantisation — the replacement of a continuous range of energies by a discrete set. Every other quantised system (atoms, molecules, quantum dots) uses the same mechanism: the wave function is forced to satisfy boundary conditions, and only certain energies produce wave functions that satisfy those conditions.

A general state of the particle is not restricted to a single rung. The particle can be in a superposition of several energy eigenstates, each oscillating at its own frequency. The coefficients in the superposition determine the probability of measuring each energy. This time-dependent picture — the general solution as a superposition of stationary states — is what connects the static energy ladder to actual dynamics.

Visual [Beginner]

The picture has two parts. On the left, the energy ladder: equally spaced labels $n=1,2,3,\dots$ but with energies $E_n\propto n^2$ — the rungs get farther apart as you go up. On the right, the corresponding wave functions ${\psi }_n(x)$ drawn as standing waves inside the box. Each wave function has $n$ half-wavelengths and $n-1$ interior nodes — points where the probability of finding the particle is exactly zero.

The ground state ($n=1$) has no interior node: the particle can be found anywhere, with highest probability near the centre. The first excited state ($n=2$) has one node at $x=L/2$: the particle is never found at the exact midpoint. Higher states have more nodes. This node-counting pattern — more nodes means higher energy — holds for every one-dimensional quantum system, not just the box.

Worked example [Beginner]

An electron ($m=9.109\times 10^{-31}$ kg) is confined to a box of length $L=1$ nm $=10^{-9}$ m.

Ground-state energy. The formula gives

$$E_1=\frac{{\pi }^2{\hslash }^2}{2m{L}^2}=\frac{{\pi }^2(1.055\times 10^{-34}{)}^2}{2\times 9.109\times 10^{-31}\times (10^{-9}{)}^2}\approx 6.024\times 10^{-20}\text{ J}.$$

Converting to electron-volts ($1\text{ eV}=1.602\times 10^{-19}$ J):

$$E_1\approx \frac{6.024\times 10^{-20}}{1.602\times 10^{-19}}\approx 0.376\text{ eV}.$$

First three energy levels. Since $E_n=n^2{E}_1$:

$$E_1\approx 0.376\text{ eV},E_2=4E_1\approx 1.504\text{ eV},E_3=9E_1\approx 3.384\text{ eV}.$$

Probability in the left half. For the ground state, the probability density $|{\psi }_1(x){|}^2=(2/L){\sin }^2(\pi x/L)$ is symmetric about $x=L/2$, so the probability of finding the electron in the left half $[0,L/2]$ is exactly $1/2$.

For the first excited state, $|{\psi }_2(x){|}^2=(2/L){\sin }^2(2\pi x/L)$. This has a node at $x=L/2$, and the density is again symmetric, so the probability in the left half is also exactly $1/2$. The shape differs — $|{\psi }_2{|}^2$ has two equal peaks, one in each half — but the total probability on each side matches.

The difference shows up if you ask about smaller regions. The probability of finding the ground-state electron in the central quarter $[3L/8,5L/8]$ is substantially higher than the probability of finding the first-excited-state electron in the same region, because the ground-state density peaks at the centre while the first excited state has a node there.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A particle of mass $m$ moves in one dimension under the potential

$$V(x)=\{\begin{array}{ll}0& \text{if }0<x<L,\\ +\infty & \text{otherwise.}\end{array}$$

The infinite potential at the walls means the wave function must vanish outside the box and be continuous at the boundaries. The Hilbert space is $\mathcal{H}=L^2([0,L])$, the space of square-integrable functions on $[0,L]$ with the standard inner product $⟨\varphi |\psi ⟩={\int }_0^L{\varphi }^{\ast }(x)\psi (x)dx$.

Inside the box the Schrödinger equation reduces to the free-particle equation (unit 12.03.01):

$$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi .$$

This is a second-order linear ODE with constant coefficients. The general solution is

$$\psi (x)=A\sin (kx)+B\cos (kx),k=\frac{\sqrt{2mE}}{\hslash }.$$

Boundary conditions. The infinite walls impose Dirichlet conditions $\psi (0)=0$ and $\psi (L)=0$.

At $x=0$: $\psi (0)=B=0$, so $B=0$ and $\psi (x)=A\sin (kx)$.

At $x=L$: $\psi (L)=A\sin (kL)=0$. The solution $A=0$ gives $\psi =0$ everywhere — not a valid state. The nonzero solution requires $\sin (kL)=0$, hence

$$kL=n\pi ,n=1,2,3,\dots$$

(The value $n=0$ gives $\psi =0$ again; negative $n$ gives the same physical state as positive $n$ up to a sign.) Substituting $k=\sqrt{2mE}/\hslash$ and solving for $E$:

$$E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2},n=1,2,3,\dots$$

The constant $A$ is fixed by normalisation: ${\int }_0^L|A{|}^2{\sin }^2(n\pi x/L)dx=|A{|}^{2}L/2=1$, so $A=\sqrt{2/L}$ (choosing the phase convention $A>0$). The energy eigenfunctions are

$${\psi }_n(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right).$$

Orthonormality and completeness. The eigenfunctions satisfy

$$⟨{\psi }_m|{\psi }_n⟩={\int }_0^L{\psi }_m^{\ast }(x){\psi }_n(x)dx={\delta }_{mn}.$$

They form a complete orthonormal basis of $L^2([0,L])$: every square-integrable function on $[0,L]$ can be expanded as a convergent series $f(x)={\sum }_{n=1}^{\infty }{c}_n{\psi }_n(x)$. Completeness follows from the general theory of Sturm-Liouville problems (the functions ${\psi }_n$ are eigenfunctions of a self-adjoint operator on $[0,L]$ with Dirichlet boundary conditions).

