QNT-07

Quantum harmonic oscillator

En = ħ ω (n + ½).

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OscillatorSchrödinger / Dirac

Governing equation

E_n=\hbar\omega\left(n+\tfrac12\right)

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

where

n: n (—)
f: Classical frequency (THz)
E_n: 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=\hbar\omega\left(n+\tfrac12\right)

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

$n = 0$ $E_n = 0.021$

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=\hbar\omega\left(n+\tfrac12\right)

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

$n = 0$ $E_n = 0.021$

Along the curve

④

Rate of change — amber

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

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

$n = 0.126$ $E_n^{\prime} = 0.041$

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-07 — Quantum harmonic oscillator) is the form associated with Schrödinger / Dirac. Working symbols: $n$ , $f$ $\rightarrow$ $E_n$ . The quadratic potential has evenly spaced levels with a zero-point ½ ħω, the origin of vacuum fluctuations in every harmonic mode.

Purpose

Purpose: compute

E_n

from

n

f

in Quantum mechanics via

E_n=\hbar\omega\left(n+\tfrac12\right)

En = ħ ω (n + ½). 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 = 0.000\,\mathrm{—}

f = 10.000\,\mathrm{THz}

, the governing relation

E_n=\hbar\omega\left(n+\tfrac12\right)

yields

E_n = 0.0207\,\mathrm{eV}

. 1-D QHO. Enter f in THz. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

n n0 —Classical frequency f10 THz

Outputs

Energy E_n0.0207 eV

Reading speed

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Narration of this film

1-D QHO. Enter f in THz.

The quadratic potential has evenly spaced levels with a zero-point ½ ħω, the origin of vacuum fluctuations in every harmonic mode.

Reading speed

Watch on YouTube