View source for Physics:Heisenberg's microscope

{{Short description|Thought experiment establishing quantum ideas}}
'''Heisenberg's microscope''' is a [[Thought experiment|thought experiment]] proposed by [[Biography:Werner Heisenberg|Werner Heisenberg]] that has served as the nucleus of some commonly held ideas about quantum mechanics. In particular, it provides an argument for the [[Physics:Uncertainty principle|uncertainty principle]] on the basis of the principles of classical [[Physics:Optics|optics]]. 

The concept was criticized{{clarify|date=April 2020}} by Heisenberg's mentor [[Biography:Niels Bohr|Niels Bohr]], and theoretical and experimental developments have suggested that Heisenberg's intuitive explanation{{clarify|date=April 2020}} of his mathematical result might be misleading.<ref>{{cite journal
 | author=Lee A. Rozema|display-authors=0
 | title =Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements
 | journal =Phys. Rev. Lett.
 | volume =109
 | issue =18
 | page =100404
 | date =6 Sep 2012
 | doi =10.1103/PhysRevLett.109.100404
 | pmid=23005268
| arxiv =1208.0034
 | bibcode =2012PhRvL.109j0404R
 |s2cid=37576344
 }}</ref><ref>
{{cite web
 | url =https://www.sciencedaily.com/releases/2012/09/120907125154.htm
 | title =Scientists cast doubt on Heisenberg's uncertainty principle
 | date =7 Sep 2012
 | website =Science Daily
}}</ref> While the act of measurement does lead to uncertainty, the loss of precision is less than that predicted by Heisenberg's argument when measured at the level of an individual [[Physics:Quantum state|state]]. The formal mathematical result remains valid, however, and the original intuitive argument has also been vindicated mathematically when the notion of disturbance{{clarify|date=April 2020}} is expanded to be independent of any specific state.<ref>{{cite journal
 | author1=Paul Busch
 | author2=Pekka Lahti
 | author3=Richard Werner
 | title =Proof of Heisenberg's error-disturbance relation
 | journal =Physical Review Letters 
 | volume =111
 | issue =16
 | pages =160405
 | date =Oct 2013
 | doi =10.1103/PhysRevLett.111.160405
 | bibcode=2013PhRvL.111p0405B
 | pmid=24182239
| arxiv=1306.1565
 | s2cid=24507489
 }}</ref><ref>{{cite web
 | url =http://www.york.ac.uk/news-and-events/news/2013/research/heisenberg/
 | title =Scientists prove Heisenberg's intuition correct
 | last =Lett
 | first =Caron
 | date =17 Oct 2013
 | publisher =University of York
}}</ref>

==Heisenberg's argument==
[[File:Heisenberg microscope with wavefronts and electron scatter.svg|thumb|140px|The electron is illuminated from below by light depicted as both photons and waves, with wavefronts  shown as blue lines. Photons that enter the microscope deviate from the vertical by an angle less than ''&epsilon;/2'' and impart momentum to the electron as they scatter off it. The depiction of the wavefronts inside the microscope is unphysical due to diffraction effects that produce a blurred image and hence uncertainty in position.]]
Heisenberg supposes that an electron is like a classical particle, moving in the <math>x</math> direction along a line below the microscope. Let the [[Astronomy:Light cone|cone]] of light rays leaving the microscope lens and focusing on the electron make an angle  <math>\varepsilon</math> with the electron. Let <math>\lambda</math> be the [[Wavelength|wavelength]] of the light rays. Then, according to the laws of classical optics, the microscope can only [[Physics:Optical resolution|resolve]] the position of the electron up to an accuracy of<ref name=Heisenberg1949>{{cite book|author=Werner Heisenberg|title=The Physical Principles of the Quantum Theory|year=1949|publisher=Courier Dover Publications|isbn=978-0-486-60113-7|url-access=registration|url=https://archive.org/details/physicalprincipl00heis}}</ref>{{rp|21}}<ref name=Richmond2016>{{cite web
 | url =http://spiff.rit.edu/classes/phys314/lectures/heis/heis.html
 | title =Heisenberg's Microscope
 | last =Richmond
 | first =Michael
 | access-date =1 Sep 2016}}</ref>

