QNT-11

Compton wavelength shift

Δλ = λc (1 − cos θ), λc = h /(me c).

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PhotonsCompton 1923

Governing equation

\Delta\lambda=\lambda_c(1-\cos\theta),\quad\lambda_c=\dfrac{h}{m_e c}

$\theta \in \left[0, 180\right]$

where

\theta: Scatter angle (°)
\lambda: Incident wavelength (pm)
\Delta\lambda: Shift (pm)
\lambda': Scattered wavelength (pm)

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

\Delta\lambda=\lambda_c(1-\cos\theta),\quad\lambda_c=\dfrac{h}{m_e c}

$\theta \in \left[0, 180\right]$

$\theta = 0$ $\Delta\lambda = 0$

Along the curve

②

Domain of the variable — blue

\theta_{\min} = 0,\quad \theta_{\max} = 180

$\theta \in \left[0, 180\right]$

$\theta = 0$ $\theta = 0$

Along the curve

③

Response curve — green

\Delta\lambda=\lambda_c(1-\cos\theta),\quad\lambda_c=\dfrac{h}{m_e c}

$\theta \in \left[0, 180\right]$

$\theta = 0$ $\Delta\lambda = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta\lambda}{\mathrm{d}\theta}

$\theta \in \left[0, 180\right]$

$\theta = 1.132$ $\Delta\lambda^{\prime} = 8.37\times 10^{-4}$

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-11 — Compton wavelength shift) is the form associated with Compton 1923. Working symbols: $\theta$ , $\lambda$ $\rightarrow$ $\Delta\lambda$ , $\lambda'$ . Treating the photon as a particle of energy hf and momentum h/λ colliding elastically with a free electron yields Compton's shift.

Purpose

Purpose: compute

\Delta\lambda

\lambda'

from

\theta

\lambda

in Quantum mechanics via

\Delta\lambda=\lambda_c(1-\cos\theta),\quad\lambda_c=\dfrac{h}{m_e c}

Δλ = λc (1 − cos θ), λc = h /(me c). 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

\theta = 90.000\,\mathrm{^{\circ}}

\lambda = 10.000\,\mathrm{pm}

, the governing relation

\Delta\lambda=\lambda_c(1-\cos\theta),\quad\lambda_c=\dfrac{h}{m_e c}

yields

\Delta\lambda = 2.4263\,\mathrm{pm}

\lambda' = 12.4263\,\mathrm{pm}

. Free electron at rest, unpolarised photon. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Scatter angle \theta90 °Incident wavelength \lambda10 pm

Outputs

Shift \Delta\lambda2.4263 pm
Scattered wavelength \lambda'12.4263 pm

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Narration of this film

Free electron at rest, unpolarised photon.

Treating the photon as a particle of energy hf and momentum h/λ colliding elastically with a free electron yields Compton's shift.

Reading speed

Watch on YouTube