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)

where

n: n (—)
f: Classical frequency (THz)
E_n: 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-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

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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