Time evolution. Each energy eigenstate picks up a phase at its own frequency:

$${\Psi }_n(x,t)={\psi }_n(x)e^{-i{E}_{n}t/\hslash }.$$

A general state is a superposition

$$\Psi (x,t)=\sum _{n=1}^{\infty }{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash },\sum _n|c_n{|}^2=1,$$

where $c_n=⟨{\psi }_n|\Psi (0)⟩$ are set by the initial condition.

Domain considerations

The Hamiltonian $H=-{\hslash }^2/(2m)\cdot d^2/d{x}^2$ is an unbounded operator. Its domain is not all of $L^2([0,L])$ but the dense subspace

$$D(H)=\{\psi \in H^2([0,L]):\psi (0)=\psi (L)=0\},$$

where $H^2$ is the Sobolev space of functions with square-integrable second derivatives. On this domain $H$ is self-adjoint (not merely symmetric), which guarantees a real spectrum and unitary time evolution via Stone's theorem. The Dirichlet boundary conditions are part of the domain specification, not an afterthought.

Counterexamples to common slips

• The energy $E_n$ is purely kinetic — there is no potential energy inside the box. The kinetic energy is not $\frac{1}{2}m{v}^2$ with a classical bouncing velocity; it is the eigenvalue of the operator $-{\hslash }^2/(2m)d^2/d{x}^2$.
• The wave function ${\psi }_n(x)$ is not a snapshot of a particle trajectory. It is a standing wave, the amplitude for position measurements. The probability density $|{\psi }_n(x){|}^2$ has $n-1$ interior nodes — locations where the particle is never found.
• The ground-state energy is nonzero. The value $E_0=0$ would require $n=0$ and hence $\psi =0$, which is not a valid state. The uncertainty principle ($\Delta x\sim L$ forces $\Delta p\gtrsim \hslash /(2L)$, hence $E\gtrsim {\hslash }^2/(2m{L}^2)$) gives the correct order of magnitude.

Key theorem with proof [Intermediate+]

Theorem (Spectrum of the Dirichlet Laplacian on $[0,L]$). The operator $H=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}$ on $L^2([0,L])$ with domain $D(H)=\{\psi \in H^2([0,L]):\psi (0)=\psi (L)=0\}$ is self-adjoint. Its spectrum is purely discrete:

$$\sigma (H)=\left\{E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}:n=1,2,3,\dots \right\},$$

with corresponding eigenfunctions ${\psi }_n(x)=\sqrt{2/L}\sin (n\pi x/L)$ forming a complete orthonormal basis of $L^2([0,L])$.

Proof.

Step 1: Self-adjointness. For $\psi ,\varphi \in D(H)$, integrate by parts twice:

$$⟨\varphi ,H\psi ⟩=-\frac{{\hslash }^2}{2m}{\int }_0^L{\varphi }^{\ast }{\psi }^{\prime \prime }dx=-\frac{{\hslash }^2}{2m}[{\varphi }^{\ast }{\psi }^{\prime }{]}_0^L+\frac{{\hslash }^2}{2m}{\int }_0^L({\varphi }^{\ast }{)}^{\prime }{\psi }^{\prime }dx.$$

The boundary term vanishes because $\varphi (0)=\varphi (L)=0$ (both functions are in $D(H)$). A second integration by parts gives $\frac{{\hslash }^2}{2m}{\int }_0^L({\varphi }^{\ast }{)}^{\prime \prime }\psi dx=⟨H\varphi ,\psi ⟩$, again with no boundary contribution. So $⟨\varphi ,H\psi ⟩=⟨H\varphi ,\psi ⟩$ — $H$ is symmetric on $D(H)$.

To confirm self-adjointness (not merely symmetry), we verify $D(H)=D(H^{\ast })$. Since $H$ is a Sturm-Liouville operator with real coefficient $p(x)={\hslash }^2/(2m)>0$ and Dirichlet boundary conditions on a compact interval, it is a standard result of Sturm-Liouville theory that $H$ is self-adjoint on $D(H)$. Concretely: if $\eta \in L^2([0,L])$ satisfies $|⟨\eta ,H\psi ⟩|\le C\Vert \psi \Vert$ for all $\psi \in D(H)$, then $\eta \in D(H)$ and $H^{\ast }\eta =H\eta$.

Step 2: The eigenvalue equation. Setting $H\psi =E\psi$ with $\psi \in D(H)$ gives the ODE ${\psi }^{\prime \prime }+k^2\psi =0$ where $k^2=2mE/{\hslash }^2$. The general solution is $\psi =A\sin (kx)+B\cos (kx)$.

