1010:
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5708:, disputed these claims. Winful argued that the wave packet of a tunnelling particle propagates locally, so a particle can't tunnel through the barrier non-locally. Winful also argued that the experiments that are purported to show non-local propagation have been misinterpreted. In particular, the group velocity of a wave packet does not measure its speed, but is related to the amount of time the wave packet is stored in the barrier. Moreover, if quantum tunneling is modeled with the relativistic
1331:, where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more energy than the barrier, through random collisions with other particles. When the electric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to a current that varies approximately exponentially with the electric field. These materials are important for flash memory, vacuum tubes, and some electron microscopes.
9014:
1750:
1517:, may allow imaging of individual atoms on the surface of a material. It operates by taking advantage of the relationship between quantum tunnelling with distance. When the tip of the STM's needle is brought close to a conduction surface that has a voltage bias, measuring the current of electrons that are tunnelling between the needle and the surface reveals the distance between the needle and the surface. By using
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1443:. When this energy level is higher than that of the electrons, no tunnelling occurs and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like an open wire. As the voltage further increases, tunnelling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable.
1014:
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1491:, the reflected part of the wave packet interferes uniformly with the transmitted one between all barriers so that 100% transmission becomes possible. The theory predicts that if positively charged nuclei form a perfectly rectangular array, electrons will tunnel through the metal as free electrons, leading to extremely high
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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
5765:
In real life, most systems are not integrable and display various degrees of chaos. Classical dynamics is then said to be mixed and the system phase space is typically composed of islands of regular orbits surrounded by a large sea of chaotic orbits. The existence of the chaotic sea, where transport
5108:
Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the two coefficients on one side of a classical turning point, the two coefficients on the other side of a classical turning point can be determined by using this local
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As shown in the animation, a wave packet impinges on the barrier, most of it is reflected and some is transmitted through the barrier. The wave packet becomes more de-localized: it is now on both sides of the barrier and lower in maximum amplitude, but equal in integrated square-magnitude, meaning
1426:
Because the tunnelling current drops off rapidly, tunnel diodes can be created that have a range of voltages for which current decreases as voltage increases. This peculiar property is used in some applications, such as high speed devices where the characteristic tunnelling probability changes as
954:
tunneling through microscopically narrow barriers. Tunneling is readily detectable with barriers of thickness about 1â3 nm or smaller for electrons, and about 0.1 nm or smaller for heavier particles such as protons or hydrogen atoms. Some sources describe the mere penetration of a wave
5794:
is small in front of the size of the regular islands, the fine structure of the classical phase space plays a key role in tunnelling. In particular the two symmetric tori are coupled "via a succession of classically forbidden transitions across nonlinear resonances" surrounding the two islands.
1521:
that change in size when voltage is applied, the height of the tip can be adjusted to keep the tunnelling current constant. The time-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor. STMs are accurate to 0.001 nm, or about 1% of
1728:
Spontaneous mutation occurs when normal DNA replication takes place after a particularly significant proton has tunnelled. A hydrogen bond joins DNA base pairs. A double well potential along a hydrogen bond separates a potential energy barrier. It is believed that the double well potential is
1154:
and reflection of electrons from metals. He assumed a surface potential barrier that confines the electrons within the metal and showed that the electrons have a finite probability of tunneling through or reflecting from the surface barrier when their energies are close to the barrier energy.
1625:
and cooled. The trap was then filled with hydrogen. At the temperatures used in the experiment, the energy barrier for reaction would not allow the reaction to succeed with classical dynamics alone. Quantum tunneling allowed reactions to happen in rare collisions. It was calculated from the
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is difficult, except in special cases that usually do not correspond to physical reality. A full mathematical treatment appears in the 1965 monograph by Fröman and Fröman. Their ideas have not been incorporated into physics textbooks, but their corrections have little quantitative effect.
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Classical mechanics predicts that particles that do not have enough energy to classically surmount a barrier cannot reach the other side. Thus, a ball without sufficient energy to surmount the hill would roll back down. In quantum mechanics, a particle can, with a small probability,
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makes use of quantum tunnelling in a very different manner to achieve a similar result. This diode has a resonant voltage for which a current favors a particular voltage, achieved by placing two thin layers with a high energy conductance band near each other. This creates a quantum
4409:. Away from the potential hill, the particle acts similar to a free and oscillating wave; beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour at these limits and classical turning points a global solution can be made.
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1297:(VLSI) electronics and results in a substantial power drain and heating effects that plague such devices. It is considered the lower limit on how microelectronic device elements can be made. Tunnelling is a fundamental technique used to program the floating gates of
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is classically allowed, between the two symmetric tori then assists the quantum tunnelling between them. This phenomenon is referred as chaos-assisted tunnelling. and is characterized by sharp resonances of the tunnelling rate when varying any system parameter.
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makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunnelling to explain the nature of the electron's collisions. When a free electron wave packet encounters a long array of uniformly spaced
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of an element with a heavier one typically results in a slower reaction rate. This is generally attributed to differences in the zero-point vibrational energies for chemical bonds containing the lighter and heavier isotopes and is generally modeled using
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The concept of quantum tunneling can be extended to situations where there exists a quantum transport between regions that are classically not connected even if there is no associated potential barrier. This phenomenon is known as dynamical tunnelling.
