Physics:Quantum Adiabatic theorem - Revision history

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	== Introduction ==		== Introduction ==
	The theorem in its original form was given by [[Biography:Max Born\|Max Born]] and Vladimir Fock in 1928.<ref name="Born-Fock" /> It is one of the central results of quantum mechanics because it explains how quantum states behave when the Hamiltonian changes slowly in time. If the evolution is sufficiently gradual and the relevant eigenvalue remains separated from the rest of the spectrum, the system stays in the corresponding instantaneous eigenstate, acquiring only phase factors.		The theorem in its original form was given by [[Biography:Max Born\|Max Born]] and Vladimir Fock in 1928.<ref name="Born-Fock">{{cite journal \|author=Born \|first1=M. \|last2=Fock \|first2=V. A. \|name-list-style=and \|year=1928 \|title=Beweis des Adiabatensatzes \|journal=Zeitschrift für Physik A \|volume=51 \|issue=3–4 \|pages=165–180 \|bibcode=1928ZPhy...51..165B \|doi=10.1007/BF01343193 \|s2cid=122149514}}</ref> It is one of the central results of quantum mechanics because it explains how quantum states behave when the Hamiltonian changes slowly in time. If the evolution is sufficiently gradual and the relevant eigenvalue remains separated from the rest of the spectrum, the system stays in the corresponding instantaneous eigenstate, acquiring only phase factors.

	The theorem is closely related to the notion of adiabatic invariant in the old quantum theory, but its meaning in quantum mechanics differs from the thermodynamic meaning of the word ''adiabatic''. In quantum mechanics, adiabatic generally means ''slow'' variation compared with the system’s internal timescales.<ref name=Griffiths>{{cite book \|last=Griffiths \|first=David J. \|title=Introduction to Quantum Mechanics \|year=2005 \|publisher=Pearson Prentice Hall \|isbn=0-13-111892-7 \|chapter=10 }}</ref>		The theorem is closely related to the notion of adiabatic invariant in the old quantum theory, but its meaning in quantum mechanics differs from the thermodynamic meaning of the word ''adiabatic''. In quantum mechanics, adiabatic generally means ''slow'' variation compared with the system’s internal timescales.<ref name=Griffiths>{{cite book \|last=Griffiths \|first=David J. \|title=Introduction to Quantum Mechanics \|year=2005 \|publisher=Pearson Prentice Hall \|isbn=0-13-111892-7 \|chapter=10 }}</ref>
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	== References ==		= References =
	{{reflist\|3}}		{{reflist\|3}}

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	'''Adiabatic theorem''' ~~is a Book I topic in~~ the ~~Quantum Collection. The~~ adiabatic theorem is a concept in quantum mechanics stating that a physical system remains in its instantaneous eigenstate if a given perturbation acts on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. In simpler terms, a quantum-mechanical system subjected to gradually changing external conditions adapts its functional form, whereas under rapidly varying conditions there is insufficient time for the state to adapt, so the spatial probability density remains unchanged. The adiabatic theorem is a concept in quantum mechanics stating that a physical system remains in its instantaneous eigenstate if a given perturbation acts on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.		'''Adiabatic theorem''' the adiabatic theorem is a concept in quantum mechanics stating that a physical system remains in its instantaneous eigenstate if a given perturbation acts on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. In simpler terms, a quantum-mechanical system subjected to gradually changing external conditions adapts its functional form, whereas under rapidly varying conditions there is insufficient time for the state to adapt, so the spatial probability density remains unchanged. The adiabatic theorem is a concept in quantum mechanics stating that a physical system remains in its instantaneous eigenstate if a given perturbation acts on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. The theorem in its original form was given by Max Born and Vladimir Fock in 1928.
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	The sudden approximation is valid when		The sudden approximation is valid when
	<math display="block">\tau \ll {\hbar \over \Delta\bar{H}},</math>		<math display="block">\tau \ll {\hbar \over \Delta\bar{H}},</math>
	which is a form of the ~~[[Heisenberg uncertainty principle#Energy-time uncertainty principle\|~~time-energy uncertainty relation]].<ref name=Messiah>{{cite book \|last=Messiah \|first=Albert \|title=Quantum Mechanics \|year=1999 \|publisher=Dover Publications \|isbn=0-486-40924-4 \|chapter=XVII }}</ref>		which is a form of the time-energy uncertainty relation.<ref name=Messiah>{{cite book \|last=Messiah \|first=Albert \|title=Quantum Mechanics \|year=1999 \|publisher=Dover Publications \|isbn=0-486-40924-4 \|chapter=XVII }}</ref>