Dirichlet condition at $x=0$: $\psi (0)=B=0$.

Dirichlet condition at $x=L$: $A\sin (kL)=0$. For a nonzero solution, $\sin (kL)=0$, giving $kL=n\pi$ for $n\in {\mathbb{Z}}^{+}$. So $k_n=n\pi /L$ and $E_n={\hslash }^2{k}_n^2/(2m)=n^2{\pi }^2{\hslash }^2/(2m{L}^2)$.

Step 3: Orthonormality. For $m\ne n$:

$$⟨{\psi }_m|{\psi }_n⟩=\frac{2}{L}{\int }_0^L\sin \left(\frac{m\pi x}{L}\right)\sin \left(\frac{n\pi x}{L}\right)dx=\frac{2}{L}\cdot \frac{L}{2}{\delta }_{mn}={\delta }_{mn},$$

using the standard integral ${\int }_0^L\sin (m\pi x/L)\sin (n\pi x/L)dx=(L/2){\delta }_{mn}$.

Step 4: Completeness. The set $\{\sqrt{2/L}\sin (n\pi x/L){\}}_{n=1}^{\infty }$ is the eigenbasis of a regular Sturm-Liouville problem on a compact interval. By the Sturm-Liouville theorem, the eigenfunctions of a regular Sturm-Liouville operator are complete in $L^2([0,L])$: the finite linear combinations are dense. Equivalently, this is the Fourier sine series on $[0,L]$, whose completeness is a classical result of analysis (converges in $L^2$ norm for any $f\in L^2([0,L])$).

Step 5: The spectrum is purely discrete. A compact interval with Dirichlet boundary conditions produces a self-adjoint operator with compact resolvent (since the domain $D(H)=H^2([0,L])\cap H_0^1([0,L])$ embeds compactly into $L^2([0,L])$ by the Rellich-Kondrachov theorem). The spectral theorem for self-adjoint operators with compact resolvent guarantees a purely discrete spectrum consisting of eigenvalues of finite multiplicity accumulating only at infinity. Here all eigenvalues are simple (multiplicity one). $\square$

Corollary (Energy measurement probabilities). If the particle is in state $\Psi (x,t)={\sum }_n{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$, a measurement of energy yields $E_n$ with probability $|c_n{|}^2$, independent of time.

The time-dependent phases cancel in $|c_n{e}^{-i{E}_{n}t/\hslash }{|}^2=|c_n{|}^2$. This is the defining property of a stationary state: energy eigenstates have time-independent measurement probabilities for the energy observable.

Bridge. The self-adjointness of the Dirichlet Laplacian and the completeness of its sine-basis eigenfunctions build toward every subsequent exactly solvable quantum system. The same proof structure — ODE solution, boundary quantisation, orthonormality, Sturm-Liouville completeness — appears again in the harmonic oscillator 12.04.02 pending with Hermite functions replacing sine functions, and in the hydrogen atom 12.06.01 pending with spherical Bessel functions and spherical harmonics. The central insight is that confinement by boundary conditions forces a discrete spectrum; this is exactly the mechanism that replaces the classical continuum. The bridge is between the Sturm-Liouville theory of the 19th century and the quantum-mechanical eigenvalue problem that gives these mathematical objects physical meaning as energy levels and probability amplitudes.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

The particle-in-a-box spectrum is a special case of the Sturm-Liouville eigenvalue problem. In Lean 4, the path to formalization runs through Mathlib's ODE and analysis libraries.

The key objects are:

• The Hilbert space $L^2([0,L])$, formalizable as MeasureTheory.Lp MeasureTheory.MeasureSpace.volume 2 restricted to the interval.
• The Laplacian $-d^2/d{x}^2$ as an unbounded operator on this space with Dirichlet domain $H^2([0,L])\cap H_0^1([0,L])$.
• The eigenvalues $E_n=n^2{\pi }^2/L^2$ (setting $\hslash =2m=1$ for the normalized problem).
• The eigenfunctions $\sqrt{2/L}\sin (n\pi x/L)$ and their orthonormality and completeness.

Mathlib has the sine and cosine functions, integration over intervals, and the basics of $L^p$ spaces. It has the Sturm-Liouville theory for regular problems and the spectral theorem for compact self-adjoint operators. What is missing: the specific identification of the Dirichlet-Laplacian spectrum on $[0,L]$ as $\{n^2{\pi }^2/L^2\}$, the completeness of the sine basis stated as a Fourier-series result, and the quantum-mechanical measurement postulates (Born rule, collapse) applied to this concrete system. A formalization would proceed by proving the eigenvalue equation and orthonormality directly, then invoking Mathlib's Sturm-Liouville completeness theorem.

