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{{Quantum book backlink|Timeline}}
'''Quantum mechanics''' (QM – also known as '''quantum physics''', or '''quantum theory''') is a branch of [[Wikipedia:physics|physics]] which deals with physical phenomena at microscopic scales. It departs from [[Wikipedia:classical mechanics|classical mechanics]] primarily at the ''[[Wikipedia:quantum realm|quantum realm]]'' of [[Wikipedia:atomic spacing|atomic]] and [[Wikipedia:subatomic scale|subatomic]] length scales. Quantum mechanics provides a mathematical description of much of the dual ''particle-like'' and ''wave-like'' behavior and interactions of [[Wikipedia:energy|energy]] and [[Wikipedia:matter|matter]]. In most contexts, the term ''''Quantum Mechanics'''' does not refer to theories that incorporate Einstein's theories of Relativity.  <ref>https://en.wikipedia.org/wiki/Quantum_mechanics</ref>

[[File:Quantum mechanics Timeline.jpg|thumb|250px]]
==What is Quantum Mechanics and what is Old Quantum Mechanics?==
[[File:Solvay conference 1927.jpg|The 1927 [[Wikipedia:Solvay Conference|Solvay Conference]] in [[Wikipedia:Brussels|Brussels]].]]

'''Old quantum theory''' is a collection of results from the years 1900–1925 which predate modern [[Physics:Quantum mechanics|Quantum mechanics]]. Although the theory was never complete or self-consistent, it yielded enough successes to establish that the classical Newtonian view of particle motion was insufficient. In 1926 [[Wikipedia:Erwin Schrödinger|Erwin Schrödinger]] found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. <ref>https://en.wikipedia.org/wiki/Old_quantum_theory</ref> At approximately the same time a matrix version of the theory was developed, and eventually the two versions merged to become equivalent.  This essay focuses on the more accessible ''wave'' version.<ref>https://en.wikipedia.org/w/index.php?title=Matrix_mechanics&oldid=577613976</ref>

===When is a system in the quantum realm?===
[[File:Modernphysicsfields.svg|thumb|right|468px|[[Wikipedia:Classical physics|Classical physics]] is usually concerned with everyday conditions: speeds much lower than the [[Wikipedia:speed of light|speed of light]], and sizes much greater than that of atoms. Modern physics is usually concerned with high velocities and small distances.]]
The '''[[Wikipedia: Planck constant| Planck constant]]''' (denoted <math>h</math>) is a [[Wikipedia: physical constant| physical constant]] that appears throughout the theory of quantum mechanics, but that has no basis in the classical physics of Isaac Newton.    The [[Wikipedia:Dimensional analysis|units]] of '''Plank's constant''' can be expressed as either, [[Wikipedia:energy|[energy]]]·[[Wikipedia:time|[time]]], or as [[Wikipedia:mass|[mass]]]·[[Wikipedia:velocity|[velocity]]]·[[Wikipedia:length|[length]]]; (both are called [[Wikipedia:Action (physics)|action]]). A particle is likely to be in the quantum realm if either the '''mass''', '''speed''', or '''length''' is small:

* [[Wikipedia:Electrons|Electrons]] have '''less mass''' than [[Wikipedia:proton|protons]] or [[Wikipedia:neutron|neutrons]].  For this reason, the electrons in an atom must be modeled by a quantum theory, but we have the option of treating the [[Wikipedia:nucleus|nucleus]] as a stationary and ''classical'' object at the center.<ref>This analysis assumes that the speed is much less than that of light (i.e. [[Wikipedia:nonrelativistic]]).</ref>

* Electrons confined to a '''small space''', such as the atom (typically 10<sup>-10</sup> meters) will exhibit quantum effects.  

* Atoms move very slowly when they are '''very cold'''.  Quantum effects can occur at very low temperatures.<ref>There is a fourth condition for quantum mechanics to occur: when elementary particles collide at very high energy, they can interact with each in such a way as to create new particles.  This is the field of [[Wikipedia:High energy physics|High Energy Physics]] (or ''particle physics''), and will not be part of our discussion, except in one minor way.  In the Compton effect, the x-ray was "high energy" (in those days) and acts as a classical particle (like a billiard ball) that rips apart the atom.</ref>

==Timeline==
* 1900 - [[Wikipedia:Max Planck|Max Planck]] vaguely <ref>Planck thought (perhaps correctly?) that it is not light but the way atoms behaved that was quantized.</ref> and reluctantly <ref>https://en.wikipedia.org/wiki/History_of_quantum_mechanics</ref> introduces what we now call the photon to accurately model the experimentally observed spectrum of '''[[Wikipedia:Black body radiation|Black body radiation]]'''. 
* 1905 – [[Wikipedia:Albert Einstein|Albert Einstein]] uses Max Planck's hypothesis to explain the [[Wikipedia:photoelectric effect|photoelectric effect]]. <ref>This and Planck's work begin to establish wave-particle duality for electromagnetic radiation.</ref>
* 1909 – [[Wikipedia:Geoffrey Ingram Taylor|Geoffrey Ingram Taylor]] shows that photons '''interfere with themselves''' even if only one photon is near the slits at any given time.<ref>Taylor, G. I. (1909).</ref>
* 1913 – Bohr uses the [[Wikipedia:Rydberg formula|Rydberg formula]] to propose the '''Bohr model''' of the atom.<ref>https://en.wikipedia.org/wiki/Bohr_model</ref> 
* 1923 – [[Wikipedia:Compton scattering|Arthur Compton]] demonstrates particle-like behavior of X-rays.<ref>https://en.wikipedia.org/wiki/Compton_scattering</ref> 
* 1924 – [[Wikipedia:Louis, 7th duc de Broglie|Louis de Broglie]] proposes '''matter waves'''.<ref>https://en.wikipedia.org/wiki/Matter_wave</ref>
* 1926 – [[Wikipedia:Erwin Schrödinger|Erwin Schrödinger]] formulates the '''[[Wikipedia:Schrödinger equation|Schrödinger equation]]'''. 
* 1926 – [[Wikipedia:Max Born|Max Born]] introduces the '''probability interpretation''' of the wavefunction.<ref>https://en.wikipedia.org/wiki/Born_rule</ref>
* 1927 – [[Wikipedia:Davisson–Germer experiment|Davisson–Germer experiment]] confirms electron diffraction.<ref>https://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment</ref>
* 1927 – [[Wikipedia:Werner Heisenberg|Werner Heisenberg]] formulates the '''uncertainty principle'''.<ref>https://en.wikipedia.org/wiki/Uncertainty_principle</ref>
* 1935 – [[Wikipedia:Schrödinger's cat|Schrödinger's cat]] and the [[Wikipedia:EPR paradox|EPR paradox]] highlight foundational issues.<ref>https://en.wikipedia.org/wiki/EPR_paradox</ref>

* 1947 – [[Wikipedia:Lamb shift|Lamb shift]] measured, confirming quantum electrodynamics (QED).<ref>https://en.wikipedia.org/wiki/Lamb_shift</ref>
* 1948 – [[Wikipedia:Quantum electrodynamics|QED]] developed by Feynman, Schwinger, and Tomonaga.<ref>https://en.wikipedia.org/wiki/Quantum_electrodynamics</ref>

