Physics:Quantum tunnelling - Revision history

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	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
	{{Quantum article nav\|previous=Physics:Quantum S-matrix\|previous label=S-matrix\|next=Physics:Quantum speed limit\|next label=Speed limit}}		{{Quantum article nav\|previous=Physics:Quantum S-matrix\|previous label=S-matrix\|next=Physics:Quantum speed limit\|next label=Speed limit}}
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	* [http://nanohub.org/resources/8799 Animated illustration of quantum tunnelling in a RTD device]		* [http://nanohub.org/resources/8799 Animated illustration of quantum tunnelling in a RTD device]
	* [http://www.ps-mix.it/Pighin/Video/Quantistica/Quantum-Tunnelling.mp4 Interactive Solution of Schrodinger Tunnel Equation]		* [http://www.ps-mix.it/Pighin/Video/Quantistica/Quantum-Tunnelling.mp4 Interactive Solution of Schrodinger Tunnel Equation]
	~~[[Category:Articles containing video clips]]~~
	~~[[Category:Particle physics]]~~
	~~[[Category:Quantum mechanics]]~~

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|Physics:Quantum tunnelling\|1}}		{{Sourceattribution\|Physics:Quantum tunnelling\|1}}

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	* [http://nanohub.org/resources/8799 Animated illustration of quantum tunnelling in a RTD device]		* [http://nanohub.org/resources/8799 Animated illustration of quantum tunnelling in a RTD device]
	* [http://www.ps-mix.it/Pighin/Video/Quantistica/Quantum-Tunnelling.mp4 Interactive Solution of Schrodinger Tunnel Equation]		* [http://www.ps-mix.it/Pighin/Video/Quantistica/Quantum-Tunnelling.mp4 Interactive Solution of Schrodinger Tunnel Equation]

	~~{{Quantum mechanics topics}}~~

	[[Category:Articles containing video clips]]		[[Category:Articles containing video clips]]
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	A classical wave-particle association was originally analysed as analogous to quantum tunnelling,<ref>{{cite journal\|last1=Eddi\|first1=A.\|last2=Fort\|first2=E.\|last3=Moisy\|first3=F.\|last4=Couder\|first4=Y.\|title=Unpredictable Tunneling of a Classical Wave-Particle Association\|journal=Physical Review Letters\|date=16 June 2009\|volume=102\|issue=24\|doi=10.1103/PhysRevLett.102.240401\|url=http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf\|access-date=1 May 2016\|bibcode=2009PhRvL.102x0401E\|pmid=19658983\|article-number=240401\|archive-date=10 March 2016\|archive-url=https://web.archive.org/web/20160310083947/http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf}}</ref> but subsequent analysis found a fluid dynamics cause related to the vertical momentum imparted to particles near the barrier.<ref>{{Cite journal \|last1=Bush \|first1=John W M \|last2=Oza \|first2=Anand U \|date=2021-01-01 \|title=Hydrodynamic quantum analogs \|url=https://iopscience.iop.org/article/10.1088/1361-6633/abc22c \|journal=Reports on Progress in Physics \|volume=84 \|issue=1 \|page=017001 \|doi=10.1088/1361-6633/abc22c \|pmid=33065567 \|issn=0034-4885\|url-access=subscription }}</ref>		A classical wave-particle association was originally analysed as analogous to quantum tunnelling,<ref>{{cite journal\|last1=Eddi\|first1=A.\|last2=Fort\|first2=E.\|last3=Moisy\|first3=F.\|last4=Couder\|first4=Y.\|title=Unpredictable Tunneling of a Classical Wave-Particle Association\|journal=Physical Review Letters\|date=16 June 2009\|volume=102\|issue=24\|doi=10.1103/PhysRevLett.102.240401\|url=http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf\|access-date=1 May 2016\|bibcode=2009PhRvL.102x0401E\|pmid=19658983\|article-number=240401\|archive-date=10 March 2016\|archive-url=https://web.archive.org/web/20160310083947/http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf}}</ref> but subsequent analysis found a fluid dynamics cause related to the vertical momentum imparted to particles near the barrier.<ref>{{Cite journal \|last1=Bush \|first1=John W M \|last2=Oza \|first2=Anand U \|date=2021-01-01 \|title=Hydrodynamic quantum analogs \|url=https://iopscience.iop.org/article/10.1088/1361-6633/abc22c \|journal=Reports on Progress in Physics \|volume=84 \|issue=1 \|page=017001 \|doi=10.1088/1361-6633/abc22c \|pmid=33065567 \|issn=0034-4885\|url-access=subscription }}</ref>