The WKB approximation and semiclassical limits [Master]

The WKB method (unit 12.09.01) gives a semiclassical approximation to the eigenvalues and eigenfunctions. For the particle in a box, the WKB quantisation condition is

$${\int }_0^{L}p(x)dx=\left(n-\frac{1}{2}\right)\pi \hslash ,p(x)=\sqrt{2mE}.$$

(The $-1/2$ accounts for the two hard-wall turning points, each contributing $-1/4$.) Evaluating the integral: $pL=(n-1/2)\pi \hslash$, giving $E_n^{\text{WKB}}=(n-1/2{)}^2{\pi }^2{\hslash }^2/(2m{L}^2)$. This disagrees with the exact answer $E_n=n^2{\pi }^2{\hslash }^2/(2m{L}^2)$ for finite $n$. The discrepancy arises because the standard WKB connection formulae assume smooth turning points, but infinite walls are discontinuous. For the box, the correct WKB matching at hard walls uses the boundary condition $\psi =0$ at the wall, which restores the exact quantisation condition ${\int }_0^{L}kdx=n\pi$, giving $kL=n\pi$ and the exact eigenvalues. The lesson: WKB is exact for the box, but only when the boundary conditions are applied correctly at hard walls rather than through the standard Airy-function connection formulae.

The WKB wave functions inside the box are the oscillatory solutions ${\psi }_{\text{WKB}}(x)\approx A/\sqrt{k}\sin ({\int }_0^{x}kd{x}^{\prime })$, which at the hard-wall corrected quantisation condition reproduce the exact sine functions (the amplitude factor $1/\sqrt{k}$ is constant because $k$ is constant inside the box). The particle in a box is one of the rare potentials where WKB gives exact results — a consequence of the free-particle kinetic energy being independent of position inside the box.

For comparison, the harmonic oscillator 12.04.02 pending with its parabolic potential has smooth turning points where the standard Airy-function connection formulae apply correctly. The WKB approximation gives $E_n^{\text{WKB}}=(n-1/2)\hslash \omega$, which matches the exact oscillator spectrum exactly. Both the box and the oscillator are "WKB-exact," but for different reasons: the box because the constant momentum makes the semiclassical approximation exact inside the well, and the oscillator because the smooth turning points are correctly treated by the connection formulae. Most other potentials are not WKB-exact — the WKB energies are approximations whose accuracy improves with increasing $n$.

The semiclassical limit $n\to \infty$ recovers classical mechanics in two senses. First, the fractional energy spacing $\Delta E_n/E_n=(2n+1)/n^2\to 0$, so the spectrum becomes quasi-continuous and the quantum number $n$ labels an effectively continuous energy. Second, the probability density $|{\psi }_n(x){|}^2=(2/L){\sin }^2(n\pi x/L)$ oscillates with period $L/n$, which vanishes in the large-$n$ limit, converging to the uniform classical distribution $1/L$ when averaged over any finite interval. This is Bohr's correspondence principle for the box.

The three-dimensional box and degeneracy [Master]

The generalisation to a rectangular box $[0,L_x]\times [0,L_y]\times [0,L_z]$ is immediate. The Hamiltonian is $H=-\frac{{\hslash }^2}{2m}{\nabla }^2$ with Dirichlet conditions on all faces. Separation of variables gives eigenfunctions

$${\psi }_{n_x{n}_y{n}_z}(x,y,z)=\sqrt{\frac{8}{L_x{L}_y{L}_z}}\sin \left(\frac{n_x\pi x}{L_x}\right)\sin \left(\frac{n_y\pi y}{L_y}\right)\sin \left(\frac{n_z\pi z}{L_z}\right),$$

with energies

$$E_{n_x{n}_y{n}_z}=\frac{{\pi }^2{\hslash }^2}{2m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right).$$

For a cube ($L_x=L_y=L_z=L$): $E_{n_x{n}_y{n}_z}=\frac{{\pi }^2{\hslash }^2}{2m{L}^2}(n_x^2+n_y^2+n_z^2)$. The ground state $(1,1,1)$ is non-degenerate. The first excited level is three-fold degenerate: the three permutations of $(1,1,2)$, $(1,2,1)$, $(2,1,1)$ all have the same energy $\frac{{\pi }^2{\hslash }^2}{2m{L}^2}(1+1+4)=6{\pi }^2{\hslash }^2/(2m{L}^2)$. This degeneracy is a consequence of the cubic symmetry group $O_h$ acting on the quantum numbers — the representation theory of $O_h$ classifies the degeneracy pattern.

The second excited level is also three-fold degenerate: the three permutations of $(1,2,2)$, $(2,1,2)$, $(2,2,1)$ at $E=9{\pi }^2{\hslash }^2/(2m{L}^2)$. The level at $E=14{\pi }^2{\hslash }^2/(2m{L}^2)$ has six-fold degeneracy (all permutations of $(1,2,3)$). Accidental degeneracies — those not forced by symmetry — also appear: for example, $E_{331}=E_{411}=E_{222}$ at $19{\pi }^2{\hslash }^2/(2m{L}^2)$ is not a symmetry degeneracy but a number-theoretic coincidence (different representations of 19 as a sum of three squares).