* 1957 – [[Wikipedia:Many-worlds interpretation|Many-worlds interpretation]] proposed by Hugh Everett.<ref>https://en.wikipedia.org/wiki/Many-worlds_interpretation</ref>

* 1964 – [[Wikipedia:Bell's theorem|Bell's theorem]] establishes testable limits on local hidden variables.<ref>https://en.wikipedia.org/wiki/Bell%27s_theorem</ref>

* 1981 – [[Wikipedia:Quantum computing|Richard Feynman]] proposes quantum simulation.<ref>https://en.wikipedia.org/wiki/Quantum_computing</ref>
* 1984 – [[Wikipedia:Quantum cryptography|BB84 protocol]] introduces quantum cryptography.<ref>https://en.wikipedia.org/wiki/BB84</ref>

* 1994 – [[Wikipedia:Shor's algorithm|Shor's algorithm]] shows exponential speedup for factoring.<ref>https://en.wikipedia.org/wiki/Shor%27s_algorithm</ref>
* 1995 – [[Wikipedia:Quantum error correction|Quantum error correction]] developed.<ref>https://en.wikipedia.org/wiki/Quantum_error_correction</ref>

* 1998 – First experimental [[Wikipedia:Quantum teleportation|quantum teleportation]].<ref>https://en.wikipedia.org/wiki/Quantum_teleportation</ref>

* 2001 – First experimental demonstration of Shor's algorithm.<ref>https://en.wikipedia.org/wiki/Shor%27s_algorithm</ref>

* 2012 – [[Wikipedia:Higgs boson|Higgs boson]] discovered at CERN.<ref>https://en.wikipedia.org/wiki/Higgs_boson</ref>

* 2015 – Loophole-free [[Wikipedia:Bell test experiments|Bell test experiments]] performed.<ref>https://en.wikipedia.org/wiki/Bell_test_experiments</ref>

* 2019 – Google claims [[Wikipedia:Quantum supremacy|quantum supremacy]].<ref>https://en.wikipedia.org/wiki/Quantum_supremacy</ref>

* 2022 – Nobel Prize awarded for experimental tests of Bell inequalities.<ref>https://www.nobelprize.org/prizes/physics/2022/summary/</ref>

* 2023 – Advances in [[Wikipedia:Quantum error correction|fault-tolerant quantum computing]].<ref>https://en.wikipedia.org/wiki/Quantum_error_correction</ref>

* 2024 – Continued progress in scalable quantum processors and error correction.<ref>https://en.wikipedia.org/wiki/Quantum_computing</ref>

* 2025 – Ongoing development of fault-tolerant quantum computers and quantum networks.<ref>https://en.wikipedia.org/wiki/Quantum_technology</ref>

==Black-body radiation 1900==
[[File:Black_body_realization.svg|thumb|upright=0.7|A [[Wikipedia:black body|black body]].]]

[[Wikipedia:Old quantum theory|Old quantum theory]] came into existence in 1900 with a calculation by [[Wikipedia:Max Planck|Max Planck]] that precisely matched the observed patterns of [[Wikipedia:Black body radiation|black-body radiation]].  He used the hypothesis that energy is radiated and absorbed in discrete "quanta" (or "energy elements").  In Planck's theory, the constant emerged as a relation between the [[energy]] (<math>E</math>) and the [[Wikipedia:frequency|frequency]] (<math>f</math>) associated with the interaction of light with the walls of the back body. This relation is called the [[Wikipedia:Planck's relation|Planck relation]]:

{{NumBlk|:|<math>E = hf \,.</math>|{{EquationRef|1|Eq. 1}}}}

Planck (cautiously) insisted that this was simply an aspect of the ''processes'' of absorption and emission of radiation and had nothing to do with the ''physical reality'' of the radiation itself.<ref>{{cite book |authorlink=Thomas Samuel Kuhn |first=T. S. |last=Kuhn |title=Black-body theory and the quantum discontinuity 1894-1912 |publisher=Clarendon Press |location=Oxford |year=1978 |isbn=0195023838 }}</ref> In fact, he considered his [[Wikipedia:quantum hypothesis|quantum hypothesis]] a mathematical trick to get the right answer rather than a sizeable discovery. 

[[file:Wavelength_in_sine_wave.JPG|right|180px]]This equation predicts that photon energy is directly proportional to frequency; if you double ''f'' then ''E'' doubles.  Frequency is inversely proportional to wavelength (since the speed of light is <math>c=f\lambda</math> where &lambda; ([[Wikipedia:lambda|lambda]]) is the length of a wave.


How Planck constructed his model and performed his calculation is beyond the scope of this essay.

==Photoelectric Effect 1905==
[[File: Photoelectric effect in a solid - diagram.svg|thumb|left|upright=.70|This depiction of photons (red) striking a metal plate and emitting '''''photoelectrons''''' serves to illustrate '''what cannot be seen'''.]]

In 1905 [[Wikipedia:Albert Einstein|Albert Einstein]] published a paper that
explained experimental data from the photoelectric effect that occurs when light strikes a metal surface.  The '''[[Wikipedia:Photoelectric effect|photoelectric effect]]''' is the emission of an electron from a substance when light is absorbed.  electrons are emitted from solids, liquids or gases when they absorb energy from light. Electrons emitted in this manner may be called '''''photoelectrons'''''.  His hypothesis was that the photon's energy frequecy obeyed <math>E=hf</math> 
(i.e. Eq. ({{EquationNote|1}}) ). Each photon was assumed to give all of its energy to a single electron, and the energy of this electron could be measured by measuring the voltage required to keep the electron from escaping from the metal.

====More links on the Photoelectric effect====
* http://phet.colorado.edu/en/simulation/photoelectric ''(Excellent simulation lab by [http://phet.colorado.edu/en/simulations/category/physics University of Colorado Boulder's PhET effort].)''
* https://en.wikiversity.org/wiki/Photoelectric_Effect_for_Beginners ''(A discussion similar to this one.}''

==Taylor makes things spooky with very dim light in 1909==
[[File:Wave-particle duality.gif|thumb|right|upright = 1|Interference occurs even when particles pass one at a time.]]
[[File:Young_Diffraction.png|thumb|right|upright = 1|Young's original drawing was used to make specfic predictions of the diffraction pattern for a given wavelength.]]
[[File:Doubleslit3Dspectrum.gif|thumb|left|upright = .70|We know light is a wave because it interferes with itself when passing through two slits.]]
While studying medicine at Göttingen in the 1790s, [[Wikipedia:Thomas Young (scientist)|Thomas Young]] wrote a thesis on the physical and mathematical properties of sound<ref name = "Mason">
{{cite book
 |last=Mason |first=P.
 |year=1981
 |title=The Light Fantastic
 |publisher=[[Penguin Books]]
 |isbn=978-0-14-006129-1
}}</ref> and in 1799, he presented a paper to the [[Wikipedia:Royal Society|Royal Society]] where he argued that light was also a wave.  His idea was furiously opposed because it contradicted Newton, whose views were considered sacred.  Nonetheless, he continued to develop his ideas. 
In 1801, he presented a famous paper to the Royal Society entitled "On the Theory of Light and Colours" <ref name=Young1802>
{{cite journal
 |last1=Young |first1=T. |author1-link=Thomas Young (scientist)
 |year=1802
 |title=The Bakerian Lecture: On the Theory of Light and Colours
 |journal=[[Philosophical Transactions of the Royal Society of London]]
 |volume=92 |issue= |pages=12–48
 |jstor=107113
}}</ref> which described various interference phenomena, and in 1803 he performed his famous double-slit experiment (strictly speaking, a double hole experiment).

