Identical Particles in a Box — The Full Treatment

From one particle to twenty. We derive the box states, build the many-body wavefunction by symmetrization, and read off the two objects an experiment sees: the density $n(x)$ of where the particles are, and the joint density $\rho_2(x_1,x_2)$ of finding one here and another there. Whether the particles are distinguishable, bosons, or fermions reshapes both — with no force between them.

Schrödinger 1926 · Pauli 1925 · Slater 1929 · standard many-body QM (Griffiths §5.1 · Ashcroft–Mermin §17 · Morrison §7.2)

HONESTY TIER A · DIRECT every density, correlation and surface is evaluated live on the grid from the equations shown — up to N = 20 particles

› Quantum Mechanics › Identical Particles in a Box

A single particle in a box has a wavefunction over a line. Two particles share one wavefunction over a plane. Twenty particles share one wavefunction over a twenty-dimensional space — which we cannot draw, but whose two physically measurable shadows we can: the one-body density and the two-body pair density. This page derives all of it in full, then lets you dial the particle number, the pair of levels, and the particle statistics, and watch the exchange hole and the boson pile-up appear out of nothing but the symmetry of $\Psi$.

I · One Particlestates & energies II · Two Particlessymmetrization, the joint density III · N ParticlesSlater determinant, ρ₁, n(x), ρ₂ IV · Two-Particle SurfaceP(x₁,x₂), live V · Up to 20 Particlesdensity & pair density, live

I · One Particle in the Box

The single-particle ladder

Inside a hard-walled box $0\le x\le L$ the potential is zero and $\psi$ must vanish at both walls. The time-independent Schrödinger equation and its quantised solutions are

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}=E\psi,\qquad \psi_n(x)=\sqrt{\frac{2}{L}}\sin\!\Big(\frac{n\pi x}{L}\Big),\qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2},\quad n=1,2,3,\dots$$

orthonormal: $\int_0^L \psi_m\psi_n\,dx=\delta_{mn}$ · we use natural units $\hbar=m=L=1$ below, so $\psi_n=\sqrt2\sin(n\pi x)$

These $\{\psi_n\}$ are the single-particle orbitals. Every many-particle state on this page is assembled from them; the only question is how identical particles are allowed to share them.

II · Two Identical Particles

Symmetrization and the joint density

With two particles the state is a function of both coordinates, $\Psi(x_1,x_2)$, and $|\Psi(x_1,x_2)|^2$ is the joint probability of particle 1 at $x_1$ and particle 2 at $x_2$. If they were distinguishable, particle 1 in orbital $a$ and particle 2 in $b$, the state would be a plain product:

$$\Psi_{\text{dist}}(x_1,x_2)=\psi_a(x_1)\psi_b(x_2),\qquad P=|\psi_a(x_1)|^2\,|\psi_b(x_2)|^2.$$

But identical particles are indistinguishable: swapping the labels cannot change any prediction, so $|\Psi|^2$ must be invariant under $x_1\leftrightarrow x_2$. Only the symmetric ($+$) and antisymmetric ($-$) combinations qualify:

$$\Psi_{\pm}(x_1,x_2)=\frac{1}{\sqrt2}\Big[\psi_a(x_1)\psi_b(x_2)\pm\psi_b(x_1)\psi_a(x_2)\Big],\qquad \begin{cases}+ & \text{bosons (symmetric)}\\[2pt]- & \text{fermions (antisymmetric)}\end{cases}$$

setting x₁=x₂ in Ψ₋ gives 0 identically — the exchange hole; and if a=b, Ψ₋≡0 — the Pauli exclusion principle

The consequences need no interaction. Fermions vanish on the diagonal $x_1=x_2$ (they are never found together); bosons are enhanced there (they are found together more than chance). One sign flips attraction into avoidance. Section IV draws $|\Psi|^2$ for any pair $(a,b)$ and any statistics.

III · N Identical Particles

The determinant, the density matrix, and the pair density

For $N$ particles the antisymmetric state is the Slater determinant of $N$ occupied orbitals $\{\psi_{n_1},\dots,\psi_{n_N}\}$; the symmetric (boson) state is the corresponding permanent:

$$\Psi_{\text{F}}(x_1,\dots,x_N)=\frac{1}{\sqrt{N!}}\begin{vmatrix}\psi_{n_1}(x_1)&\cdots&\psi_{n_1}(x_N)\\ \vdots&\ddots&\vdots\\ \psi_{n_N}(x_1)&\cdots&\psi_{n_N}(x_N)\end{vmatrix}.$$

a determinant vanishes if two rows (orbitals) or two columns (positions) coincide — Pauli exclusion and the exchange hole in one stroke

We never need the full $N$-dimensional object. Everything measurable follows from the one-body density matrix built from the occupied orbitals:

$$\rho_1(x,x')=\sum_{n\ \text{occ}}\psi_n(x)\,\psi_n^{*}(x'),\qquad n(x)\equiv\rho_1(x,x)=\sum_{n\ \text{occ}}|\psi_n(x)|^2,\qquad \int_0^L n(x)\,dx=N.$$

n(x) is the ordinary density — how many particles per unit length; it integrates to the particle number N

