Editing Uncertainty principle (section)

=== Heisenberg's microscope ===
[[File:Heisenberg gamma ray microscope.svg|thumb|200px|right|Heisenberg's gamma-ray microscope for locating an electron (shown in blue). The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle ''θ''. The scattered gamma-ray is shown in red. Classical [[optics]] shows that the electron position can be resolved only up to an uncertainty Δ''x'' that depends on ''θ'' and the wavelength ''λ'' of the incoming light.]]
{{Main article|Heisenberg's microscope}}

The principle is quite counter-intuitive, so the early students of quantum theory had to be reassured that naive measurements to violate it were bound always to be unworkable. One way in which Heisenberg originally illustrated the intrinsic impossibility of violating the uncertainty principle is by using the [[observer effect (physics)|observer effect]] of an imaginary microscope as a measuring device.<ref name="Heisenberg_1930"/>

He imagines an experimenter trying to measure the position and momentum of an [[electron]] by shooting a [[photon]] at it.<ref name=GreensteinZajonc2006>{{cite book|first1=George |last1=Greenstein|first2=Arthur |last2=Zajonc|authorlink2=Arthur Zajonc|title=The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics|year=2006|publisher=Jones & Bartlett Learning|isbn=978-0-7637-2470-2}}</ref>{{rp|49–50}}
* Problem 1 – If the photon has a short [[wavelength]], and therefore, a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long [[wavelength]] and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
* Problem 2 – If a large [[aperture]] is used for the microscope, the electron's location can be well resolved (see [[Angular resolution#The_Rayleigh_criterion|Rayleigh criterion]]); but by the principle of [[conservation of momentum]], the transverse momentum of the incoming photon affects the electron's beamline momentum and hence, the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.

The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to the [[Planck constant]].<ref>{{Citation |last1=Tipler |first1=Paul A. |first2=Ralph A. |last2=Llewellyn |title=Modern Physics |volume=3 |publisher=W.H. Freeman & Co. |year=1999 |isbn=978-1572591646|lccn= 98046099
|url-access=|url=https://archive.org/details/modernphysics0003tipl |page=3 }}</ref> Heisenberg did not care to formulate the uncertainty principle as an exact limit, and preferred to use it instead, as a heuristic quantitative statement, correct up to small numerical factors, which makes the radically new noncommutativity of quantum mechanics inevitable.