:<math>\Delta x = \frac{\lambda}{\sin \varepsilon}.</math>

An observer perceives an image of the particle because the light rays strike the particle and bounce back through the microscope to the observer's eye. We know from experimental evidence that when a [[Physics:Photon|photon]] strikes an electron, the latter has a [[Physics:Compton wavelength|Compton recoil]] with [[Physics:Momentum|momentum]] proportional to <math>h/\lambda</math>, where <math>h</math> is [[Physics:Planck constant|Planck's constant]]. However, the extent of "recoil cannot be exactly known, since the direction of the scattered photon is undetermined within the bundle of rays entering the microscope."{{citation needed|date=April 2020}} In particular, the electron's momentum in the <math>x</math> direction is only determined up to<ref name=Richmond2016 />

:<math>\Delta p_x \approx \frac{h}{\lambda}\sin\varepsilon.</math>

Combining the relations for <math>\Delta x</math> and <math>\Delta p_x</math>, we thus have<ref name=Richmond2016/>

:<math>\Delta x \Delta p_x \approx \left( \frac{\lambda}{\sin \varepsilon} \right)\left( \frac{h}{\lambda}\sin\varepsilon \right) = h</math>,

which is an approximate expression of Heisenberg's [[Physics:Uncertainty principle|uncertainty principle]].{{citation needed|date=April 2020}}

==Analysis of argument==

Although the thought experiment was formulated as an introduction to Heisenberg's [[Physics:Uncertainty principle|uncertainty principle]], one of the pillars of modern physics, it [[Philosophy:Reductio ad absurdum|attacks the very premises]] under which it was constructed, thereby contributing to the development of an area of physics—namely, quantum mechanics—that redefined the terms under which the original thought experiment was conceived. 

Quantum mechanics questions whether an electron actually has a determinate position before it is disturbed by the measurement used to establish said determinate position. Under a more thorough quantum mechanical analysis, an electron has some probability of showing up at any point in the universe, though the probability that it will be far from where one  expects becomes very low at great distances from the neighborhood in which it is originally found. In other words, the "position" of an electron can only be stated in terms of a [[Probability distribution|probability distribution]], as can predictions of where it may move.{{Citation needed|date=May 2011}}

==See also==
* [[Physics:Atom localization|Atom localization]]
* [[Physics:Quantum mechanics|Quantum mechanics]]
* Basics of quantum mechanics
* [[Physics:Interpretation of quantum mechanics|Interpretation of quantum mechanics]]
* [[Physics:Philosophical interpretation of classical physics|Philosophical interpretation of classical physics]]
* [[Physics:Schrödinger's cat|Schrödinger's cat]]
* [[Physics:Uncertainty principle|Uncertainty principle]]
* [[Physics:Quantum field theory|Quantum field theory]]
* [[Physics:Electromagnetic radiation|Electromagnetic radiation]]

==References==
{{reflist}}

==Sources==
* {{cite book | last=Aczel | first=Amir | title=Entanglement : the unlikely story of how scientists, mathematicians, and philosophers proved Einstein's spookiest theory | publisher=Plume | publication-place=New York | year=2003 | isbn=978-0-452-28457-9 | oclc=53378914 | page=77-79}}
* {{cite journal | last=Bohr | first=N. | title=The Quantum Postulate and the Recent Development of Atomic Theory | journal=Nature | publisher=Springer Science and Business Media LLC | volume=121 | issue=3050 | year=1928 | issn=0028-0836 | doi=10.1038/121580a0 | pages=580–590| bibcode=1928Natur.121..580B | s2cid=4097746 | doi-access=free }}
* {{cite book | last=Heisenberg | first=Werner | title=Physics & philosophy : the revolution in modern science | publisher=HarperPerennial | publication-place=New York | year=2007 | isbn=978-0-06-120919-2 | oclc=135128032 | page=46}}
* {{cite book | last=Messiah | first=Albert | title=Quantum Mechanics | publisher=Dover Publications | publication-place=| year=2014 | isbn=978-0-486-78455-7 | oclc=874097814 | volume=I|page=143}}
* {{cite book | last=Newman | first=James | title=The World of Mathematics Set | publisher=Dover Publications | publication-place=City | year=2003 | isbn=978-0-486-43268-7 | oclc=691512261 | volume=II|page=1051-1055}}

==External links==
*[http://www.aip.org/history/heisenberg/p08b.htm History of Heisenberg's Microscope] {{Webarchive|url=https://web.archive.org/web/20051202185202/http://www.aip.org/history/heisenberg/p08b.htm |date=2005-12-02 }}
*[http://spiff.rit.edu/classes/phys314/lectures/heis/heis.html Lectures on Heisenberg's Microscope]

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[[Category:Thought experiments in quantum mechanics]]

{{Sourceattribution|Heisenberg's microscope}}