	~~== See also ==~~
	* {{anl\|Dielectric barrier discharge}}
	* {{anl\|Field electron emission}}
	* {{anl\|Holstein–Herring method}}
	* Proton tunnelling – Type of quantum tunnelling
	* {{anl\|Quantum cloning}}
	* {{anl\|Superconducting tunnel junction}}
	* {{anl\|Tunnel diode}}
	* {{anl\|Tunnel junction}}
	* {{anl\|White hole}}

	== See also ==		== See also ==

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	{{Short description\|Quantum mechanical phenomenon}}		{{Short description\|Quantum mechanical phenomenon}}

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	== Concept ==		== Concept ==
	[[File:Quantum_book1_tunnelling_yellow.png\|thumb\|right\|upright=1.5\|Animation showing the tunnel effect and its application to an ~~[[Scanning tunnelling microscope\|~~STM]]]]		[[File:Quantum_book1_tunnelling_yellow.png\|thumb\|right\|upright=1.5\|Animation showing the tunnel effect and its application to an STM]]

	Quantum tunnelling falls under the domain of [[Physics:Quantum mechanics\|quantum mechanics]]. To understand the phenomenon, particles attempting to travel across a potential barrier can be compared to a ball trying to roll over a hill. Quantum mechanics and classical mechanics differ in their treatment of this scenario.		Quantum tunnelling falls under the domain of [[Physics:Quantum mechanics\|quantum mechanics]]. To understand the phenomenon, particles attempting to travel across a potential barrier can be compared to a ball trying to roll over a hill. Quantum mechanics and classical mechanics differ in their treatment of this scenario.
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	[[File:E14-V20-B1.gif\|thumb\|A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.\|alt=]]		[[File:E14-V20-B1.gif\|thumb\|A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.\|alt=]]

	The wave function of a physical system of particles specifies everything that can be known about the system.<ref>{{Cite book \|last1=Bjorken \|first1=James D. \|title=Relativistic quantum mechanics \|last2=Drell \|first2=Sidney D. \|date=1964 \|publisher=McGraw Hill \|isbn=978-0-07-005493-6 \|series=International series in pure and applied physics \|location=New York, NY \|page=2}}</ref> Therefore, problems in quantum mechanics analyse the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the time evolution of a known wave function can be deduced. The square of the ~~[[Absolute value#Complex numbers\|~~absolute value]] of this wave function is directly related to the probability distribution of the particle positions, which describes the probability that the particles would be measured at those positions.		The wave function of a physical system of particles specifies everything that can be known about the system.<ref>{{Cite book \|last1=Bjorken \|first1=James D. \|title=Relativistic quantum mechanics \|last2=Drell \|first2=Sidney D. \|date=1964 \|publisher=McGraw Hill \|isbn=978-0-07-005493-6 \|series=International series in pure and applied physics \|location=New York, NY \|page=2}}</ref> Therefore, problems in quantum mechanics analyse the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the time evolution of a known wave function can be deduced. The square of the absolute value of this wave function is directly related to the probability distribution of the particle positions, which describes the probability that the particles would be measured at those positions.