	=== Diabatic passage ===		=== Diabatic passage ===

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	The theorem in its original form was given by [[Biography:Max Born\|Max Born]] and Vladimir Fock in 1928.<ref name="Born-Fock" /> It is one of the central results of quantum mechanics because it explains how quantum states behave when the Hamiltonian changes slowly in time. If the evolution is sufficiently gradual and the relevant eigenvalue remains separated from the rest of the spectrum, the system stays in the corresponding instantaneous eigenstate, acquiring only phase factors.		The theorem in its original form was given by [[Biography:Max Born\|Max Born]] and Vladimir Fock in 1928.<ref name="Born-Fock" /> It is one of the central results of quantum mechanics because it explains how quantum states behave when the Hamiltonian changes slowly in time. If the evolution is sufficiently gradual and the relevant eigenvalue remains separated from the rest of the spectrum, the system stays in the corresponding instantaneous eigenstate, acquiring only phase factors.

	The theorem is closely related to the notion of [[adiabatic invariant]] in the old quantum theory, but its meaning in quantum mechanics differs from the thermodynamic meaning of the word ''adiabatic''. In quantum mechanics, adiabatic generally means ''slow'' variation compared with the system’s internal timescales.<ref name=Griffiths>{{cite book \|last=Griffiths \|first=David J. \|title=Introduction to Quantum Mechanics \|year=2005 \|publisher=Pearson Prentice Hall \|isbn=0-13-111892-7 \|chapter=10 }}</ref>		The theorem is closely related to the notion of adiabatic invariant in the old quantum theory, but its meaning in quantum mechanics differs from the thermodynamic meaning of the word ''adiabatic''. In quantum mechanics, adiabatic generally means ''slow'' variation compared with the system’s internal timescales.<ref name=Griffiths>{{cite book \|last=Griffiths \|first=David J. \|title=Introduction to Quantum Mechanics \|year=2005 \|publisher=Pearson Prentice Hall \|isbn=0-13-111892-7 \|chapter=10 }}</ref>

	== Adiabatic pendulum ==		== Adiabatic pendulum ==
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	== Diabatic vs. adiabatic processes ==		== Diabatic vs. adiabatic processes ==
	Rapidly changing conditions prevent the system from adapting its configuration during the process, so the spatial probability density remains unchanged. The system then generally ends in a [[linear superposition]] of eigenstates of the final Hamiltonian. By contrast, gradually changing conditions allow the probability density to change continuously, and if the system begins in an eigenstate of the initial Hamiltonian it ends in the corresponding eigenstate of the final Hamiltonian.<ref name="Kato">{{cite journal \|author=Kato \|first=T. \|year=1950 \|title=On the Adiabatic Theorem of Quantum Mechanics \|journal=Journal of the Physical Society of Japan \|volume=5 \|issue=6 \|pages=435–439 \|bibcode=1950JPSJ....5..435K \|doi=10.1143/JPSJ.5.435}}</ref>		Rapidly changing conditions prevent the system from adapting its configuration during the process, so the spatial probability density remains unchanged. The system then generally ends in a linear superposition of eigenstates of the final Hamiltonian. By contrast, gradually changing conditions allow the probability density to change continuously, and if the system begins in an eigenstate of the initial Hamiltonian it ends in the corresponding eigenstate of the final Hamiltonian.<ref name="Kato">{{cite journal \|author=Kato \|first=T. \|year=1950 \|title=On the Adiabatic Theorem of Quantum Mechanics \|journal=Journal of the Physical Society of Japan \|volume=5 \|issue=6 \|pages=435–439 \|bibcode=1950JPSJ....5..435K \|doi=10.1143/JPSJ.5.435}}</ref>