Diffraction occurs also with [http://www.google.com/imgres?imgurl=&imgrefurl=http%3A%2F%2Fwww.thestargarden.co.uk%2FWaveTheories.html&h=0&w=0&tbnid=i4BMB0v5TfrAhM&zoom=1&tbnh=201&tbnw=251&docid=JaPdDmfbSb7XyM&tbm=isch&ei=uQwbVMb8FpD4yQS-7oDgAg&ved=0CAoQsCUoAgwater water] and sound.  Diffraction allows one to measure wavelength, and the same value of wavelength occurs for diffraction through single slits, double slits, and N-slits.  The results are consistent with diffraction through a circular hole.

This experiment was repeated 100 years later by [[Wikipedia:G. I. Taylor|G. I. Taylor]] in light so dim that only one photon at a time was likely to be involved with the interference.  Amazingly, one photon can interfere with itself.  Shown in the figure to the right is a simulated animation of what is observed.  Since the photons are passing one at a time, a time gap separates when each particle strikes the screen and creates a black dot. <ref>Taylor, Geoffrey Ingram. "Interference fringes with feeble light." Proceedings of the Cambridge Philosophical Society. Vol. 15. No. 1. 1909.</ref> The experiment has been repeated in recent years with electrons, atoms, and even molecules containing over 800 atoms.<ref>https://en.wikipedia.org/w/index.php?title=Double-slit_experiment&oldid=618587827</ref>

==Bohr-Rutherford Model 1913==
[[Image:Bohr atom model (mul).svg |thumb|right|280px|Here, E = E<sub>2</sub>-E<sub>1</sub>, is the energy lost by the electron as it "falls" from high to low energy.]]

The '''Bohr model''', introduced by [[Wikipedia:Niels Bohr|Niels Bohr]] in 1913, depicts the [[atom]] as a small, positively charged [[Wikipedia:atomic nucleus|nucleus]] surrounded by [[Wikipedia:electron|electron]]s that travel in circular orbelits around the nucleus&mdash;similar in structure to the [[Wikipedia:solar system|solar system]], but with attraction provided by [[Wikipedia:electrostatic force|electrostatic force]]s rather than [[Wikipedia:gravity|gravity]]. The Bohr model is a primitive model of the [[Wikipedia:hydrogen atom|hydrogen atom]] that has been surpassed by the more sophisticated Schrodinger's equation. Nevertheless, Bohr's calculation should be viewed as more than a ''lucky guess''.   A guess based on dimensional analysis using a new and controversial [[Wikipedia:Category:Fundamental constants|fundamental constant]] is also an ''educated guess''.   The Bohr model assumes circular orbits of radius, <math>v</math>, and follows Newton's laws of physics, but with the following two embellishments to Newton's theory: 

{{NumBlk|:|<math> 
n \lambda = 2 \pi r \,\,\,(n = 1, 2, 3, \cdots)
</math>|{{EquationRef|2|Eq. 2}}}}

{{NumBlk|:|<math> 
\lambda = h/p, 
</math>|{{EquationRef|3|Eq. 3}}}}

and <math>p=mv</math> is momentum.  Lambda, or <math>\lambda</math>, is now known as the De Broglie wavelength (although Bohr did not use the De Broglie wavelength to construct this model.)  For an electron of mass ,<math>m</math> and velocity <math>v</math>, <math>p=mv</math>.<ref>Incidentally, this equation also holds for photons, where <math>p=h/\lambda</math> is the photon's momentum.</ref>  

[[Image:Electron_wave_doubling_over.png|thumb|left|430px|Only the orbit on the left satisfies Bohr's conditions for an "allowed" orbit.]]
[[File:Waves in Box.svg|thumb|right|upright=.75|These [[Wikipedia:standing waves|standing waves]] somewhat resemble electrons in the Bohr model.  The momentum, and hence the speed and energy of each "orbit" can be calculated from the wavelength.]]


The meaning of allowed orbits can be discerned from the pair of figures situated to the left. The wavelength must be such that the number that fit into a full circumference (2пr) must equal one, or two, or 3, or 4 (and so forth). In one case, the wave fits perfectly in the circles (with exactly four wavelengths).  This corresponds to an electron in the third excited state, or n =4.  The other wave does not fit into its allowed radius because it doubles over itself.  This energy level does not exist for this atom.

Planck's relation (i.e. Eq.({{EquationNote|1}})) holds for the photons that are emitted from the hydrogen atom, with the photon energy, <math> E</math>, being equal to the difference in energy between the two electron orbitals. This can be expressed as an energy conservation law:

:<math> E_2 - E_1 = hf</math> ,

where <math> E_2 </math> is the higher energy and is <math> E_2 </math> the lower energy of the atom.  These energies were calculated using Newton's laws of physics, just as one would calculate the energy of a satellite orbiting the Earth.  The frequency of the light, <math> f </math>, could be measured by passing light the very hot gas (glowing) through a prism and measuring the wavelength.

While equations({{EquationNote|1}}), ({{EquationNote|2}}), and ({{EquationNote|3}}) seem to embody all the calculations of Old Quantum Mechanics, they oversimplify the lines of reasoning actually used.  Bohr, for example, did not use equation ({{EquationNote|3}}).  And he wrote ({{EquationNote|2}}) not as a statement about "allowed" wavelengths, but about the quantization of angular momentum (''L = mvr= nh/(2π)''). In fact, the wave behavior of matter particles such as the electron (i.e., matter waves) was not suspected in 1913, and at that time Bohr did not believe in the existence of photons. <ref>https://en.wikipedia.org/wiki/Bohr_model</ref>

==Compton Effect 1923==
[[Image:Newtons cradle animation book 2.gif|left|200px|thumb|Compton scattering is like this, but in two dimensions and relativistic.]]
[[Image:Compton-scattering.svg|thumb|upright=.7|right|A photon of wavelength <math>\lambda</math> comes in from the left, collides with a target at rest, and a new photon of wavelength <math>\lambda'</math> emerges at an angle <math>\theta</math>.]]

More evidence that Eq.({{EquationNote|3}}) also holds for the momentum of an individual photon was provided by [[Wikipedia:Arthur Holly Compton|Arthur Holly Compton]] in 1923.  Since the photon is generally understood to be [[Wikipedia:Photon#Experimental checks on photon mass|massless]], the momentum, <math>p</math>, of the photon is not equal to <math>mv</math>, but equal to <math>hf/c</math>, where <math>c</math> is the speed of light. The calculation of [[Wikipedia:Compton scattering%20Compton%20scattering|Compton scattering]] is a bit more difficult than those collisions usually studied in an undergraduate physics course because it is necessary to perform a [[Wikipedia:Special relativity|relativistic]] calculation.  Nevertheless, the calculation can be performed and predictions were confirmed by experimental observation.

