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}

$L \in \left[0.2, 20\right]$

where

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

The figure, part by part

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①

Governing law — red

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

$L \in \left[0.2, 20\right]$

$L = 0.2$ $E_n = 6.85\times 10^{-9}$

Along the curve

②

Domain of the variable — blue

L_{\min} = 0.2,\quad L_{\max} = 20

$L \in \left[0.2, 20\right]$

$L = 0.2$ $L = 0.2$

Along the curve

③

Response curve — green

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

$L \in \left[0.2, 20\right]$

$L = 0.2$ $E_n = 6.85\times 10^{-9}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E_n}{\mathrm{d}L}

$L \in \left[0.2, 20\right]$

$L = 0.325$ $E_n^{\prime} = 0$

Along the curve

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

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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