The joint "one particle at $x_1$ and another at $x_2$" is the pair density $\rho_2$. For a single determinant (fermions) or permanent (bosons) it collapses to a beautifully simple closed form in terms of the same two objects:

$$\boxed{\;\rho_2(x_1,x_2)=n(x_1)\,n(x_2)\;\mp\;\big|\rho_1(x_1,x_2)\big|^2\;}\qquad\begin{cases}-&\text{fermions}\\ +&\text{bosons}\end{cases}\qquad \iint \rho_2\,dx_1dx_2=N(N-1).$$

the first term is the uncorrelated (distinguishable) product; the ∓|ρ₁|² is pure exchange

On the diagonal $\rho_1(x,x)=n(x)$, so the fermion pair density is $\rho_2(x,x)=n(x)^2-n(x)^2=0$ — exactly zero, for every $N$. That valley along $x_1=x_2$ is the Fermi exchange hole: identical fermions carve a region of avoidance around each other with no force at all, purely from antisymmetry. Bosons do the reverse, $\rho_2(x,x)=2\,n(x)^2$ — a ridge of bunching. Section V computes $n(x)$ and this $\rho_2$ surface live for any $N$ up to 20.

SECTION IV · LIVE

The Two-Particle Joint Density P(x₁, x₂)

TIER A · DIRECT

What you're looking at

A surface over the square $0\le x_1,x_2\le L$; the height is $P(x_1,x_2)=|\Psi(x_1,x_2)|^2$. Pick the two occupied orbitals $n_a,n_b$, then switch the statistics. Colour tracks height.

height = joint probability x₁ = particle 1 · x₂ = particle 2 diagonal x₁ = x₂

Watch: Distinguishable at $n_a=4,n_b=3$ is the textbook Figure 7.7. Switch to Fermions and turn on the diagonal — a valley cuts down $x_1=x_2$. Bosons raise a ridge there. Set $n_a=n_b$ on Fermions and the whole surface vanishes (Pauli).

◐ drag · scroll to zoom

orbital a nₐ4

orbital b n_b3

states: nₐ=4, n_b=3Distinguishable: P=|ψ_a(x₁)|²·|ψ_b(x₂)|².

$\displaystyle \Psi_{\pm}=\tfrac{1}{\sqrt2}\big[\psi_a(x_1)\psi_b(x_2)\pm\psi_b(x_1)\psi_a(x_2)\big],\qquad P=|\Psi|^2$

evaluated on a 110×110 grid live

SECTION V · LIVE

Up to Twenty Particles — Density & Pair Density

TIER A · DIRECT

Fill the lowest $N$ orbitals and add particles one at a time. The surface is the pair density $\rho_2(x_1,x_2)=n(x_1)n(x_2)\mp|\rho_1(x_1,x_2)|^2$ — the joint chance of one particle at $x_1$ and another at $x_2$, the direct generalisation of Figure 7.7 to $N$ particles. Below it, the curve is the ordinary density $n(x)=\sum_{n=1}^{N}|\psi_n(x)|^2$: watch $N$ fermions spread and fill the box (Pauli) while $N$ bosons pile into the ground hump.

What you're looking at

Top: the pair-density surface $\rho_2(x_1,x_2)$. Fermions cut an exact zero-valley along $x_1=x_2$ (the Fermi hole) for any $N$; bosons raise a ridge; distinguishable is the smooth product with neither. Bottom: the density $n(x)$ — blue is fermions filling $n=1\dots N$, gold is bosons all in $n=1$.

surface = ρ₂(x₁,x₂), ∫∫ = N(N−1) fermion density n(x) boson density N|ψ₁|² diagonal x₁=x₂

Watch: drag $N$ up to 20 on Fermions — the density flattens toward uniform ($n=N$) while the surface keeps an exact hole on the diagonal. Flip to Bosons: the density collapses into one tall hump and the surface grows a diagonal ridge. Same particles, opposite behaviour — only the sign of the exchange term changed.

◐ drag · scroll to zoom

particles N6

drag N 2–20

config: fill n=1…6Fermions: exchange hole on the diagonal.

density n(x) for 6 particlesblue fills the box (fermions, Pauli) · gold piles in the ground state (bosons) · dashed line = uniform n=N

$\displaystyle n(x)=\sum_{n=1}^{N}|\psi_n(x)|^2,\quad \rho_1(x_1,x_2)=\sum_{n=1}^{N}\psi_n(x_1)\psi_n(x_2),\quad \rho_2=n(x_1)n(x_2)\mp|\rho_1(x_1,x_2)|^2$

all three summed live over the filled orbitals · self-checks: …

Sources & further reading

Griffiths, D. J. Introduction to Quantum Mechanics, §5.1 — identical particles, exchange forces, the symmetric/antisymmetric wavefunction.
Slater, J. C. (1929). The Theory of Complex Spectra. Phys. Rev. 34, 1293 — the determinant.
Ashcroft, N. W. & Mermin, N. D. Solid State Physics, Ch. 17 — the exchange hole and the pair correlation of a free-fermion gas.
Morrison, M. A. Understanding Quantum Physics, §7.2 — Figure 7.7, the two-particle asymmetric density reproduced in Section IV.
Pauli, W. (1925). Z. Phys. 31, 765 — the exclusion principle, here the $a=b$ collapse and the determinant's vanishing.