==de Broglie Waves 1924==
{{cquote|When I conceived the first basic ideas of wave mechanics in 1923–24, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905.|20px|20px|De Broglie<ref>Louis de Broglie [http://www.springerlink.com/content/n170347gr6h82147/ "The Reinterpretation of Wave Mechanics" Foundations of Physics, Vol. 1 No. 1 (1970)]</ref>}}
[[File:Plane Wave 3D Animation 300x216 255Colors.gif|thumb|left|200px|How would these structures appear to move if an observer were moving alongside it?  How big would they be?  How fast would they be moving?.]]

Today, the de Broglie relations<ref>L. de Broglie, ''Recherches sur la théorie des quanta'' (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, ''Ann. Phys.'' (Paris) '''3''', 22 (1925).</ref>   are more likely to be expressed using different variables. Instead of <math>h</math>, it is more common to use
<math>\hbar=h/2\pi </math> (pronounced "h-bar").  Also,
<math>k= 2\pi/\lambda</math> is defined as the [[Wikipedia:wavenumber#In wave equations|wavenumber]],
<math>\omega=2\pi f</math> is the [[Wikipedia:angular frequency|angular frequency]]. In these variables, we have
<math> p = mv = \hbar k </math>, and <math>E = \hbar \omega</math>.  


Although we are using the de Broglie relations within the context of the non-relativistic quantum theory, much of de Broglie's argument was based on Einstein's theory of special relativity, which describes how length, time, mass and energy are perceived by observers who are moving relative to one another.   The figure on the left shows the crests and troughs of a travelling wave (in blue and red).  Einstein's special theory of relativity [[Wikipedia:Inertial reference frame|inertial reference frames]] describes how different observers (moving at different speeds) will perceive length, time, speed, momentum, and energy.  Self-consistency is possible only if frequency-wavenumber is proportional to energy-momentum.  Mathematically, De Broglie derived only the need for this proportionality, not its value.  But it was clear that ''<big>ħ</big>'' is that constant.

==Schrödinger Wave Equation 1926==
[[File:QuantumHarmonicOscillatorAnimation.gif|thumb|260px|left|Some trajectories of a [[Wikipedia:harmonic oscillator|harmonic oscillator]] (a ball attached to a [[Wikipedia:Hooke's law|spring]]) in [[Wikipedia:classical mechanics|classical mechanics]] (A–B) and [[Wikipedia:quantum mechanics|quantum mechanics]] (C–H). In quantum mechanics (C–H), the ball has a '''wave function''', which is shown with [[Wikipedia:real part|real part]] in blue and [[Wikipedia:imaginary part|imaginary part]] in red. The trajectories C,D,E,F, (but not G or H) are examples of [[Wikipedia:standing wave|standing wave]]s, (or "[[Wikipedia:stationary state|stationary state]]s"). Each standing-wave frequency is proportional to a possible [[Wikipedia:energy level|energy level]] of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have ''any'' energy.]]

[[File:frontgroupphase.gif|thumb|right|upright=0.5|This illustration illustrates how a simple wavepacket moves through space.]]

Schrödinger put forth a wave equation that gave the correct energy levels of the hydrogen atom in 1926.  <ref>https://en.wikipedia.org/wiki/Schrodinger</ref>  At last we have an equation from which a fundamental wave theory of quantum mechanics might be constructed.  

{{NumBlk|:|<math>i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)</math>|{{EquationRef|1|Eq. 4}}}}

Although Heisenberg, Born, and Jordan had already developed a parallel effort using matrices that would prove to be a mathematically equivalent theory<ref>https://en.wikipedia.org/wiki/Quantum_mechanics</ref>), we shall take this as the starting point in our quest for a fully developed theory.  While this equation looks intimidating to a novice, it is a standard wave equation that is in many respects simpler than [[Wikipedia:Maxwell's Equations|Maxwell's Equations]]. It is completely [[Wikipedia:Deterministic system|deterministic]], meaning that it allows one to predict how a wave will evolve (given [[Wikipedia: initial value problem|initial conditions]]).  '''In other words, like Newton's ''<math>F=ma</math>'', equation ('''{{EquationNote|4}}) '''predicts the future of the wave amplitude''' (We shall soon discover that predicting future values of ψ is not necessarily the same as predicting the future behavior of the particle.)   

[[Image:Hydrogen Density Plots.png|The electron probability density for the first few [[hydrogen atom]] electron [[atomic orbital|orbital]]s shown as cross-sections. These orbitals form an [[orthonormal basis]] for the wave function of the electron. Different orbitals are depicted with different scale.]]

To the left are some computer generated solutions Schrödinger's equations for a (very tiny) mass attached to a spring.  The symbol, ψ, is spelled psi, but often pronounced "sigh", with a hint of "p" at the beginning. It can also be pronounced "psee". It is called the "wavefunction", and is essentially the "amplitude", analogous to the "height" of the wave.  The wavefunctions C through F represent states of known energy not unlike the "allowed" orbits of the Bohr atom.  Wavefunctions G and H are more complicated and have no counterpart in Bohr's model of "allowed" states. They are known as "mixed energy" states. 
====What is the nature of the Wavefunction?====
The Schrödinger equation details the behavior of ψ but says nothing of its ''nature''.  Schrödinger unsuccessfully tried to interpret it as a charge density. <ref>{{cite book
 |last=Moore |first=W.J.
 |year=1992
 |title=Schrödinger: Life and Thought
 |publisher=[[Cambridge University Press]]
 |page=219
 |isbn=0-521-43767-9
}}</ref>   More specifically it seemed plausible to interpret the square of the amplitude, |ψ|<sup>2</sup>, as a charge density, since the total charge can be shown to remain constant if ψ obeys Schrödinger's equation.  Schrödinger always opposed a [[Wikipedia:statistical|statistical]] or probabilistic approach, with its associated [[Wikipedia:wavefunction collapse|discontinuities]]—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying [[Wikipedia:determinism|deterministic theory]]— and never reconciled with the [[Wikipedia:Copenhagen interpretation|Copenhagen interpretation]].  In 1926,  [[Wikipedia:Max Born|Max Born]] successfully interpreted ''ψ'' as the [[Wikipedia:probability amplitude|probability amplitude]], whose absolute square,|ψ|<sup>2</sup>, is equal to [[Wikipedia:Probability density function|probability density]]. <ref>
{{cite book
 |last=Moore |first=W.J.
 |year=1992
 |title=Schrödinger: Life and Thought
 |publisher=[[Cambridge University Press]]
 |page=220
 |isbn=0-521-43767-9
}}</ref> .<ref>It is clear that even in his last year of life, as shown in a letter to Max Born, that Schrödinger never accepted the Copenhagen interpretation (cf. p. 220).
 {{cite book
 |last=Moore |first=W.J.
 |year=1992
 |title=Schrödinger: Life and Thought
 |publisher=[[Cambridge University Press]]
 |page=479
 |isbn=0-521-43767-9
}}</ref> Like charge, probability also obeys a conservation law, namely that the sum of all probabilities always adds to 1 (i.e., 100%)

Even though  Schrödinger is (correctly) credited with inventing this wave equation, a connection between waves and particle motion was understood as early as 1834 when '''[[Wikipedia:Topic:Advanced Classical Mechanics/The Eikonal Approximation and Classical Particle Motion|Hamilton]]''' wrote equations that describe the path taken by light in the approximation that the wavelength is very small.

