Wave function - Revision history

imported>Transmittibal spongioform encephalopathy: /* top */Fixed typo

2025-12-18T12:35:31Z

top: Fixed typo

← Previous revision		Revision as of 12:35, 18 December 2025
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	\| direction = vertical		\| direction = vertical
	\| width = 402		\| width = 402
	\| footer = The [[Real and imaginary parts\|real parts]] of position wave function {{math\|Ψ(''x'')}} and momentum wave function {{math\|Φ(''p'')}}, and corresponding probability densities {{math\|{{!}}Ψ(''x''){{!}}<sup>2</sup>}} and {{math\|{{!}}Φ(''p''){{!}}<sup>2</sup>}}, for one spin-0 particle in one {{mvar\|x}} or {{mvar\|p}} dimension. The ~~colour~~ opacity of the particles corresponds to the probability density (''not'' the wave function) of finding the particle at position {{mvar\|x}} or momentum {{math\|''p''}}.		\| footer = The [[Real and imaginary parts\|real parts]] of position wave function {{math\|Ψ(''x'')}} and momentum wave function {{math\|Φ(''p'')}}, and corresponding probability densities {{math\|{{!}}Ψ(''x''){{!}}<sup>2</sup>}} and {{math\|{{!}}Φ(''p''){{!}}<sup>2</sup>}}, for one spin-0 particle in one {{mvar\|x}} or {{mvar\|p}} dimension. The color opacity of the particles corresponds to the probability density (''not'' the wave function) of finding the particle at position {{mvar\|x}} or momentum {{math\|''p''}}.
	\| image1 = Quantum mechanics standing wavefunctions.svg		\| image1 = Quantum mechanics standing wavefunctions.svg
	\| caption1 = [[Standing wave]]s for a [[particle in a box]], examples of [[stationary state]]s.		\| caption1 = [[Standing wave]]s for a [[particle in a box]], examples of [[stationary state]]s.

imported>Dan100 at 04:19, 9 September 2025

2025-09-09T04:19:56Z

← Previous revision		Revision as of 04:19, 9 September 2025
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	[[File:Indeterminacy principle.gif\|thumb\|The wave function of an initially very localized free particle.]]		[[File:Indeterminacy principle.gif\|thumb\|The wave function of an initially very localized free particle.]]

	In [[quantum physics]], a '''wave function''' (or '''wavefunction''') is a mathematical description of the [[quantum state]] of an isolated [[quantum system]]. The most common symbols for a wave function are the Greek letters {{math\|''ψ''}} and {{math\|Ψ}} (lower-case and capital [[psi (letter)\|psi]], respectively). Wave functions are [[complex number\|complex-valued]]. For example, a wave function might assign a complex number to each point in a region of space. The [[Born rule]]<ref name=Born_1926_A /><ref name="Born_1926_B" /><ref>[[Max Born\|Born, M.]] (1954).</ref> provides the means to turn these complex [[probability amplitude]]s into actual probabilities. In one common form, it says that the [[squared modulus]] of a wave function that depends upon position is the [[probability density function\|probability density]] of [[measurement in quantum mechanics\|measuring]] a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply [[Operator (quantum mechanics)\|quantum operators]], whose eigenvalues correspond to sets of possible results of measurements, to the wave function {{math\|''ψ''}} and calculate the statistical distributions for measurable quantities.		In [[quantum physics]], a '''wave function''' (or '''wavefunction''') is a mathematical description of the [[quantum state]] of an isolated [[quantum system]]. The most common symbols for a wave function are the Greek letters {{math\|''ψ''}} and {{math\|Ψ}} (lower-case and capital [[psi (letter)\|psi]], respectively).

			According to the [[superposition principle]] of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a [[Hilbert space]]. The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the [[Born rule]], relating transition probabilities to inner products. The [[Schrödinger equation]] determines how wave functions evolve over time, and a wave function behaves qualitatively like other [[wave]]s, such as [[water wave]]s or waves on a string, because the Schrödinger equation is mathematically a type of [[wave equation]]. This explains the name "wave function", and gives rise to [[wave–particle duality]]. However, whether the wave function in quantum mechanics describes a kind of physical phenomenon is still open to different [[Interpretations of quantum mechanics\|interpretations]], fundamentally differentiating it from [[classic mechanical]] waves.{{sfn\|Born\|1927\|pp=354–357}}{{sfn\|Heisenberg\|1958\|p=143}}<ref>[[Werner Heisenberg\|Heisenberg, W.]] (1927/1985/2009). Heisenberg is translated by {{harvnb\|Camilleri\|2009\|p=71}}, (from {{harvnb\|Bohr\|1985\|p=142}}).</ref>{{sfn\|Murdoch\|1987\|p=43}}{{sfn\|de Broglie\|1960\|p=48}}{{sfn\|Landau\|Lifshitz\|1977\|p=6}}{{sfn\|Newton\|2002\|pp=19–21}}