	As shown in the animation, when a wave packet impinges on the barrier, most of it is reflected and some is transmitted through the barrier. The wave packet becomes more delocalised: it is now on both sides of the barrier and lower in maximum amplitude, but equal in integrated square-magnitude, meaning that the probability the particle is ''somewhere'' remains unity. The wider the barrier and the higher the barrier energy, the lower the probability of tunnelling.		As shown in the animation, when a wave packet impinges on the barrier, most of it is reflected and some is transmitted through the barrier. The wave packet becomes more delocalised: it is now on both sides of the barrier and lower in maximum amplitude, but equal in integrated square-magnitude, meaning that the probability the particle is ''somewhere'' remains unity. The wider the barrier and the higher the barrier energy, the lower the probability of tunnelling.
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	{{Main\|Physics:Radioactive decay}}		{{Main\|Physics:Radioactive decay}}

	Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunnelling into the nucleus is electron capture). This was the first application of quantum tunnelling. Radioactive decay is a relevant issue for astrobiology as this consequence of quantum tunnelling creates a constant energy source over a large time interval for environments outside the circumstellar habitable zone where insolation would not be possible (~~[[Astronomy:Planetary oceanography#Extraterrestrial water oceans\|~~subsurface oceans]]) or effective.<ref name="Trixler2013" />		Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunnelling into the nucleus is electron capture). This was the first application of quantum tunnelling. Radioactive decay is a relevant issue for astrobiology as this consequence of quantum tunnelling creates a constant energy source over a large time interval for environments outside the circumstellar habitable zone where insolation would not be possible (subsurface oceans) or effective.<ref name="Trixler2013" />

	Quantum tunnelling may be one of the mechanisms of hypothetical proton decay.<ref name="url[nucl-th/9809006] Time-dependent properties of proton decay from crossing single-particle metastable states in deformed nuclei">{{cite journal \|title=Time-dependent properties of proton decay from crossing single-particle metastable states in deformed nuclei \|year=1998 \|doi=10.1103/PhysRevC.58.3280 \|arxiv=nucl-th/9809006 \|last1=Talou \|first1=P. \|last2=Carjan \|first2=N. \|last3=Strottman \|first3=D. \|journal=Physical Review C \|volume=58 \|issue=6 \|pages=3280–3285 \|bibcode=1998PhRvC..58.3280T \|s2cid=119075457 }}</ref><ref name="urladsabs.harvard.edu">{{cite web \|url=http://adsabs.harvard.edu/pdf/1982ApJ...252....1D \|title=adsabs.harvard.edu \|format= }}</ref>		Quantum tunnelling may be one of the mechanisms of hypothetical proton decay.<ref name="url[nucl-th/9809006] Time-dependent properties of proton decay from crossing single-particle metastable states in deformed nuclei">{{cite journal \|title=Time-dependent properties of proton decay from crossing single-particle metastable states in deformed nuclei \|year=1998 \|doi=10.1103/PhysRevC.58.3280 \|arxiv=nucl-th/9809006 \|last1=Talou \|first1=P. \|last2=Carjan \|first2=N. \|last3=Strottman \|first3=D. \|journal=Physical Review C \|volume=58 \|issue=6 \|pages=3280–3285 \|bibcode=1998PhRvC..58.3280T \|s2cid=119075457 }}</ref><ref name="urladsabs.harvard.edu">{{cite web \|url=http://adsabs.harvard.edu/pdf/1982ApJ...252....1D \|title=adsabs.harvard.edu \|format= }}</ref>
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	==== Astrochemistry in interstellar clouds ====		==== Astrochemistry in interstellar clouds ====
	By including quantum tunnelling, the astrochemical syntheses of various molecules in interstellar clouds can be explained, such as the synthesis of ~~[[Software:Hydrogen#Spin isomers\|~~molecular hydrogen]], water (ice) and the prebiotic important formaldehyde.<ref name="Trixler2013" /> Tunnelling of molecular hydrogen has been observed in the lab.<ref name="ionTunneling" />		By including quantum tunnelling, the astrochemical syntheses of various molecules in interstellar clouds can be explained, such as the synthesis of molecular hydrogen, water (ice) and the prebiotic important formaldehyde.<ref name="Trixler2013" /> Tunnelling of molecular hydrogen has been observed in the lab.<ref name="ionTunneling" />