==Davisson–Germer's accidental diffraction of electrons 1927 ==
[[File:Davisson_and_Germer.jpg|thumb|right|upright=.70|Davisson (left) and Germer (right)]]
While attempting to clean an oxide film off of nickel surface, Davisson and Germer heated the specimen to high temperature, not knowing that this would create large single crystal big enough to caused diffraction of their electron beam.<ref name="Hugh D. Young 2004">Hugh D. Young, Roger A. Freedman: ''University Physics, Ed. 11.'' Pearson Education, Addison Wesley, San Francisco 2004, 0-321-20469-7, S. 1493-1494.</ref><ref>https://en.wikipedia.org/w/index.php?title=Davisson%E2%80%93Germer_experiment&oldid=580087519</ref>

==Heisenberg's uncertainty principle 1927==
Have you ever noticed that you can't ask a person what they are thinking without changing what they are thinking about?  Here is what one of the founders of Quantum Mechanics had to say:

{{cquote|If a person tries to observe what he is thinking about at the very moment that he is reflecting on a particular subject, it is generally agreed that he introduces unpredictable and uncontrollable changes in the his thoughts proceed thereafter.  Why this happens is not definitely known at the present, but some plausible explanations will be suggested later.|20px|20px|David Bohm<ref>Bohm, David.  Quantum Theory, Dourier Dover Publications, 1951]</ref>}}

Heisenberg came up with more convincing understanding that learning about something changes it in an unpredictable way.  His argument imagined a powerful but [[Wikipedia:Heisenberg%27s microscope|hypothetical microscope]] that could see the electron as it orbited around the nucleus.  Combining his knowledge of optics with the Compton Effect, Heisenberg concluded that the light required to see the electron would drastically change its orbit.  And if the light were gentle, with low-energy photons, the microscope would yield an image so blurred that it would be impossible to track the electron's path.
===== Measuring a particle at rest in a dark room =====
[[File:Wavelength=slitwidthspectrum.gif|thumb|left|Numerical approximation of diffraction pattern from a slit of width equal to wavelength of an incident plane wave in 3D spectrum visualization]]
[[File:5wavelength=slitwidthsprectrum.gif|thumb|right|Numerical approximation of diffraction pattern from a slit of width equal to five times the wavelength of an incident plane wave in 3D spectrum visualization]]

Suppose you are in large dark room with an electron that is known to be at rest.  You want to measure its location.  The room must be dark because light would interact with and induce motion in the electron.  So you put a small hole in a large flat sheet and wave the sheet past where you think the electron is (there is of course no air in the room.)  If the electron hits the sheet, your measurement was a failure.  But if you repeat the experiment often enough, you will eventually succeed and have knowledge about where the electron is located.  Since your sheet never touched the electron, is still at rest, right? '''Wrong!--No force ever acted on the electron; but nevertheless the electron has now been set in motion!'''  Moreover, you don't know exactly how it is moving.

To understand why, put yourself reference frame of your flat sheet.  The sheet is now at rest and an electron is moving towards it.   By the de Broglie hypothesis, this electron will have wavelength (given by Eq. ({{EquationNote|3}}) to be ''h/mv'').  But if the electron acts as a wave, it  will undergo single slit diffraction as shown in the figures to the left and right.  Notice how decreasing the diameter of the hole increases the motion the spreading of the wave.  The more you know about the position, the less you know about the motion, and vice versa.   With a bit of basic algebra, the well-known equation for [[Wikipedia:Diffraction#Single-slit_diffraction|single slit diffraction]] can be shown to be consistent with the following inequality:

{{NumBlk|:|<math> \sigma_{x}\sigma_{p} \geq \frac{\hbar}{2}</math>|{{EquationRef|1|Eq. 5}}}}

where <math> \sigma_{x}</math>, is the uncertainty in position, and <math> \sigma_{p} </math> is the uncertainty in momentum, <math> p=mv </math>.  (The symbol <math> \sigma </math> is called "sigma").  Equation ({{EquationNote|5}}) is known as [[Wikipedia:Uncertainty principle|Heisenberg's uncertainty principle]], and renowned theoretical physicists have spent hours attempting to violate it.  Under the [[Wikipedia:Copenhagen interpretation|Copenhagen interpretation]] of quantum mechanics, it can be proven as a mathematical theorem.

====Copenhagen interpretation====
[[File:Partial_transmittance.gif|thumb|left|Delocalization of waves: What attribute of a particle can be in two places at  the same time?  Probability!]]
The image to the left illustrates a simple argument for a probabilistic interpretation.  Although the wave shown is not a solution to Schrodinger's equation.  Wavepackets can become delocalized, hear by impinging on some sort of barrier.  What attribute of a particle can be in two places at  the same time?  Probability! To create this situation in a classroom, tape a wooden block to a stretched snaky spring and strike it.

[[File:Snaky spring quantum collision.png|thumb|right|Create the splitting of a pulse in the classroom with a stretched snaky spring. After doing this the instructor holds up a coin, buts both hands behind his back and interchanges them so nobody knows which hand has the coin when both hands are held up.  "What property of a quarter can be in both hands at the same time?"]]
A detailed set of rules for interpreting ψ that is beyond our scope.  But for reference, three fundamental features of these rules are stated: 

# |ψ|<sup>2</sup>ΔV is the probability of the particle being inside a volume ΔV.
# If a wave function is a superposition of two or more states of known energy, the particle has a probability of being in any of those energy states.  In particular, if ψ<sub>1</sub> and ψ<sub>2</sub> are [[Wikipedia:Wave function#Mathematical introduction|normalized wavefunctions]]( ʃ|ψ|<sup>2</sup>dV=1), and if the (complex) amplitudes a<sub>1</sub> and a<sub>2</sub> are also normalized (i.e., |a<sub>1</sub>|<sup>2</sup>+|a<sub>2</sub>|<sup>2</sup>=1), then |a<sub>1</sub>|<sup>2</sup> is the probability of having energy E<sub>1</sub> and |a<sub>2</sub>|<sup>2</sup> is the probability of having energy E<sub>2</sub>.
# If a measurement changes our understanding of the nature of any of the particle's attributes (energy, position, velocity, ...), then the wavefunction immediately rearranges itself in a process called the "[[Wikipedia:Wave function collapse|collapsing of the wavefunction]]" .