	At an initial time <math>t_0</math> a quantum-mechanical system has Hamiltonian <math>\hat{H}(t_0)</math> and is in an eigenstate <math>\psi(x,t_0)</math>. A continuous change in conditions modifies the Hamiltonian to <math>\hat{H}(t_1)</math> at time <math>t_1</math>. The theorem states that the behavior depends critically on the interval <math>\tau = t_1 - t_0</math> over which the change occurs.		At an initial time <math>t_0</math> a quantum-mechanical system has Hamiltonian <math>\hat{H}(t_0)</math> and is in an eigenstate <math>\psi(x,t_0)</math>. A continuous change in conditions modifies the Hamiltonian to <math>\hat{H}(t_1)</math> at time <math>t_1</math>. The theorem states that the behavior depends critically on the interval <math>\tau = t_1 - t_0</math> over which the change occurs.
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	=== Comparison with the adiabatic concept in thermodynamics ===		=== Comparison with the adiabatic concept in thermodynamics ===
	In thermodynamics, an [[adiabatic process]] is one in which there is no exchange of heat between system and environment. In practice, such processes are often relatively fast compared with the timescale of heat transfer.		In thermodynamics, an adiabatic process is one in which there is no exchange of heat between system and environment. In practice, such processes are often relatively fast compared with the timescale of heat transfer.

	In classical and quantum mechanics, however, ''adiabatic'' means something closer to a [[quasistatic process]]: a process slow enough that the system remains near equilibrium or stays in the corresponding instantaneous eigenstate.<ref name=Griffiths />		In classical and quantum mechanics, however, ''adiabatic'' means something closer to a quasistatic process: a process slow enough that the system remains near equilibrium or stays in the corresponding instantaneous eigenstate.<ref name=Griffiths />

	Thus, in quantum mechanics the adiabatic theorem states that quantum jumps are suppressed when the Hamiltonian changes sufficiently slowly, and the system tends to preserve its state label and quantum numbers.<ref name=":1">{{cite web \|author=Zwiebach \|first=Barton \|date=Spring 2018 \|title=L15.2 Classical adiabatic invariant \|url=https://www.youtube.com/watch?v=qxBhW2DRnPg&t=254s?t=03m00s \|url-status=live \|archive-url=https://ghostarchive.org/varchive/youtube/20211221/qxBhW2DRnPg \|archive-date=2021-12-21 \|publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref>		Thus, in quantum mechanics the adiabatic theorem states that quantum jumps are suppressed when the Hamiltonian changes sufficiently slowly, and the system tends to preserve its state label and quantum numbers.<ref name=":1">{{cite web \|author=Zwiebach \|first=Barton \|date=Spring 2018 \|title=L15.2 Classical adiabatic invariant \|url=https://www.youtube.com/watch?v=qxBhW2DRnPg&t=254s?t=03m00s \|url-status=live \|archive-url=https://ghostarchive.org/varchive/youtube/20211221/qxBhW2DRnPg \|archive-date=2021-12-21 \|publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref>
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	== Example systems ==		== Example systems ==
	=== Simple pendulum ===		=== Simple pendulum ===
	A useful classical analogy is a [[pendulum]] whose support is moved slowly. If the change is gradual enough, the mode of oscillation relative to the support remains essentially unchanged, showing how a slow external variation allows the system to adapt continuously.<ref name=":2">{{cite web \|author=Zwiebach \|first=Barton \|date=Spring 2018 \|title=Classical analog: oscillator with slowly varying frequency \|url=https://www.youtube.com/watch?v=DYJM_P4sG-c \|url-status=live \|archive-url=https://ghostarchive.org/varchive/youtube/20211221/DYJM_P4sG-c \|archive-date=2021-12-21 \|publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref>		A useful classical analogy is a pendulum whose support is moved slowly. If the change is gradual enough, the mode of oscillation relative to the support remains essentially unchanged, showing how a slow external variation allows the system to adapt continuously.<ref name=":2">{{cite web \|author=Zwiebach \|first=Barton \|date=Spring 2018 \|title=Classical analog: oscillator with slowly varying frequency \|url=https://www.youtube.com/watch?v=DYJM_P4sG-c \|url-status=live \|archive-url=https://ghostarchive.org/varchive/youtube/20211221/DYJM_P4sG-c \|archive-date=2021-12-21 \|publisher=MIT 8.06 Quantum Physics III}}{{cbignore}}</ref>

	=== Quantum harmonic oscillator ===		=== Quantum harmonic oscillator ===
	A more specifically quantum example is the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]] with increasing [[spring constant]] <math>k</math>. If <math>k</math> increases adiabatically, the system remains in the corresponding instantaneous eigenstate of the current Hamiltonian. For a state initially in the ground state, the wavefunction narrows as the potential steepens, but the quantum number remains unchanged.		A more specifically quantum example is the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]] with increasing spring constant <math>k</math>. If <math>k</math> increases adiabatically, the system remains in the corresponding instantaneous eigenstate of the current Hamiltonian. For a state initially in the ground state, the wavefunction narrows as the potential steepens, but the quantum number remains unchanged.