			Wave functions are [[complex number\|complex-valued]]. For example, a wave function might assign a complex number to each point in a region of space. The [[Born rule]]<ref name="Born_1926_A" /><ref name="Born_1926_B" /><ref>[[Max Born\|Born, M.]] (1954).</ref> provides the means to turn these complex [[probability amplitude]]s into actual probabilities. In one common form, it says that the [[squared modulus]] of a wave function that depends upon position is the [[probability density function\|probability density]] of [[measurement in quantum mechanics\|measuring]] a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply [[Operator (quantum mechanics)\|quantum operators]], whose eigenvalues correspond to sets of possible results of measurements, to the wave function {{math\|''ψ''}} and calculate the statistical distributions for measurable quantities.

	Wave functions can be [[function (mathematics)\|functions]] of variables other than position, such as [[momentum]]. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a [[Fourier transform]]. Some particles, like [[electron]]s and [[photon]]s, have nonzero [[Spin (physics)\|spin]], and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as [[isospin]]. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a [[column matrix]] (e.g., a {{math\|2 × 1}} column vector for a non-relativistic electron with spin {{math\|{{frac\|1\|2}}}}).		Wave functions can be [[function (mathematics)\|functions]] of variables other than position, such as [[momentum]]. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a [[Fourier transform]]. Some particles, like [[electron]]s and [[photon]]s, have nonzero [[Spin (physics)\|spin]], and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as [[isospin]]. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a [[column matrix]] (e.g., a {{math\|2 × 1}} column vector for a non-relativistic electron with spin {{math\|{{frac\|1\|2}}}}).

	According to the [[superposition principle]] of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a [[Hilbert space]]. The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the [[Born rule]], relating transition probabilities to inner products. The [[Schrödinger equation]] determines how wave functions evolve over time, and a wave function behaves qualitatively like other [[wave]]s, such as [[water wave]]s or waves on a string, because the Schrödinger equation is mathematically a type of [[wave equation]]. This explains the name "wave function", and gives rise to [[wave–particle duality]]. However, whether the wave function in quantum mechanics describes a kind of physical phenomenon is still open to different [[Interpretations of quantum mechanics\|interpretations]], fundamentally differentiating it from [[classic mechanical]] waves.{{sfn\|Born\|1927\|pp=354–357}}{{sfn\|Heisenberg\|1958\|p=143}}<ref>[[Werner Heisenberg\|Heisenberg, W.]] (1927/1985/2009). Heisenberg is translated by {{harvnb\|Camilleri\|2009\|p=71}}, (from {{harvnb\|Bohr\|1985\|p=142}}).</ref>{{sfn\|Murdoch\|1987\|p=43}}{{sfn\|de Broglie\|1960\|p=48}}{{sfn\|Landau\|Lifshitz\|1977\|p=6}}{{sfn\|Newton\|2002\|pp=19–21}}

	==Historical background==		==Historical background==

imported>Erkcan: fix a sentence that got broken by an anonymous edit on 16 May 2016 (can't believe no one has fixed this in 9 years)

2025-06-21T18:24:51Z

fix a sentence that got broken by an anonymous edit on 16 May 2016 (can't believe no one has fixed this in 9 years)

← Previous revision		Revision as of 18:24, 21 June 2025
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	Wave functions can be [[function (mathematics)\|functions]] of variables other than position, such as [[momentum]]. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a [[Fourier transform]]. Some particles, like [[electron]]s and [[photon]]s, have nonzero [[Spin (physics)\|spin]], and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as [[isospin]]. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a [[column matrix]] (e.g., a {{math\|2 × 1}} column vector for a non-relativistic electron with spin {{math\|{{frac\|1\|2}}}}).		Wave functions can be [[function (mathematics)\|functions]] of variables other than position, such as [[momentum]]. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a [[Fourier transform]]. Some particles, like [[electron]]s and [[photon]]s, have nonzero [[Spin (physics)\|spin]], and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as [[isospin]]. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a [[column matrix]] (e.g., a {{math\|2 × 1}} column vector for a non-relativistic electron with spin {{math\|{{frac\|1\|2}}}}).

