QNT-15

Infinite well energy

En = n² π² ħ² / (2 m L²). Standing waves on a length L.

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QuantaParticle in a box

Governing equation

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

$n \in \left[1, 8\right]$

where

n: Quantum number (—)
L: Width (nm)
E_n: Level energy (eV)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$n \in \left[1, 8\right]$

$n = 1$ $E_n = 0.376$

Along the curve

②

Domain of the variable — blue

n_{\min} = 1,\quad n_{\max} = 8

$n \in \left[1, 8\right]$

$n = 1$ $n = 1$

Along the curve

③

Response curve — green

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

$n \in \left[1, 8\right]$

$n = 1$ $E_n = 0.376$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E_n}{\mathrm{d}n}

$n \in \left[1, 8\right]$

$n = 1.044$ $E_n^{\prime} = 0.785$

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-15 — Infinite well energy) is the form associated with Particle in a box. Working symbols: $n$ , $L$ $\rightarrow$ $E_n$ . ψ(0)=ψ(L)=0 forces n = 1,2,3,… The ground state is not zero.

Purpose

Purpose: compute

E_n

from

n

L

in Quantum mechanics via

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

En = n² π² ħ² / (2 m L²). Standing waves on a length L. 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}

, the governing relation

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

yields

E_n = 0.376\,\mathrm{eV}

. A well, n humps, energy rungs. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Quantum number n1 —Width L1 nm

Outputs

Level energy E_n0.376 eV

Reading speed

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Narration of this film

A well, n humps, energy rungs.

ψ(0)=ψ(L)=0 forces n = 1,2,3,… The ground state is not zero.

Reading speed

Watch on YouTube