	If <math>k</math> increases rapidly, the process becomes diabatic. The system then has no time to adapt to the changing potential and the final state becomes a superposition of many eigenstates of the new Hamiltonian, whose combination reproduces the original probability density.		If <math>k</math> increases rapidly, the process becomes diabatic. The system then has no time to adapt to the changing potential and the final state becomes a superposition of many eigenstates of the new Hamiltonian, whose combination reproduces the original probability density.

	=== Avoided curve crossing ===		=== Avoided curve crossing ===
	~~{{main\|~~Avoided crossing}}		''Related topic:'' Avoided crossing
	A particularly important example is a two-level atom in an external [[magnetic field]]. The system may be described in terms of diabatic states <math>\|1\rangle</math> and <math>\|2\rangle</math> with wavefunction		A particularly important example is a two-level atom in an external magnetic field. The system may be described in terms of diabatic states <math>\|1\rangle</math> and <math>\|2\rangle</math> with wavefunction
	<math display="block">\|\Psi\rangle = c_1(t)\|1\rangle + c_2(t)\|2\rangle.</math>		<math display="block">\|\Psi\rangle = c_1(t)\|1\rangle + c_2(t)\|2\rangle.</math>

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	\end{align}</math>		\end{align}</math>

	Because of the off-diagonal coupling, the eigenvalues do not cross but exhibit an [[avoided crossing]]. Under slow variation of the magnetic field, the system follows an adiabatic eigenstate. Under rapid variation it instead follows a diabatic path, giving a finite probability of transition between the two eigenstates.<ref name="Stenholm">{{cite journal \|author=Stenholm \|first=Stig \|author-link=Stig Stenholm \|year=1994 \|title=Quantum Dynamics of Simple Systems \|journal=The 44th Scottish Universities Summer School in Physics \|pages=267–313}}</ref>		Because of the off-diagonal coupling, the eigenvalues do not cross but exhibit an avoided crossing. Under slow variation of the magnetic field, the system follows an adiabatic eigenstate. Under rapid variation it instead follows a diabatic path, giving a finite probability of transition between the two eigenstates.<ref name="Stenholm">{{cite journal \|author=Stenholm \|first=Stig \|author-link=Stig Stenholm \|year=1994 \|title=Quantum Dynamics of Simple Systems \|journal=The 44th Scottish Universities Summer School in Physics \|pages=267–313}}</ref>

	== Mathematical statement ==		== Mathematical statement ==
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	This helps explain a variety of phenomena in:		This helps explain a variety of phenomena in:
	* '''thermodynamics''': temperature dependence of [[specific heat]], [[thermal expansion]], and [[melting]]		* '''thermodynamics''': temperature dependence of specific heat, thermal expansion, and melting
	* '''transport phenomena''': temperature dependence of [[electric resistivity]] in conductors, temperature dependence of [[electric conductivity]] in ~~[[insulator (electricity)\|~~insulators]], and properties of low-temperature [[superconductivity]]		* '''transport phenomena''': temperature dependence of electric resistivity in conductors, temperature dependence of electric conductivity in insulators, and properties of low-temperature superconductivity
	* '''optics''': infrared absorption in ionic crystals, Brillouin scattering, and Raman scattering		* '''optics''': infrared absorption in ionic crystals, Brillouin scattering, and Raman scattering

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	== Calculating adiabatic passage probabilities ==		== Calculating adiabatic passage probabilities ==
	=== The Landau–Zener formula ===		=== The Landau–Zener formula ===
	~~{{main\|~~Landau–Zener formula}}		''Related topic:'' Landau–Zener formula
	For a two-level system with a linearly changing perturbation, the probability of a diabatic transition can be estimated using the Landau–Zener formula. The key quantity is the Landau–Zener velocity		For a two-level system with a linearly changing perturbation, the probability of a diabatic transition can be estimated using the Landau–Zener formula. The key quantity is the Landau–Zener velocity
	<math display="block">v_\text{LZ} = {\frac{\partial}{\partial t}\|E_2 - E_1\| \over \frac{\partial}{\partial q}\|E_2 - E_1\|} \approx \frac{dq}{dt} ,</math>		<math display="block">v_\text{LZ} = {\frac{\partial}{\partial t}\|E_2 - E_1\| \over \frac{\partial}{\partial q}\|E_2 - E_1\|} \approx \frac{dq}{dt} ,</math>