	=== Biology ===		=== Biology ===
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	=== Schrödinger equation ===		=== Schrödinger equation ===
	The ~~[[Schrödinger equation#Time independent equation 2\|~~time-independent Schrödinger equation]] for one particle in one dimension can be written as		The time-independent Schrödinger equation for one particle in one dimension can be written as
	<math display="block">-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x)</math>		<math display="block">-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x)</math>
	or		or
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	Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:		Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:
	<math display="block">A'(x) + A(x)^2 - B(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right).</math>[[File:Quantum_book1_tunnelling_yellow.png\|upright=1.2\|thumb\|Quantum tunnelling in the [[phase space formulation]] of quantum mechanics. ~~[[Wigner quasiprobability distribution\|~~Wigner function]] for tunnelling through the potential barrier <math>U(x) = 8e^{-0.25 x^2}</math> in atomic units (a.u.). The solid lines represent the ~~[[Level set\|~~level set]] of the ~~[[Physics:Hamiltonian mechanics\|~~Hamiltonian]] <math>H(x,p) = p^2 / 2 + U(x) </math>.]]		<math display="block">A'(x) + A(x)^2 - B(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right).</math>[[File:Quantum_book1_tunnelling_yellow.png\|upright=1.2\|thumb\|Quantum tunnelling in the [[phase space formulation]] of quantum mechanics. Wigner function for tunnelling through the potential barrier <math>U(x) = 8e^{-0.25 x^2}</math> in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian <math>H(x,p) = p^2 / 2 + U(x) </math>.]]

	To solve this equation using the semiclassical approximation, each function must be expanded as a power series in <math>\hbar</math>. From the equations, the power series must start with at least an order of <math>\hbar^{-1}</math> to satisfy the real part of the equation; for a good classical limit starting with the highest power of the Planck constant possible is preferable, which leads to		To solve this equation using the semiclassical approximation, each function must be expanded as a power series in <math>\hbar</math>. From the equations, the power series must start with at least an order of <math>\hbar^{-1}</math> to satisfy the real part of the equation; for a good classical limit starting with the highest power of the Planck constant possible is preferable, which leads to
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	== Related phenomena ==		== Related phenomena ==
	Several phenomena have the same behaviour as quantum tunnelling. Two examples are evanescent wave coupling<ref>{{Cite journal \|last1=Martin \|first1=Th. \|last2=Landauer \|first2=R. \|date=1992-02-01 \|title=Time delay of evanescent electromagnetic waves and the analogy to particle tunneling \|url=https://link.aps.org/doi/10.1103/PhysRevA.45.2611 \|journal=Physical Review A \|language=en \|volume=45 \|issue=4 \|pages=2611–2617 \|doi=10.1103/PhysRevA.45.2611 \|pmid=9907285 \|bibcode=1992PhRvA..45.2611M \|issn=1050-2947\|url-access=subscription }}</ref> (the application of Maxwell's wave-equation to light) and the application of the non-dispersive wave-equation from acoustics applied to ~~[[Wave#Waves on strings\|~~"waves on strings"]].		Several phenomena have the same behaviour as quantum tunnelling. Two examples are evanescent wave coupling<ref>{{Cite journal \|last1=Martin \|first1=Th. \|last2=Landauer \|first2=R. \|date=1992-02-01 \|title=Time delay of evanescent electromagnetic waves and the analogy to particle tunneling \|url=https://link.aps.org/doi/10.1103/PhysRevA.45.2611 \|journal=Physical Review A \|language=en \|volume=45 \|issue=4 \|pages=2611–2617 \|doi=10.1103/PhysRevA.45.2611 \|pmid=9907285 \|bibcode=1992PhRvA..45.2611M \|issn=1050-2947\|url-access=subscription }}</ref> (the application of Maxwell's wave-equation to light) and the application of the non-dispersive wave-equation from acoustics applied to "waves on strings".
	These effects are modelled similarly to the rectangular potential barrier. In these cases, one transmission medium through which the wave propagates that is the same or nearly the same throughout, and a second medium through which the wave travels differently. This can be described as a thin region of medium B between two regions of medium A. The analysis of a rectangular barrier by means of the Schrödinger equation can be adapted to these other effects provided that the wave equation has travelling wave solutions in medium A but real exponential solutions in medium B.		These effects are modelled similarly to the rectangular potential barrier. In these cases, one transmission medium through which the wave propagates that is the same or nearly the same throughout, and a second medium through which the wave travels differently. This can be described as a thin region of medium B between two regions of medium A. The analysis of a rectangular barrier by means of the Schrödinger equation can be adapted to these other effects provided that the wave equation has travelling wave solutions in medium A but real exponential solutions in medium B.