====Example 1: Minimum uncertainty within Copenhagen interpretation from repetitive measurement====

We consider the perfect measurement when the dense grid of [[Wikipedia:Maxwell's_demon|Maxwell's demons]] is used to detect the 
presence of the particle at certain coordinate at certain time who can catch the particle at the coordinates
when they see it near.
Let us consider the free particle with the mass <math>m</math> in one spatial dimension in spreading Gaussian wave packet 
state:
:<math>|\psi(x,t)|^2 = [2 \pi \Delta x(t)^2]^{-1/2} e^{-\frac{x^2}{2 \Delta x(t)^2 }}</math>
where the spreading :<math>\Delta x^2(t)</math> evolves in time
:<math>\Delta x(t)^2=\Delta x(0)^2 + \frac{\hbar^2 t^2}{4 m^2 \Delta x(0)^2} </math>
According to the Copenhagen interpretation the position measurement will collapse the wave 
function to the eigenstate of the position operator (or back prepare it in this state)
:<math>x \delta (x-x_0) = x_0 \delta (x-x_0) </math>
To measure the momentum of the particle two consecutive measurements of the position must me done separated in time by
<math>\tau</math> to find the velocity first:
:<math>p_n=m \frac{x_n-x_{n-1}}{\tau}</math>
While the delta function may be well approximated by a Gaussian with the infinitesimal spread <math>\epsilon</math>
:<math>\Delta x(0)^2=\epsilon^2</math> (the exact wave function is proportional to it) 
the time evolution of the collapsed state between repetitive measurement which will be therefore
:<math>\Delta x(\tau)^2=\frac{\hbar^2 \tau^2}{4 m^2 \epsilon^2} </math>
Therefore after each measurement the wave functions reconstructs itself by the free evolution 
of the spreading near-delta function and is only space shifted so the demolition measurements are
nonetheless done effectively on the same quantum state.
We can reset the center of the coordinate system to 0 after each measurement so if <math>|\psi(x)|^2</math> is the 
probability density distribution of each square of position measurement  the average 
over many measurements is 
:<math>\int |\psi(x)|^2 x^2 = \Delta x(\tau)^2</math>
The corresponding average of the momentum dispersion is immediately
:<math>\int |\psi(x)|^2 m^2 x^2/\tau^2 = \Delta p(\tau)^2</math>
or
:<math>\Delta p(\tau)^2=\frac{\hbar^2 }{4  \epsilon^2} + \frac{\epsilon^2 m^2}{\tau^2} </math>
Putting the between-measurement time <math>\tau</math> infinitesimal before <math>\epsilon</math> we get 
:<math>\Delta x(\tau)^2 \Delta p(\tau)^2=\frac{\hbar^2}{4} </math>
or the minimum uncertainty.

====Example 2: Minimum uncertainty of position and momentum in 3 dimensions====

The uncertainty relation in more dimensions is exotic in the sense that coefficients appear at the 
front of <math>\hbar/2 </math>. We want the uncertainty relation for <math>\sigma_{r}\sigma_{p}</math>

We will use the general relation for the uncertainty for the operators <math>A, B</math>
::: <math>\sigma_{A}\sigma_{B} \geq  \frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle \right|.</math>
We want the uncertainty relation for <math>\sigma_{r}\sigma_{p}</math> i.e. for the operators
::: <math>r^2=x^2+y^2+z^2</math>
::: <math>p^2=p_x^2+p_y^2+p_z^2</math>

The first step is to find the auxiliary operators <math>\tilde r,  \tilde p</math> such that this relation
can be used directly. We must make therefore the same trick that Dirac made to calculate the 
square root of the Klein-Gordon operator to get the [[Wikipedia:Dirac equation|Dirac equation]]:

::: <math>\tilde p = \alpha_1 p_x + \alpha_2 p_y + \alpha_3 p_z </math>
::: <math>\tilde r = \alpha_1 x + \alpha_2 y + \alpha_3 z </math>

where <math>\alpha_i</math> are matrices from the Dirac equation:

::: <math>\alpha_i^2=1</math>
::: <math>\alpha_i \alpha_k + \alpha_k \alpha_i= 2  \delta_{ik} </math>

Therefore we have 

::: <math>\tilde p^2 = p^2</math>
::: <math>\tilde r^2 = r^2</math>

We can now readily calculate the commutator while using the anty-comutation properties 
of <math>\alpha_i</math> matrixes and noticing that the symmetric Gaussian state 
<math>e^{-a r^2}</math> is annihilating in average the terms containing mixed variable like
<math>x p_y</math>

Calculating 9 commutators (mixed may be zero by Gaussian example) we obtain 

::: <math>\left|\langle[\tilde r,\tilde p]\rangle \right|= 3 \hbar .</math>

Therefore 

<math>\sigma_{r}\sigma_{p}\geq \frac{3}{2}\hbar</math>

which is 3 times (space dimension) more then for the one dimension.

==Schrodinger's cat 1935==
[[File:Schrodingers cat.svg|thumb|right|A cat, a flask of poison, and a [[Wikipedia:radioactive|radioactive]] source are placed in a sealed box. If an internal monitor detects a single atom decaying, the flask is shattered, releasing the poison that kills the cat.  The Copenhagen interpretation implies that after a while, the cat is [[Wikipedia:quantum superposition|''simultaneously'' alive ''and'' dead]]. |180x180px]]

The idea that an object can be in two places at the same time is unsettling, although there seems to be little evidence that this philosophical mystery has interfered with the pursuit of experimental or even theoretical physics.  Three ideas have been put forth that have become cultural icons of Quantum Mechanics.  They are the EPR Paradox, Schrodingers Cat, and Bell's Theorem (which has been justifiably called “the most profound discovery of science”.<ref>[[Henry P. Stapp]], "Bell's Theorem and World Process", ''[[Nuovo Cimento]]'', Vol. 29B, No. 2, p. 270 (1975). [http://link.springer.com/article/10.1007/BF02728310 (Quote on p. 271)]</ref>)  Of these three cultural icons, Schrodinger's cat is easiest to grasp. Schrödinger wrote:<ref>[http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html#sect5 Schroedinger: "The Present Situation in Quantum Mechanics." 5. Are the Variables Really Blurred?]</ref>

{{cquote|One can even set up quite ridiculous cases.  A [[Wikipedia:cat|cat]] is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a [[Wikipedia:Geiger counter|Geiger counter]], there is a tiny bit of [[Wikipedia:radioactive|radioactive]] substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the [[Wikipedia:Geiger counter|counter tube]] discharges and through a relay releases a hammer that shatters a small flask of [[Wikipedia:hydrocyanic acid|hydrocyanic acid]]. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has [[Wikipedia:radioactive decay|decayed]]. The [[Wikipedia:wave function|psi-function]] of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts..|20px|20px|Erwin Schrödinger<ref>''Die gegenwärtige Situation in der Quantenmechanik (The present situation in quantum mechanics)'', Naturwissenschaften<br>(translated by John D. Trimmer in Proceedings of the American Philosophical Society)]</ref>}}

====Developments====

* 1964 – [[Wikipedia:John Stewart Bell|John Stewart Bell]] puts forth '''[[Wikipedia:Bell's theorem|Bell's theorem]]'''
* 1982 - Bell's theorem was '''experimentally verified''' by [[Wikipedia:Alain Aspect|Alain Aspect]]<ref>in what might be '''the most philosophically profound experiment ever performed'''</ref>
* 2010 - Cleland puts '''an object visible to the naked eye''' (but smaller than a cat) into a '''mixed energy state'''.  <ref>http://www.nature.com/news/2010/100317/full/news.2010.130.html Andrew Cleland at the University of California, Santa Barbara, and his team cooled a tiny metal paddle until it reached its quantum mechanical 'ground state' — the lowest-energy state permitted by quantum mechanics. They then used the weird rules of quantum mechanics to simultaneously set the paddle moving while leaving it standing still.</ref>
==Recent Dvelopments==
==Lamb shift 1947==
[[File:Hydrogen fine structure2.svg|thumb|Hydrogen energy level comparison between Bohr's model and Dirac's equation, including fine structure and denoting the Lyman alpha transitions]]

In 1947 [[Wikipedia:Willis Lamb|Willis Lamb]] and [[Wikipedia:Robert Retherford|Robert Retherford]] measured a tiny shift in the hydrogen spectrum now called the '''[[Wikipedia:Lamb shift|Lamb shift]]'''.<ref>https://en.wikipedia.org/wiki/Lamb_shift</ref> The result was important because it showed that a simple relativistic wave equation for the electron was not enough.  Something else was affecting the atom.

That "something else" was eventually understood to be the interaction of the electron with the quantized electromagnetic field.  The Lamb shift therefore became one of the early great successes of modern [[Wikipedia:quantum electrodynamics|quantum electrodynamics]].<ref>https://en.wikipedia.org/wiki/Quantum_electrodynamics</ref>

==Quantum electrodynamics 1948==
[[File:Feynmann Diagram Gluon Radiation.svg|thumb|right|upright=.8|A [[Wikipedia:Feynman diagram|Feynman diagram]].  Such diagrams organize calculations in quantum field theory.]]

By the late 1940s [[Wikipedia:Richard Feynman|Richard Feynman]], [[Wikipedia:Julian Schwinger|Julian Schwinger]], and [[Wikipedia:Sin-Itiro Tomonaga|Sin-Itiro Tomonaga]] had developed the theory now called '''[[Wikipedia:quantum electrodynamics|quantum electrodynamics]]''' (QED).<ref>https://en.wikipedia.org/wiki/Quantum_electrodynamics</ref> QED combines quantum mechanics with the electromagnetic field and describes how light and charged matter interact.

QED was revolutionary because it produced extremely accurate predictions, including corrections associated with the Lamb shift and the magnetic moment of the electron.  It also introduced calculational tools that became central to later quantum field theories.

==Many-worlds interpretation 1957==
[[File:Schroedingers cat film.svg|thumb|250px|The quantum-mechanical "[[Wikipedia:Schrödinger's cat|Schrödinger's cat]]".]]

In 1957 [[Wikipedia:Hugh Everett III|Hugh Everett III]] proposed the '''[[Wikipedia:Many-worlds interpretation|many-worlds interpretation]]''' of quantum mechanics.<ref>https://en.wikipedia.org/wiki/Many-worlds_interpretation</ref> Rather than assuming that a measurement causes the wavefunction to collapse, Everett suggested that the wavefunction always evolves according to Schrödinger's equation.

In this view, a measurement correlates observer and system, producing effectively separate branches corresponding to different outcomes.  Whether this is the best interpretation remains controversial, but Everett's proposal permanently changed the discussion of the foundations of quantum mechanics.

[[Wikipedia:Schrödinger's cat|Schrödinger's cat]]" paradox according to the many-worlds interpretation. In this interpretation, every quantum event is a branch point; the cat is both alive and dead, even after the box is opened, but the "alive" and "dead" cats are in different branches of the multiverse, both of which are equally real, but which do not interact with each other.

==Bell's theorem 1964==
[[File:Bell-test-photon-analyer.svg]]
In 1964 [[Wikipedia:John Stewart Bell|John Bell]] derived what is now called '''[[Wikipedia:Bell's theorem|Bell's theorem]]'''.<ref>https://en.wikipedia.org/wiki/Bell%27s_theorem</ref> Bell showed that any theory based on ''local hidden variables'' must satisfy certain inequalities.  Quantum mechanics predicts situations in which those inequalities are violated.

Bell's theorem transformed philosophical debate into experimental science.  It showed that the strange correlations predicted by quantum mechanics could be tested in the laboratory, and that "completing" quantum mechanics with local hidden variables would not work.

==Quantum simulation 1981==
[[File:Bloch sphere.svg|thumb|right|upright=.8|A [[Wikipedia:qubit|qubit]] can be represented geometrically on a Bloch sphere.]]

In 1981 [[Wikipedia:Richard Feynman|Richard Feynman]] argued that classical computers are poorly suited for simulating quantum systems, and suggested that one should instead build '''quantum simulators'''.<ref>https://en.wikipedia.org/wiki/Quantum_computing</ref> This idea is one of the conceptual origins of quantum computing.

The basic insight is simple: if nature is quantum mechanical, then a controllable quantum system may efficiently imitate another quantum system.  This turned the weirdness of quantum mechanics into a possible computational resource.

== Quantum cryptography 1984==
[[File:BB84_QKD_protocol_-_Virtual_Lab_by_Quantum_Flytrap.png|An interactive simulation of an [[Quantum_optics|optical implementation]] of the BB84 quantum key distribution protocol in the Virtual Lab by Quantum Flytrap,<ref>{{cite journal|last1=Migdał|first1=Piotr|last2=Jankiewicz|first2=Klementyna|last3=Grabarz|first3=Paweł|last4=Decaroli|first4=Chiara|last5=Cochin|first5=Philippe|title=Visualizing quantum mechanics in an interactive simulation - Virtual Lab by Quantum Flytrap|journal=Optical Engineering|volume=61|issue=8|article-number=081808|year=2022|doi=10.1117/1.OE.61.8.081808|arxiv=2203.13300}}</ref> [https://lab.quantumflytrap.com/lab/bb84 available online]. In this optical setup, bits are encoded using orthogonal [[Polarization_(waves)|polarization]] states of [[photon]]s. Alice and Bob select their measurement bases by rotating the polarization by 0 or 45 degrees using [[Faraday_rotator|Faraday rotators]]. [[Photon_counting|Single-photon detectors]] measure the output after the photons pass through a [[Beam_splitter|polarizing beam splitter]], which separates the polarizations.]]

In 1984 [[Wikipedia:Charles Bennett (physicist)|Charles Bennett]] and [[Wikipedia:Gilles Brassard|Gilles Brassard]] proposed the '''[[Wikipedia:BB84|BB84]]''' protocol.<ref>https://en.wikipedia.org/wiki/BB84</ref> It is the first major protocol for [[Wikipedia:quantum cryptography|quantum cryptography]], especially for quantum key distribution.

Its importance lies in the fact that eavesdropping on unknown quantum states generally disturbs them.  This makes it possible, in principle, to detect interception of the key.  BB84 showed that quantum mechanics is not only a theory of matter and radiation, but also a resource for information security.

==Shor's algorithm 1994==
[[File:Asymmetric encryption scheme.png|thumb|450px|]]

In 1994 [[Wikipedia:Peter Shor|Peter Shor]] discovered '''[[Wikipedia:Shor's algorithm|Shor's algorithm]]''', a quantum algorithm for factoring large integers in polynomial time.<ref>https://en.wikipedia.org/wiki/Shor%27s_algorithm</ref> This was a turning point in the history of quantum computing.

Its significance is practical as well as theoretical.  Many classical cryptographic systems rely on the difficulty of factoring.  Shor's work showed that a sufficiently powerful quantum computer would threaten such systems and therefore gave the field a major strategic and scientific push.

Diagram presenting the encryption and the decryption of a document using asymmetric cryptography. Some forms of encryption (including asymmetric cryptography) are at risk of being broken by future quantum computers.

==Quantum error correction 1995==
[[File:Shore code.svg|thumb|Quantum circuit to encode a single logical qubit with the Shor code. ''E'' indicates an error and the rest of the circuit to the right decodes the state.]]

