QNT-29

HO quanta

Quantum harmonic oscillator.

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GoverningHO quanta

Governing equation

E=(n+\\tfrac12)\\hbar\\omega

$n \in \left[0, 20\right]$

where

n: n (—)
w: w (PHz)
E: HO quanta (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+\\tfrac12)\\hbar\\omega

$n \in \left[0, 20\right]$

$n = 0$ $E = 0.329$

Along the curve

②

Domain of the variable — blue

n_{\min} = 0,\quad n_{\max} = 20

$n \in \left[0, 20\right]$

$n = 0$ $n = 0$

Along the curve

③

Response curve — green

E=(n+\\tfrac12)\\hbar\\omega

$n \in \left[0, 20\right]$

$n = 0$ $E = 0.329$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E}{\mathrm{d}n}

$n \in \left[0, 20\right]$

$n = 0.126$ $E^{\prime} = 0.658$

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-29 — HO quanta) is the form associated with HO quanta. Working symbols: $n$ , $w$ $\rightarrow$ $E$ . Quantum harmonic oscillator. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

E

from

n

w

in Quantum mechanics via

E=(n+\\tfrac12)\\hbar\\omega

Quantum harmonic oscillator. 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 = 3.000\,\mathrm{—}

w = 1.000\,\mathrm{PHz}

, the governing relation

E=(n+\\tfrac12)\\hbar\\omega

yields

E = 2.304\,\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 n3 —w w1 PHz

Outputs

HO quanta E2.304 eV

Reading speed

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Narration of this film

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

Quantum harmonic oscillator. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube