QNT-24

1D Fermi energy

1-D Fermi gas snapshot.

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Governing1D Fermi energy

Governing equation

E_F=(\\pi^2\\hbar^2 n^2)/(2m)

$n \in \left[0.1, 8\right]$

where

n: n (1/nm)
Ef: 1D Fermi 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_F=(\\pi^2\\hbar^2 n^2)/(2m)

$n \in \left[0.1, 8\right]$

$n = 0.1$ $Ef = 0.004$

Along the curve

②

Domain of the variable — blue

n_{\min} = 0.1,\quad n_{\max} = 8

$n \in \left[0.1, 8\right]$

$n = 0.1$ $n = 0.1$

Along the curve

③

Response curve — green

E_F=(\\pi^2\\hbar^2 n^2)/(2m)

$n \in \left[0.1, 8\right]$

$n = 0.1$ $Ef = 0.004$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Ef}{\mathrm{d}n}

$n \in \left[0.1, 8\right]$

$n = 0.15$ $Ef^{\prime} = 0.113$

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-24 — 1D Fermi energy) is the form associated with 1D Fermi energy. Working symbols: $n$ $\rightarrow$ $Ef$ . 1-D Fermi gas snapshot. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

Ef

from

n

in Quantum mechanics via

E_F=(\\pi^2\\hbar^2 n^2)/(2m)

1-D Fermi gas snapshot. 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{1/nm}

, the governing relation

E_F=(\\pi^2\\hbar^2 n^2)/(2m)

yields

Ef = 0.376\,\mathrm{eV}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

n n1 1/nm

Outputs

1D Fermi energy Ef0.376 eV

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

1-D Fermi gas snapshot. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube