QNT-01

Particle in an infinite well

En = n² π² ħ² /(2 m L²), n = 1, 2, …

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Bound statesSchrödinger 1926

Governing equation

E_n=\dfrac{n^2\pi^2\hbar^2}{2m L^2}

where

n: Quantum number (—)
L: Well width (nm)
m: Mass (electron units) (m_e)
E_n: Energy (eV)

Lecture brief

Historical brief

Planck (1900), Einstein’s photoelectric law, Bohr, de Broglie, Heisenberg and Schrödinger’s 1926 equation rebuilt matter as amplitude. These sheets are the first solvable models: wells, spin, tunneling, uncertainty. This sheet (QNT-01 — Particle in an infinite well) is the form associated with Schrödinger 1926. Working symbols: $n$ , $L$ , $m$ $\rightarrow$ $E_n$ . Schrödinger's equation with ψ(0)=ψ(L)=0 has sinusoidal eigenfunctions whose energies scale as n²/L², the prototype bound spectrum.

Purpose

Purpose: compute

E_n

from

n

L

m

in Quantum mechanics via

E_n=\dfrac{n^2\pi^2\hbar^2}{2m L^2}

En = n² π² ħ² /(2 m L²), n = 1, 2, … Use it when a real quantum mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 1.000\,\mathrm{—}

L = 1.000\,\mathrm{nm}

m = 1.000\,\mathrm{m_e}

, the governing relation

E_n=\dfrac{n^2\pi^2\hbar^2}{2m L^2}

yields

E_n = 6.851e-9\,\mathrm{eV}

. 1-D infinite walls, electron mass default via u = me if m=1. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Quantum number n1 —Well width L1 nmMass (electron units) m1 m_e

Outputs

Energy E_n0.0000 eV

Reading speed

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Narration of this film

1-D infinite walls, electron mass default via u = me if m=1.

Schrödinger's equation with ψ(0)=ψ(L)=0 has sinusoidal eigenfunctions whose energies scale as n²/L², the prototype bound spectrum.

Reading speed

Watch on YouTube