The density of states for the 3D box — the number of energy levels per unit energy — scales as $\sqrt{E}$ at high energies. For a cube of side $L$, the cumulative count of states with energy $\le E$ is

$$N(E)=\frac{1}{8}\cdot \frac{4\pi }{3}{\left(\frac{L\sqrt{2mE}}{\pi \hslash }\right)}^3=\frac{V}{6{\pi }^2}{\left(\frac{2mE}{{\hslash }^2}\right)}^{3/2},$$

where the factor $1/8$ restricts to the positive octant ($n_x,n_y,n_z\ge 1$) and $V=L^3$. Differentiating gives the density of states

$$g(E)=\frac{dN}{dE}=\frac{V}{4{\pi }^2}{\left(\frac{2m}{{\hslash }^2}\right)}^{3/2}\sqrt{E}.$$

This is the free-electron-gas density of states used in solid-state physics 11.05.02 pending. For non-cubic boxes ($L_x\ne L_y\ne L_z$), the symmetry degeneracies are lifted — only accidental degeneracies remain. The density of states acquires step-like features at low energies and smooths to the same $\sqrt{E}$ envelope at high energies, independent of the aspect ratio (Weyl's law for the Laplacian on a bounded domain).

The finite square well and tunnelling [Master]

When the walls have finite height $V_0>0$, the boundary conditions change: $\psi$ is no longer zero outside the box but decays exponentially in the classically forbidden region. The quantisation condition becomes a transcendental equation. For even-parity states (about the well centre $x=L/2$), the condition is

$$k\tan (kL/2)=\kappa ,k=\frac{\sqrt{2mE}}{\hslash },\kappa =\frac{\sqrt{2m(V_0-E)}}{\hslash }.$$

For odd-parity states, $\tan$ is replaced by $-\cot$. Key differences from the infinite well:

1. The number of bound states is finite, not infinite. For $V_0<{\pi }^2{\hslash }^2/(2m{L}^2)$, there is exactly one bound state.
2. The eigenfunctions leak into the classically forbidden region, with a penetration depth $\delta =1/\kappa =\hslash /\sqrt{2m(V_0-E)}$.
3. The eigenvalues are lower than the corresponding infinite-well values, because the effective box is wider — the wave function extends beyond the walls.
4. In the limit $V_0\to \infty$ the finite-well solutions converge to the infinite-well solutions.

Proposition (Existence of at least one bound state). For any $V_0>0$ and any $L>0$, the finite square well of width $L$ and depth $V_0$ has at least one bound state.

Proof. Define $z=kL/2$ and $z_0=L\sqrt{2m{V}_0}/(2\hslash )$. The even-parity quantisation condition is $z\tan z=\sqrt{z_0^2-z^2}$. The left-hand side starts at $z=0$ with value $0$ and slope $1$, increasing through a singularity at $z=\pi /2$. The right-hand side starts at $z_0>0$ and decreases monotonically to $0$ at $z=z_0$. For any $z_0>0$, however small, the left-hand side starts below the right-hand side and the curves must cross before $z=\pi /2$ (since the left-hand side diverges to $+\infty$ there). Hence at least one even-parity bound state exists for any $V_0>0$. $\square$

The number of bound states is $N=\lfloor 2z_0/\pi \rfloor +1$. For the infinite well, $z_0\to \infty$ and $N\to \infty$, recovering the infinite discrete spectrum.

The tunnelling into the classically forbidden region has a measurable consequence: the probability of finding the particle outside the nominal box is

$$P_{\text{out}}={\int }_{-\infty }^0|\psi (x){|}^{2}dx+{\int }_L^{\infty }|\psi (x){|}^{2}dx,$$

which vanishes as $V_0\to \infty$ and increases toward the continuum limit as $V_0$ decreases. The penetration depth $\delta$ sets the scale of the evanescent tail. In one dimension, unlike three, there is always at least one bound state regardless of how shallow the well — a qualitative difference between 1D and higher-dimensional quantum mechanics.

Quantum dots and nanostructure confinement [Master]

Quantum dots are semiconductor nanostructures that confine electrons in all three dimensions on length scales of 1–10 nm. The particle-in-a-box model gives a zeroth-order description. For a spherical dot of radius $R$, the radial equation in spherical coordinates gives energy levels involving spherical Bessel functions $j_{\ell }(kr)$. The ground state ($\ell =0$, $n=1$) has $kR=\pi$, giving energy $E={\hslash }^2{\pi }^2/(2m^{\ast }{R}^2)$ where $m^{\ast }$ is the effective mass of the electron in the semiconductor (typically $0.01$–$0.1m_e$).

The key observable — the optical absorption edge — scales as

$$E_{\text{abs}}=E_g+\frac{{\hslash }^2{\pi }^2}{2m^{\ast }{R}^2},$$

where $E_g$ is the bulk band gap. The $1/R^2$ scaling produces the quantum confinement effect: smaller dots absorb at higher energies (bluer light), which is why colloidal quantum dots of different sizes fluoresce in different colours. CdSe quantum dots of radius 2 nm emit in the blue ($\sim 470$ nm), while dots of radius 5 nm emit in the red ($\sim 630$ nm).

Proposition (Confinement energy scaling). For a spherical quantum dot of radius $R$ with effective mass $m^$,theconfinementenergyofthegroundstatesatisfies$E_{\text{conf}} \propto 1/R^2$.Thefirstexcitedstate($\ell = 1$)hasconfinementenergy$E_{\ell=1} = \hbar^2\beta_{1,1}^2/(2m^R^2)$where$\beta_{1,1} \approx 4.493$isthefirstzeroof$j_1$.