	According to the [[superposition principle]] of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a [[Hilbert space]]. The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the [[Born rule]], relating transition probabilities to inner products. The [[Schrödinger equation]] determines how wave functions evolve over time, and a wave function behaves qualitatively like other [[wave]]s, such as [[water wave]]s or waves on a string, because the Schrödinger equation is mathematically a type of [[wave equation]]. This explains the name "wave function", and gives rise to [[wave–particle duality]]. However, the wave function in quantum mechanics describes a kind of physical phenomenon~~, as of 2023~~ still open to different [[Interpretations of quantum mechanics\|interpretations]], ~~which~~ fundamentally ~~differs~~ from ~~that of~~ [[classic mechanical]] waves.{{sfn\|Born\|1927\|pp=354–357}}{{sfn\|Heisenberg\|1958\|p=143}}<ref>[[Werner Heisenberg\|Heisenberg, W.]] (1927/1985/2009). Heisenberg is translated by {{harvnb\|Camilleri\|2009\|p=71}}, (from {{harvnb\|Bohr\|1985\|p=142}}).</ref>{{sfn\|Murdoch\|1987\|p=43}}{{sfn\|de Broglie\|1960\|p=48}}{{sfn\|Landau\|Lifshitz\|1977\|p=6}}{{sfn\|Newton\|2002\|pp=19–21}}		According to the [[superposition principle]] of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a [[Hilbert space]]. The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the [[Born rule]], relating transition probabilities to inner products. The [[Schrödinger equation]] determines how wave functions evolve over time, and a wave function behaves qualitatively like other [[wave]]s, such as [[water wave]]s or waves on a string, because the Schrödinger equation is mathematically a type of [[wave equation]]. This explains the name "wave function", and gives rise to [[wave–particle duality]]. However, whether the wave function in quantum mechanics describes a kind of physical phenomenon is still open to different [[Interpretations of quantum mechanics\|interpretations]], fundamentally differentiating it from [[classic mechanical]] waves.{{sfn\|Born\|1927\|pp=354–357}}{{sfn\|Heisenberg\|1958\|p=143}}<ref>[[Werner Heisenberg\|Heisenberg, W.]] (1927/1985/2009). Heisenberg is translated by {{harvnb\|Camilleri\|2009\|p=71}}, (from {{harvnb\|Bohr\|1985\|p=142}}).</ref>{{sfn\|Murdoch\|1987\|p=43}}{{sfn\|de Broglie\|1960\|p=48}}{{sfn\|Landau\|Lifshitz\|1977\|p=6}}{{sfn\|Newton\|2002\|pp=19–21}}

	==Historical background==		==Historical background==

imported>OriolLopezMassaguer at 10:15, 17 June 2025

2025-06-17T10:15:14Z

← Previous revision		Revision as of 10:15, 17 June 2025
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	==External links==		==External links==
	* [~~http~~://~~www~~.eng.fsu.edu/~dommelen/quantum/~~style_a/complexs.html~~ Quantum Mechanics for Engineers]		* [https://web1.eng.famu.fsu.edu/~dommelen/quantum/ Quantum Mechanics for Engineers]
	* [http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_9/node2.html Spin wave functions NYU]		* [http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_9/node2.html Spin wave functions NYU]
	* [http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm Identical Particles Revisited, Michael Fowler]		* [http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm Identical Particles Revisited, Michael Fowler]

imported>Leonidlednev: Reverted 1 edit by Bailey6969 (talk) to last revision by MrGumballs

2025-05-14T21:25:25Z

Reverted 1 edit by Bailey6969 (talk) to last revision by MrGumballs

Show changes

← Previous revision		Revision as of 04:19, 9 September 2025
Line 4:		Line 4:
	[[File:Indeterminacy principle.gif\|thumb\|The wave function of an initially very localized free particle.]]		[[File:Indeterminacy principle.gif\|thumb\|The wave function of an initially very localized free particle.]]

	In [[quantum physics]], a '''wave function''' (or '''wavefunction''') is a mathematical description of the [[quantum state]] of an isolated [[quantum system]]. The most common symbols for a wave function are the Greek letters {{math\|''ψ''}} and {{math\|Ψ}} (lower-case and capital [[psi (letter)\|psi]], respectively). Wave functions are [[complex number\|complex-valued]]. For example, a wave function might assign a complex number to each point in a region of space. The [[Born rule]]<ref name=Born_1926_A /><ref name="Born_1926_B" /><ref>[[Max Born\|Born, M.]] (1954).</ref> provides the means to turn these complex [[probability amplitude]]s into actual probabilities. In one common form, it says that the [[squared modulus]] of a wave function that depends upon position is the [[probability density function\|probability density]] of [[measurement in quantum mechanics\|measuring]] a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply [[Operator (quantum mechanics)\|quantum operators]], whose eigenvalues correspond to sets of possible results of measurements, to the wave function {{math\|''ψ''}} and calculate the statistical distributions for measurable quantities.		In [[quantum physics]], a '''wave function''' (or '''wavefunction''') is a mathematical description of the [[quantum state]] of an isolated [[quantum system]]. The most common symbols for a wave function are the Greek letters {{math\|''ψ''}} and {{math\|Ψ}} (lower-case and capital [[psi (letter)\|psi]], respectively).