One of the apparent obstacles to quantum computing is that quantum states are easily disturbed by noise and decoherence.  In 1995, schemes for '''[[Wikipedia:Quantum error correction|quantum error correction]]''' were developed that showed this obstacle was not fatal.<ref>https://en.wikipedia.org/wiki/Quantum_error_correction</ref>

Quantum error correction works by encoding one logical qubit into a larger entangled state of several physical qubits.  Although unknown quantum states cannot be copied, they can still be protected by cleverly distributing information so that certain errors can be detected and corrected.

==Quantum teleportation 1998==
[[File:GlycoNMR simulation.jpg|thumb|Comparative prediction of the 13C NMR spectrum of sucrose using various methods.]]

In 1998 experimental groups demonstrated early forms of '''[[Wikipedia:quantum teleportation|quantum teleportation]]'''.<ref>https://en.wikipedia.org/wiki/Quantum_teleportation</ref> Quantum teleportation does not move matter itself from place to place.  Instead, it transfers the quantum state of one system to another distant system.

Teleportation requires two ingredients: entanglement and an ordinary classical communication channel.  It is one of the clearest demonstrations that quantum information behaves differently from classical information.

Comparative prediction of the 13C NMR spectrum of sucrose using various methods. Experimental spectrum is in the middle. Upper spectrum (black) was obtained by empirical routine. Lower spectra (red and green) were obtained by quantum-chemical calculations in PRIRODA and GAUSSIAN respectively. Included information: used theory level/basis set/solvent model, accuracy of prediction (linear correlation factor and root mean square deviation), calculation time on personal computer (blue).

==Experimental Shor demonstration 2001==
[https://commons.wikimedia.org/wiki/File:Quantum_teleportation_video.ogg?embedplayer=true  Schematic video demonstrating individual steps of quantum teleportation. A quantum state Q is sent from station A to station B using a pair of entangled particles created by source S. Station A measures its two particles and communicates the result to station B, which chooses an appropriate device based on the received message. Due to the action of the device, the state of the particle of station B turns into Q.] <br>

In 2001 a small-scale experimental implementation of '''[[Wikipedia:Shor's algorithm|Shor's algorithm]]''' was reported.<ref>https://en.wikipedia.org/wiki/Shor%27s_algorithm</ref> The number factored was tiny, but the importance of the experiment was symbolic: it showed that nontrivial quantum algorithms could be realized in the laboratory.

Such demonstrations did not yet threaten real-world cryptography, but they helped establish quantum computing as an experimental discipline rather than merely a theoretical possibility.

==Higgs boson 2012==
[[File:CMS Higgs-event.jpg|thumb|right|100px|[[Wikipedia:Higgs boson|Higgs boson]] observed at CERN.]]

In 2012 experiments at CERN announced the discovery of the '''[[Wikipedia:Higgs boson|Higgs boson]]'''.<ref>https://en.wikipedia.org/wiki/Higgs_boson</ref> Strictly speaking, this belongs more naturally to high-energy particle physics and quantum field theory than to nonrelativistic quantum mechanics.

Nevertheless, it is historically relevant because it confirmed a central part of the quantum field-theoretic description of fundamental particles.  It also illustrates how the quantum viewpoint ultimately extends far beyond atoms and light.

==Loophole-free Bell tests 2015==
[[File:Two channel bell test.svg|300px|thumb|right|'''Scheme of a "two-channel" Bell test''']]
In 2015 several groups reported '''[[Wikipedia:Bell test experiments|loophole-free Bell test experiments]]'''.<ref>https://en.wikipedia.org/wiki/Bell_test_experiments</ref> Earlier Bell tests strongly supported quantum mechanics, but skeptics could point to experimental "loopholes" such as imperfect detection or possible communication between devices.

The 2015 experiments were designed to close the major loopholes simultaneously.  Their results further strengthened the conclusion that nature violates Bell inequalities in the way predicted by quantum mechanics.
The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a two-channel polariser whose orientation can be set by the experimenter. Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM.

==Quantum supremacy claim 2019==
[[Commons:File:Demonstrating Quantum Supremacy.webm|Demonstrating Quantum Supremacy]]<br>
In 2019 Google announced that its quantum processor had performed a specific computation faster than a known classical alternative, and described the result as '''[[Wikipedia:quantum supremacy|quantum supremacy]]'''.<ref>https://en.wikipedia.org/wiki/Quantum_supremacy</ref>

The term itself is controversial, and the practical importance of the particular benchmark was debated.  Even so, the announcement marked an important moment in public awareness of quantum computing and intensified work on useful quantum advantage.

==Entanglement Nobel Prize 2022==
[[File:Key witness helps scientists detect ‘spooky’ quantum entanglement in solid materials (53804603193).png|thumb|200px|‘Spooky’ quantum entanglement in solid materials]]

In 2022 the Nobel Prize in Physics was awarded to [[Wikipedia:Alain Aspect|Alain Aspect]], [[Wikipedia:John Clauser|John Clauser]], and [[Wikipedia:Anton Zeilinger|Anton Zeilinger]] for experiments involving entangled photons and tests of Bell inequalities.<ref>https://www.nobelprize.org/prizes/physics/2022/summary/</ref>

This award was historically important because it recognized that foundational questions once treated as philosophical had become central experimental physics.  Entanglement is now also a practical resource in quantum information science.

<br><br>
==Fault-tolerant progress 2023==
[[File:不同量子计算范式的概念区别.png|thumb|Illustration of different quantum computing paradigms]]
In 2023 researchers reported further progress in '''fault-tolerant quantum computing''' and in experiments showing that logical qubits can be protected better than the underlying physical qubits.<ref>https://en.wikipedia.org/wiki/Quantum_error_correction</ref>

This matters because large-scale quantum computing requires not merely a few good qubits, but a scalable architecture in which errors can be suppressed faster than they accumulate.  Quantum error correction therefore remains central to the field.
<br><br>

==Scalable processors and error correction 2024==
In 2024 work continued on larger and more reliable quantum processors, improved control systems, and better implementations of error-correcting codes.<ref>https://en.wikipedia.org/wiki/Quantum_computing</ref> These advances did not yet amount to fully general fault-tolerant quantum computing, but they represented steady progress toward that goal.

By this stage, the field had clearly moved beyond proof-of-principle demonstrations and into engineering questions involving architecture, fabrication, calibration, and error management.

==Quantum computing and networking 2025==
In 2025 research continued on '''fault-tolerant quantum computers''' and on the development of '''quantum networks''', including protocols for distributed entanglement and secure communication.<ref>https://en.wikipedia.org/wiki/Quantum_technology</ref>

Whether these efforts soon produce broadly useful machines remains uncertain, but they show that quantum mechanics has evolved from a theory explaining atomic spectra into a framework supporting new technologies for computation, communication, and sensing.
==See also==
{{:Physics:Quantum basics/See also}}
==[[Physics:Quantum_mechanics/Timeline/Quiz/]]==
[[Physics:Quantum_mechanics/Timeline/Quiz/|See the subpage for a quiz on this topic.]]

==References and Endnotes==
* Most of material was lifted out of [http://www.wikipedia.org/ Wikipedia]
{{reflist|3}}
{{Author|Harold Foppele}}

[[Category:Quantum mechanics]]
[[Category:Science]]
[[Category:History of Physics]]