Proof. The radial Schrödinger equation for $\ell =0$ is $-(R^2/2m^{\ast })(d^{2}u/d{r}^2)=Eu$ with $u(r)=r\psi (r)$, Dirichlet conditions $u(0)=u(R)=0$. The solution is $u(r)=A\sin (kr)$ with $kR=n\pi$. For the ground state $n=1$: $k=\pi /R$, giving $E={\hslash }^2{\pi }^2/(2m^{\ast }{R}^2)$. For $\ell =1$, the centrifugal term adds $\ell (\ell +1){\hslash }^2/(2m^{\ast }{r}^2)$, and the radial equation has solutions $u(r)=Ar{j}_1(kr)$ with $j_1({\beta }_{1,1})=0$. The $1/R^2$ scaling follows from dimensional analysis: the only energy scale constructible from $\hslash$, $m^{\ast }$, and $R$ is ${\hslash }^2/(m^{\ast }{R}^2)$. $\square$

Quantum dots also provide the cleanest experimental verification of the particle-in-a-box energy-level structure. Scanning tunnelling spectroscopy (STS) measures the tunnelling current as a function of bias voltage, directly probing the discrete density of states. The measured peak spacing follows the $1/R^2$ prediction to within experimental uncertainty for dots larger than about 3 nm; below this size, the effective-mass approximation breaks down and atomistic models are required.

The quantum-dot model also extends to quantum wires (1D confinement, free in 2D) and quantum wells (confinement in 1D, free in 2D). A quantum wire of cross-section $L\times L$ has a density of states that is step-like, with each step corresponding to a transverse mode ${\psi }_{n_x{n}_y}$. The energy of each transverse mode is $E_{n_x{n}_y}^{\perp }={\pi }^2{\hslash }^2(n_x^2+n_y^2)/(2m{L}^2)$, and the 1D free-electron dispersion $E^{\parallel }={\hslash }^2{k}_z^2/(2m)$ adds continuously on top. The resulting sub-band structure is the basis for the design of semiconductor lasers and high-electron-mobility transistors (HEMTs).

Time-dependent dynamics and selection rules [Master]

The box eigenstates form a natural basis for time-dependent perturbation theory 12.07.01 pending. A general superposition $\Psi (x,t)={\sum }_n{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$ evolves periodically whenever the coefficients $c_n$ are nonzero for only finitely many $n$. The revival time — the time after which the wave packet returns to its initial form — is determined by the energy spacing. For the box, the level spacings $\Delta E_n=E_{n+1}-E_n=(2n+1){\pi }^2{\hslash }^2/(2m{L}^2)$ are incommensurate, so exact revivals require the common period

$$T_{\text{rev}}=\frac{4m{L}^2}{\pi \hslash },$$

derived from the condition that all phase differences $(E_n-E_1)T_{\text{rev}}/\hslash$ are integer multiples of $2\pi$.

If a time-dependent perturbation $W(x,t)$ is applied to the box, the transition rate from state $n$ to state $m$ is governed by Fermi's golden rule 12.07.02:

$${\Gamma }_{n\to m}=\frac{2\pi }{\hslash }|W_{mn}{|}^2\rho (E_m),$$

where $W_{mn}=⟨{\psi }_m|W|{\psi }_n⟩$ is the matrix element and $\rho (E_m)$ is the density of final states. For the 1D box the density of states is discrete, so Fermi's golden rule applies to transitions into a continuum (e.g., ionisation from a box). The matrix elements $W_{mn}$ inherit the selection rules of the perturbation.

Proposition (Parity selection rule for the particle in a box). Let ${\psi }_n(x)=\sqrt{2/L}\sin (n\pi x/L)$ be the box eigenfunctions. Define the parity operator about the box centre $x_c=L/2$: $P\psi (x)=\psi (L-x)$. Then ${\psi }_n$ has parity $(-1{)}^{n+1}$: even for $n$ odd, odd for $n$ even. If $W(x)$ has definite parity (either even or odd about $x_c$), then the matrix element $W_{mn}=⟨{\psi }_m|W|{\psi }_n⟩$ vanishes unless $m-n$ is even (for even-parity $W$) or $m-n$ is odd (for odd-parity $W$).

Proof. Under the parity operator $P:x\mapsto L-x$, the eigenfunctions transform as ${\psi }_n(L-x)=\sqrt{2/L}\sin (n\pi (L-x)/L)=\sqrt{2/L}\sin (n\pi -n\pi x/L)=(-1{)}^{n+1}{\psi }_n(x)$. If $W$ is even about $x_c$, then $W(x)=W(L-x)$ and the integrand ${\psi }_m^{\ast }(x)W(x){\psi }_n(x)$ has overall parity $(-1{)}^{m+1}(-1{)}^{n+1}=(-1{)}^{m+n}$ under $x\mapsto L-x$. The integral over $[0,L]$ vanishes unless the integrand is even, requiring $m+n$ even, i.e., $m-n$ even. For odd-parity $W$, the additional sign from $W(L-x)=-W(x)$ flips the condition to $m-n$ odd. $\square$

The position operator $x$ itself has no definite parity about the box centre, but the shifted operator $x^{\prime }=x-L/2$ is odd-parity. Hence $⟨{\psi }_m|x^{\prime }|{\psi }_n⟩$ vanishes unless $m-n$ is odd — the box analogue of the electric-dipole selection rule. This is the mechanism behind the oscillating $⟨x⟩(t)$ computed in Exercise 7: only states differing by an odd quantum number contribute to the dipole matrix element.

