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)

where

n_x: nx (—)
n_y: ny (—)
n_z: nz (—)
L: Cube side (nm)
E: 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-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