QNT-10

Particle in a 3-D box

E = (π² ħ² / 2m) (nx²/Lx² + ny²/Ly² + nz²/Lz²).

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Bound statesSchrödinger

Governing equation

E=\dfrac{\pi^2\hbar^2}{2m}\left(\dfrac{n_x^2}{L_x^2}+\dfrac{n_y^2}{L_y^2}+\dfrac{n_z^2}{L_z^2}\right)

$L \in \left[0.2, 10\right]$

where

n_x: nx (—)
n_y: ny (—)
n_z: nz (—)
L: Cube side (nm)
E: 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=\dfrac{\pi^2\hbar^2}{2m}\left(\dfrac{n_x^2}{L_x^2}+\dfrac{n_y^2}{L_y^2}+\dfrac{n_z^2}{L_z^2}\right)

$L \in \left[0.2, 10\right]$

$L = 0.2$ $E = 28.2$

Along the curve

②

Domain of the variable — blue

L_{\min} = 0.2,\quad L_{\max} = 10

$L \in \left[0.2, 10\right]$

$L = 0.2$ $L = 0.2$

Along the curve

③

Response curve — green

E=\dfrac{\pi^2\hbar^2}{2m}\left(\dfrac{n_x^2}{L_x^2}+\dfrac{n_y^2}{L_y^2}+\dfrac{n_z^2}{L_z^2}\right)

$L \in \left[0.2, 10\right]$

$L = 0.2$ $E = 28.2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E}{\mathrm{d}L}

$L \in \left[0.2, 10\right]$

$L = 0.262$ $E^{\prime} = -141.2$

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-10 — Particle in a 3-D box) is the form associated with Schrödinger. Working symbols: $n_x$ , $n_y$ , $n_z$ , $L$ $\rightarrow$ $E$ . Separation of variables in a rectangular well multiplies three 1-D spectra. Degeneracy appears when two sides are equal.

Purpose

Purpose: compute

E

from

n_x

n_y

n_z

L

in Quantum mechanics via

E=\dfrac{\pi^2\hbar^2}{2m}\left(\dfrac{n_x^2}{L_x^2}+\dfrac{n_y^2}{L_y^2}+\dfrac{n_z^2}{L_z^2}\right)

E = (π² ħ² / 2m) (nx²/Lx² + ny²/Ly² + nz²/Lz²). 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_x = 1.000\,\mathrm{—}

n_y = 1.000\,\mathrm{—}

n_z = 1.000\,\mathrm{—}

L = 1.000\,\mathrm{nm}

, the governing relation

E=\dfrac{\pi^2\hbar^2}{2m}\left(\dfrac{n_x^2}{L_x^2}+\dfrac{n_y^2}{L_y^2}+\dfrac{n_z^2}{L_z^2}\right)

yields

E = 1.1281\,\mathrm{eV}

. Infinite walls, electron, lengths in nm. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

nx n_x1 —ny n_y1 —nz n_z1 —Cube side L1 nm

Outputs

Energy E1.1281 eV

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Narration of this film

Infinite walls, electron, lengths in nm.

Separation of variables in a rectangular well multiplies three 1-D spectra. Degeneracy appears when two sides are equal.

Reading speed

Watch on YouTube