Full proof set [Master]

Proposition (Minimum-energy gap). For a particle in a box of length $L$ with mass $m$, the energy gap $\Delta E_n=E_{n+1}-E_n$ satisfies $\Delta E_n\ge \Delta E_1=3{\pi }^2{\hslash }^2/(2m{L}^2)$ for all $n\ge 1$, with equality only at $n=1$.

Proof. $\Delta E_n=[(n+1{)}^2-n^2]{\pi }^2{\hslash }^2/(2m{L}^2)=(2n+1){\pi }^2{\hslash }^2/(2m{L}^2)$. This is an increasing function of $n$ (since the coefficient $2n+1$ grows linearly). At $n=1$: $\Delta E_1=3{\pi }^2{\hslash }^2/(2m{L}^2)$. For $n>1$, $\Delta E_n>\Delta E_1$ because $2n+1>3$. $\square$

Proposition (Feynman-Hellmann relation for the box). For the $n$-th energy eigenstate of the particle in a box of length $L$, the derivative of the energy with respect to the inverse-square length parameter is $d{E}_n/d(1/L^2)=n^2{\pi }^2{\hslash }^2/(2m)=E_n{L}^2$.

Proof. By the Feynman-Hellmann theorem, $d{E}_n/d\lambda =⟨{\psi }_n|dH/d\lambda |{\psi }_n⟩$ for any parameter $\lambda$ appearing in the Hamiltonian. Setting $\lambda =1/L^2$ and rescaling $u=x/L$, the Hamiltonian becomes $H=-({\hslash }^2/(2m{L}^2))(d^2/d{u}^2)$ with domain $[0,1]$. So $dH/d(1/L^2)=-{\hslash }^2/(2m)(d^2/d{u}^2)$, and $⟨{\psi }_n|dH/d(1/L^2)|{\psi }_n⟩=⟨{\psi }_n|H\cdot L^2|{\psi }_n⟩=E_n{L}^2$. Since $E_n=n^2{\pi }^2{\hslash }^2/(2m{L}^2)$, direct differentiation gives $d{E}_n/d(1/L^2)=n^2{\pi }^2{\hslash }^2/(2m)$. $\square$

Proposition (Node-counting for the box). The $n$-th energy eigenfunction ${\psi }_n(x)=\sqrt{2/L}\sin (n\pi x/L)$ has exactly $n-1$ interior nodes (zeros in the open interval $(0,L)$), located at $x_k=kL/n$ for $k=1,2,\dots ,n-1$.

Proof. The zeros of $\sin (n\pi x/L)$ on $(0,L)$ occur at $n\pi x/L=k\pi$, i.e., $x=kL/n$, for integer $k$. The endpoints $k=0$ and $k=n$ give the boundary zeros $x=0$ and $x=L$. The interior zeros are $k=1,2,\dots ,n-1$, giving exactly $n-1$ interior nodes. By the Sturm comparison theorem for second-order ODEs, the $n$-th eigenfunction of any Sturm-Liouville problem on $[0,L]$ with Dirichlet conditions has at least $n-1$ interior nodes; for the free-particle problem, the count is exact because ${\psi }_n$ is a pure sine with exactly $n$ half-wavelengths. $\square$

Synthesis. The particle-in-a-box spectrum, where $E_n\propto n^2$, stands at the crossroads of three structural threads. The first is the Sturm-Liouville theory of the Dirichlet Laplacian on a compact interval — the foundational reason the spectrum is discrete, complete, and real. The second is the connection to the harmonic oscillator 12.04.02 pending, where $E_n\propto n+1/2$ produces equally spaced levels — the central insight that the shape of the potential controls the level spacing through the turning-point structure. The third is the density-of-states bridge to statistical mechanics 11.05.02 pending: the 3D box puts these together to give the $\sqrt{E}$ density of states that underpins the free-electron theory of metals. This is exactly the pattern that generalises to arbitrary confining potentials via the WKB approximation: the quantum-mechanical density of states is the phase-space volume divided by $h^d$, and the bridge is between the semiclassical count of classical trajectories and the exact quantum spectrum. Putting these together identifies the particle in a box as the minimal example of every theme in one-dimensional quantum mechanics — quantisation, completeness, node-counting, uncertainty, and the semiclassical limit.

Connections [Master]

• The quantum harmonic oscillator 12.04.02 pending. Replaces the flat-bottomed box with a parabolic well, producing equally spaced energy levels $E_n=\hslash \omega (n+1/2)$ instead of the quadratically spaced $E_n\propto n^2$. Both are exactly solvable; the oscillator introduces ladder operators and coherent states that have no direct analogue in the box. The oscillator builds toward every perturbative treatment in quantum field theory, where the free field is a continuum of oscillators.

