Particle in a 1D Box

Equivalent forms

E_n = \frac{n^2 h^2}{8mL^2}

Confinement alone quantizes energy — no forces needed inside the box.

Symbol	Name	SI unit	Dimension	Range
$E_n$	Energy of level noutput Quantized energy of the nth stationary state	J	ML^2T^-2	0 – 1e-15
$n$	Quantum number Positive integer mode index	dimensionless	1	1 – 50
$hbar$	Reduced Planck constant h / (2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34
$m$	Particle mass Mass of the confined particle	kg	M	1e-31 – 1e-24
$L$	Box length Width of the 1D infinite well	m	L	1e-12 – 0.000001

Unit systems

Symbol	SI	CGS	Imperial
$E_n$	J	erg	ft*lbf
$n$	dimensionless	dimensionless	dimensionless
$hbar$	J*s	erg*s	ftlbfs
$m$	kg	g	lb
$L$	m	cm	ft

Where it holds

Valid for idealized infinite square well. Finite wells allow tunneling and have fewer bound states. 3D boxes give

E_nlm = (n^2 + l^2 + m^2) * E_1

.

Dimensional analysis

E_n \to [M*L^2*T^-2]

[M*L^2*T^-1]^2 / ([M]*[L]^2) \to [M*L^2*T^-2]

Both sides are energy.

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Discovery

Erwin Schrödinger (canonical example) · 1926

The simplest exactly-solvable bound-state problem in the new wave mechanics — a pedagogical cornerstone ever since.

Try this

What happens when you trap an electron in a nanometer-wide well?

A particle confined to a rigid 1D box has quantized energies that depend on box length L. What are the allowed energies?

Research status: stable

Real-world applications

Quantum dots and nanocrystals (tunable color by size)
Semiconductor quantum well lasers
Conjugated molecule absorption bands (free-electron model)
Nanowire transistors

Common misconceptions

The ground state is NOT zero energy — zero-point energy is a physical consequence of confinement
Wavefunctions must vanish AT the walls, not just be small
Level spacing grows with n — it is not uniform like a harmonic oscillator

Experimental verification

Quantum dots, semiconductor heterostructures, and quantum wires all exhibit discrete energy spectra following the box-confinement scaling. Observed in photoluminescence and tunneling spectroscopy.

Derivation

Inside 0 < x < L,

V = 0

so -hbar^

2/(2m)*\psi '' = E*\psi

.

General solution

\psi = A*\sin (k*x) + B*\cos (k*x)

with

k = \sqrt{2mE}

/hbar.

Boundary conditions

\psi (0) = \psi (L) = 0

force

B = 0

and

\sin (k*L) = 0 \to k*L = n*pi

.

Therefore

k_n = n*pi/L

and

E_n =

hbar^

2*k_n^2/(2m) = n^2*pi^2

*hbar^2/(2m*L^2).

Limiting cases

n = 1

⟶

E_1 = pi^2

*hbar^2/(2mL^2) — zero-point energyGround state is NOT zero; uncertainty principle forbids it.

L \to \infty

⟶

E_n \to 0

, continuous spectrumFree particle limit — quantization disappears.

Large n⟶ Level spacing

\Delta _E = (2n+1)*E_1

→ quasi-continuumCorrespondence principle — classical limit recovered.

What if…

What if L doubles?

E_n drops by factor 4 — confinement energy $\propto 1/L^2$ .

What if the walls were finite height?

Wavefunctions tunnel into the barrier; fewer bound states; energies slightly lower.

What if the particle were a proton instead of an electron?

Masses ratio 1836 — energies drop by 1836 for same L.

1

Electron in 1 nm box, ground state

Given ·

m:: 9.1093837015e-31
L:: 1e-9
n:: 1

Find · E_1

Steps

$E_1 = pi^2$ * hbar^2 / (2 * m * L^2)
$= (9.8696 * (1.054571817e-34)^2) / (2 * 9.1093837015e-31 * 1e-18)$
$\approx 6.02e-20\,\mathrm{J} \approx 0.376\,\mathrm{eV}$

Result ·

E_1 \approx 6.02e-20\,\mathrm{J} \approx 0.376\,\mathrm{eV}

2

Transition n=2 → n=1 in the same box

Given ·

m:: 9.1093837015e-31
L:: 1e-9

Find · emitted photon wavelength

Steps

$\Delta _E = (4 - 1) * E_1 = 3 * 0.376\,\mathrm{eV} \approx 1.13\,\mathrm{eV}$
$\lambda = hc/\Delta _E = 1240\,\mathrm{eV}*nm / 1.13\,\mathrm{eV}$
$\lambda \approx 1098\,\mathrm{nm}$

Result ·

\lambda \approx 1100\,\mathrm{nm}

(near-infrared)

Time-Dependent Schrödinger Equation→Heisenberg Uncertainty Principle→de Broglie Wavelength→Bohr Hydrogen Energy Levels→