= Heisenberg's microscope
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{wiki}
<Heisenberg's microscope> is a <thought experiment>, not a report of a performed measurement. In <Werner Heisenberg>'s Nobel lecture, The development of quantum mechanics, delivered on 11 December 1933, the imagined <Electron> is illuminated with very short-wavelength light and observed through a high-resolution <microscope>. A scattered <photon> entering the observer's eye would, through <Compton scattering>, eject the <Electron> from its original path. The passage challenges a classical <trajectory> picture; its microscope is hypothetical. The lecture date is 1933 even though the award is the <Nobel Prize in Physics in 1932>.<Nobel Lectures, Physics 1922–1941/The Development of Quantum Mechanics by Werner Heisenberg>[Nobel Lectures, Physics 1922–1941; Werner Heisenberg, The development of quantum mechanics, heading and printed pp. 291–292; text inspected in the Studyres transcription]{ref}
The following idealized geometry makes the tradeoff visible. Let the collected light have <wavelength> $\lambda$ and let the <microscope objective> accept outgoing directions within a half-angle $\alpha$ about the vertical axis. Take $x$ transverse to that axis and hold the incoming <photon> <momentum> fixed. A <diffraction>-limited estimate of transverse resolution has scale $\delta x\sim\lambda/\sin\alpha$. The unobserved outgoing direction admits transverse <photon> <momentum> components between approximately $-(h/\lambda)\sin\alpha$ and $+(h/\lambda)\sin\alpha$, neglecting the angular variation of the scattered <wavelength>. By <conservation of momentum>, uncertainty in that component leaves uncertainty in the transverse recoil of the <Electron>:
$$
\delta p_x\sim\frac{h}{\lambda}\sin\alpha,
\qquad
\delta x\,\delta p_x\sim h.
$$
Here $h=2\pi\hbar$ is the <Planck constant>; the two $\delta$ symbols denote heuristic resolution and recoil scales, not the <standard deviations> in the <Heisenberg uncertainty relation>. Increasing the acceptance angle improves the estimated resolution while broadening the allowed recoil range. Numerical factors depend on the resolution criterion and angular distribution; this scaling argument cannot supply the sharp constant $\hbar/2$.
\Image[/heisenberg-microscope-thought-geometry.png]
{title=Heisenberg microscope thought experiment: an imagined scattering event and collection cone, with resolution and recoil scales rather than measured data}
The <thought experiment> makes the interaction needed to obtain an image memorable. The <preparation uncertainty> theorem has broader use: it applies to the prepared <quantum state> even when the two <observables> are measured on separate copies. No claim that <Werner Heisenberg> built this <gamma ray> apparatus is needed to understand either argument.
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