• Statistical mechanics and the Fermi-Dirac distribution 11.05.02 pending. The 3D particle-in-a-box density of states $g(E)\propto \sqrt{E}$ is the input to the free-electron model of metals. Filling these states with fermions up to the Fermi energy produces the electronic specific heat, the Wiedemann-Franz law, and the Pauli paramagnetic susceptibility. The density-of-states computation is the direct bridge from this unit to statistical mechanics.

• The hydrogen atom in quantum chemistry 14.04.01 pending. The particle-in-a-box serves as a zeroth-order model for pi electrons in conjugated molecules. The box length is the molecular length, the electrons fill levels according to the Aufbau principle, and the HOMO-LUMO gap predicts the absorption wavelength. For butadiene (4 pi electrons in a box of approximately 0.56 nm), the predicted absorption is approximately 220 nm, compared with the experimental 217 nm. The hydrogen atom adds a Coulomb potential and spherical geometry, producing the $1/n^2$ Rydberg spectrum from the same Schrödinger-equation machinery.

• Time-independent perturbation theory 12.07.01 pending. The box eigenstates form a complete orthonormal basis that is the natural starting point for perturbative corrections. A perturbation $V^{\prime }(x)$ added to the box produces energy shifts $\Delta E_n=⟨{\psi }_n|V^{\prime }|{\psi }_n⟩$ and mixes the unperturbed eigenstates according to the matrix elements $⟨{\psi }_m|V^{\prime }|{\psi }_n⟩$, which are governed by the parity selection rule proved above.

Historical and philosophical context [Master]

The particle in a box was not historically the first quantum system solved — that distinction belongs to Planck's blackbody radiation (1900), Einstein's photoelectric effect (1905), and Bohr's hydrogen atom (1913). The box potential appears as a pedagogical example in every quantum mechanics textbook from about 1926 onward, after Schrodinger formulated wave mechanics ^{[Schrödinger 1926]}. Its first systematic mathematical treatment is in Courant and Hilbert's Methoden der mathematischen Physik (1924, 1937) ^{[Courant-Hilbert]}, where the eigenvalue problem for the Laplacian on a bounded domain is solved as a Sturm-Liouville problem — predating and anticipating the quantum-mechanical application.

The deeper mathematical lineage runs through Sturm and Liouville's work of the 1830s on eigenvalue problems for second-order ODEs ^{[Sturm-Liouville]}, and through Fourier's 1807 theory of heat conduction, where the sine-series expansion of a function on a finite interval first appeared. The quantum-mechanical reinterpretation — that the Fourier coefficients are probability amplitudes and the eigenvalues are energy levels — was not anticipated by these mathematicians.

The zero-point energy $E_1>0$ has philosophical weight. A classical particle at rest in a box has zero energy; the quantum particle cannot be at rest. This is not a deficiency of the model but a consequence of the uncertainty principle: confining the particle to a region of size $L$ requires a momentum uncertainty $\Delta p\gtrsim \hslash /(2L)$, and the associated kinetic energy $\Delta p^2/(2m)\sim {\hslash }^2/(8m{L}^2)$ is the same order of magnitude as $E_1$. The zero-point energy is a direct physical manifestation of the position-momentum uncertainty relation, and it appears again in quantum field theory as vacuum energy — one of the deepest and most consequential predictions of quantum mechanics. The Casimir effect, where two parallel conducting plates attract each other due to the modification of the vacuum zero-point energy between them, is the most direct experimental confirmation that this energy is physically real, not merely a mathematical artefact of the formalism.

Bibliography [Master]

• Griffiths, D. J. and Schroeter, D. F. Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, 2018. Ch. 2.1-2.2.
• Sakurai, J. J. and Napolitano, J. Modern Quantum Mechanics, 2nd ed. Cambridge University Press, 2017. Ch. 2.1.
• Messiah, A. Quantum Mechanics, Vol. 1. Dover, 1999. Ch. III.
• Cohen-Tannoudji, C., Diu, B. and Laloe, F. Quantum Mechanics, Vol. 1. Wiley, 1991. Complement A-I.
• Dirac, P. A. M. The Principles of Quantum Mechanics, 4th ed. Oxford University Press, 1958. Ch. III.
• Susskind, L. and Friedman, A. Quantum Mechanics: The Theoretical Minimum. Basic Books, 2014. Lecture 4.
• Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. Wiley-Interscience, 1989. Ch. V-VI (Sturm-Liouville theory).
• Reed, M. and Simon, B. Methods of Modern Mathematical Physics, Vol. I: Functional Analysis. Academic Press, 1980. Ch. X (self-adjoint operators).
• Tannor, D. J. Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books, 2007. Ch. 1 (particle-in-a-box dynamics).
• Bransden, B. H. and Joachain, C. J. Quantum Mechanics, 2nd ed. Pearson, 2000. Ch. 4 (one-dimensional problems).
• Landau, L. D. and Lifshitz, E. M. Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Pergamon, 1977. §22 (particle in a potential well).
• Shankar, R. Principles of Quantum Mechanics, 2nd ed. Springer, 1994. Ch. 7 (the particle in a box, solved with operator methods).
• Gottfried, K. and Yan, T.-M. Quantum Mechanics: Fundamentals, 2nd ed. Springer, 2003. Ch. 2 (one-dimensional examples).