			According to the [[superposition principle]] of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a [[Hilbert space]]. The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the [[Born rule]], relating transition probabilities to inner products. The [[Schrödinger equation]] determines how wave functions evolve over time, and a wave function behaves qualitatively like other [[wave]]s, such as [[water wave]]s or waves on a string, because the Schrödinger equation is mathematically a type of [[wave equation]]. This explains the name "wave function", and gives rise to [[wave–particle duality]]. However, whether the wave function in quantum mechanics describes a kind of physical phenomenon is still open to different [[Interpretations of quantum mechanics\|interpretations]], fundamentally differentiating it from [[classic mechanical]] waves.{{sfn\|Born\|1927\|pp=354–357}}{{sfn\|Heisenberg\|1958\|p=143}}<ref>[[Werner Heisenberg\|Heisenberg, W.]] (1927/1985/2009). Heisenberg is translated by {{harvnb\|Camilleri\|2009\|p=71}}, (from {{harvnb\|Bohr\|1985\|p=142}}).</ref>{{sfn\|Murdoch\|1987\|p=43}}{{sfn\|de Broglie\|1960\|p=48}}{{sfn\|Landau\|Lifshitz\|1977\|p=6}}{{sfn\|Newton\|2002\|pp=19–21}}

			Wave functions are [[complex number\|complex-valued]]. For example, a wave function might assign a complex number to each point in a region of space. The [[Born rule]]<ref name="Born_1926_A" /><ref name="Born_1926_B" /><ref>[[Max Born\|Born, M.]] (1954).</ref> provides the means to turn these complex [[probability amplitude]]s into actual probabilities. In one common form, it says that the [[squared modulus]] of a wave function that depends upon position is the [[probability density function\|probability density]] of [[measurement in quantum mechanics\|measuring]] a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply [[Operator (quantum mechanics)\|quantum operators]], whose eigenvalues correspond to sets of possible results of measurements, to the wave function {{math\|''ψ''}} and calculate the statistical distributions for measurable quantities.

	Wave functions can be [[function (mathematics)\|functions]] of variables other than position, such as [[momentum]]. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a [[Fourier transform]]. Some particles, like [[electron]]s and [[photon]]s, have nonzero [[Spin (physics)\|spin]], and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as [[isospin]]. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a [[column matrix]] (e.g., a {{math\|2 × 1}} column vector for a non-relativistic electron with spin {{math\|{{frac\|1\|2}}}}).		Wave functions can be [[function (mathematics)\|functions]] of variables other than position, such as [[momentum]]. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a [[Fourier transform]]. Some particles, like [[electron]]s and [[photon]]s, have nonzero [[Spin (physics)\|spin]], and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as [[isospin]]. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a [[column matrix]] (e.g., a {{math\|2 × 1}} column vector for a non-relativistic electron with spin {{math\|{{frac\|1\|2}}}}).

	According to the [[superposition principle]] of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a [[Hilbert space]]. The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the [[Born rule]], relating transition probabilities to inner products. The [[Schrödinger equation]] determines how wave functions evolve over time, and a wave function behaves qualitatively like other [[wave]]s, such as [[water wave]]s or waves on a string, because the Schrödinger equation is mathematically a type of [[wave equation]]. This explains the name "wave function", and gives rise to [[wave–particle duality]]. However, whether the wave function in quantum mechanics describes a kind of physical phenomenon is still open to different [[Interpretations of quantum mechanics\|interpretations]], fundamentally differentiating it from [[classic mechanical]] waves.{{sfn\|Born\|1927\|pp=354–357}}{{sfn\|Heisenberg\|1958\|p=143}}<ref>[[Werner Heisenberg\|Heisenberg, W.]] (1927/1985/2009). Heisenberg is translated by {{harvnb\|Camilleri\|2009\|p=71}}, (from {{harvnb\|Bohr\|1985\|p=142}}).</ref>{{sfn\|Murdoch\|1987\|p=43}}{{sfn\|de Broglie\|1960\|p=48}}{{sfn\|Landau\|Lifshitz\|1977\|p=6}}{{sfn\|Newton\|2002\|pp=19–21}}

	==Historical background==		==Historical background==