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	A classical wave-particle association was originally analysed as analogous to quantum tunnelling,<ref>{{cite journal\|last1=Eddi\|first1=A.\|last2=Fort\|first2=E.\|last3=Moisy\|first3=F.\|last4=Couder\|first4=Y.\|title=Unpredictable Tunneling of a Classical Wave-Particle Association\|journal=Physical Review Letters\|date=16 June 2009\|volume=102\|issue=24\|doi=10.1103/PhysRevLett.102.240401\|url=http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf\|access-date=1 May 2016\|bibcode=2009PhRvL.102x0401E\|pmid=19658983\|article-number=240401\|archive-date=10 March 2016\|archive-url=https://web.archive.org/web/20160310083947/http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf}}</ref> but subsequent analysis found a fluid dynamics cause related to the vertical momentum imparted to particles near the barrier.<ref>{{Cite journal \|last1=Bush \|first1=John W M \|last2=Oza \|first2=Anand U \|date=2021-01-01 \|title=Hydrodynamic quantum analogs \|url=https://iopscience.iop.org/article/10.1088/1361-6633/abc22c \|journal=Reports on Progress in Physics \|volume=84 \|issue=1 \|page=017001 \|doi=10.1088/1361-6633/abc22c \|pmid=33065567 \|issn=0034-4885\|url-access=subscription }}</ref>		A classical wave-particle association was originally analysed as analogous to quantum tunnelling,<ref>{{cite journal\|last1=Eddi\|first1=A.\|last2=Fort\|first2=E.\|last3=Moisy\|first3=F.\|last4=Couder\|first4=Y.\|title=Unpredictable Tunneling of a Classical Wave-Particle Association\|journal=Physical Review Letters\|date=16 June 2009\|volume=102\|issue=24\|doi=10.1103/PhysRevLett.102.240401\|url=http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf\|access-date=1 May 2016\|bibcode=2009PhRvL.102x0401E\|pmid=19658983\|article-number=240401\|archive-date=10 March 2016\|archive-url=https://web.archive.org/web/20160310083947/http://stilton.tnw.utwente.nl/people/eddi/Papers/PhysRevLett_TUNNEL.pdf}}</ref> but subsequent analysis found a fluid dynamics cause related to the vertical momentum imparted to particles near the barrier.<ref>{{Cite journal \|last1=Bush \|first1=John W M \|last2=Oza \|first2=Anand U \|date=2021-01-01 \|title=Hydrodynamic quantum analogs \|url=https://iopscience.iop.org/article/10.1088/1361-6633/abc22c \|journal=Reports on Progress in Physics \|volume=84 \|issue=1 \|page=017001 \|doi=10.1088/1361-6633/abc22c \|pmid=33065567 \|issn=0034-4885\|url-access=subscription }}</ref>

	~~== See also ==~~
	* {{anl\|Dielectric barrier discharge}}
	* {{anl\|Field electron emission}}
	* {{anl\|Holstein–Herring method}}
	* Proton tunnelling – Type of quantum tunnelling
	* {{anl\|Quantum cloning}}
	* {{anl\|Superconducting tunnel junction}}
	* {{anl\|Tunnel diode}}
	* {{anl\|Tunnel junction}}
	* {{anl\|White hole}}

	== See also ==		== See also ==

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	{{Quantum mechanics\|cTopic=Fundamental concepts}}		{{Quantum mechanics\|cTopic=Fundamental concepts}}