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asymmetric, with one well deeper than the other such that the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunnelled into the shallower well. The proton's movement from its regular position is called a
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of the particle and the energy of emission that depended directly on the mathematical probability of tunneling. All three researchers were familiar with the works on field emission, and Gamow was aware of
Mandelstam and Leontovich's findings.
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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 tunneling into the nucleus is
1896:
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in which the gate (channel) is controlled via quantum tunnelling rather than by thermal injection, reducing gate voltage from â1 volt to 0.2 volts and reducing power consumption by up to 100Ă. If these transistors can be scaled up into
4365:{\displaystyle \Psi (x)\approx {\frac {C_{+}e^{+\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}+C_{-}e^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}{\sqrt{{\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}}}}
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to serve its purpose. When these are heavily doped the depletion layer can be thin enough for tunnelling. When a small forward bias is applied, the current due to tunnelling is significant. This has a maximum at the point where the
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and medium B is the potential barrier. These have an incoming wave and resultant waves in both directions. There can be more mediums and barriers, and the barriers need not be discrete. Approximations are useful in this case.
5253:
1553:. Quantum tunnelling increases the probability of penetrating this barrier. Though this probability is still low, the extremely large number of nuclei in the core of a star is sufficient to sustain a steady fusion reaction.
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A classical wave-particle association was originally analyzed as analogous to quantum tunneling, but subsequent analysis found a fluid dynamics cause related to the vertical momentum imparted to particles near the barrier.
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The Schrödinger equation was published in 1926. The first person to apply the Schrödinger equation to a problem that involved tunneling between two classically allowed regions through a potential barrier was
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occur at extremely low energies. Probably the most fundamental ion-molecule reaction involves hydrogen ions with hydrogen molecules. The quantum mechanical tunnelling rate for the same reaction using the
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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
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of particles specifies everything that can be known about the system. Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the
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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.
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through a barrier decreases exponentially with the barrier height, the barrier width, and the tunneling particle's mass, so tunneling is seen most prominently in low-mass particles such as
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3097:
2935:
5858:, medium A is a vacuum while medium B is glass. In acoustics, medium A may be a liquid or gas and medium B a solid. For both cases, medium A is a region of space where the particle's
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2572:{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)={\kappa }^{2}\Psi (x),\qquad {\text{where}}\quad {\kappa }^{2}={\frac {2m}{\hbar ^{2}}}M.}
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3970:{\displaystyle \Psi (x)\approx C{\frac {e^{i\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}+\theta }}{\sqrt{{\frac {2m}{\hbar ^{2}}}\left(E-V(x)\right)}}}}
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the emission of electrons induced by strong electric fields. Nordheim and Fowler simplified
Oppenheimer's derivation and found values for the emitted currents and
4688:
1653:. However, in certain cases, large isotopic effects are observed that cannot be accounted for by a semi-classical treatment, and quantum tunnelling is required.
2367:{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)=-k^{2}\Psi (x),\qquad {\text{where}}\quad k^{2}=-{\frac {2m}{\hbar ^{2}}}M.}
6975:
Talou, P.; Carjan, N.; Strottman, D. (1998). "Time-dependent properties of proton decay from crossing single-particle metastable states in deformed nuclei".
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The concept of dynamical tunnelling is particularly suited to address the problem of quantum tunnelling in high dimensions (d>1). In the case of an
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2593:) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of
2066:{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)\Psi (x)\equiv {\frac {2m}{\hbar ^{2}}}M(x)\Psi (x),}
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Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between
8444:
1190:. The latter researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the
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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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shown, can be analysed and solved algebraically. Most problems do not have an algebraic solution, so numerical solutions are used. "
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as this consequence of quantum tunnelling creates a constant energy source over a large time interval for environments outside the
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function into the barrier, without transmission on the other side, as a tunneling effect, such as in tunneling into the walls of a
137:
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In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points
1753:
Quantum tunnelling through a barrier. The energy of the tunnelled particle is the same but the probability amplitude is decreased.
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5749:, tunnelling can be understood as the quantum transport between semi-classical states built on two distinct but symmetric tori.
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Winful, H. G. (2006). "Tunneling time, the
Hartman effect, and superluminality: A proposed resolution of an old paradox".
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are the same. As the voltage bias is increased, the two conduction bands no longer line up and the diode acts typically.
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was not used, and the effect was instead referred to as penetration of, or leaking through, a barrier. The German term
1741:. Other instances of quantum tunnelling-induced mutations in biology are believed to be a cause of ageing and cancer.
1733:. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised, causing a mutation.
1001:
The effect was predicted in the early 20th century. Its acceptance as a general physical phenomenon came mid-century.
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8171:. International series of monographs on physics (2nd ed.). Oxford ; New York: Oxford University Press.
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Classically, the electron would either transmit or reflect with 100% certainty, depending on its energy. In 1928
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5099:{\displaystyle \Psi (x)=C_{A}Ai\left({\sqrt{v_{1}}}(x-x_{1})\right)+C_{B}Bi\left({\sqrt{v_{1}}}(x-x_{1})\right)}
2956:
Substituting the second equation into the first and using the fact that the real part needs to be 0 results in:
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remains unity. The wider the barrier and the higher the barrier energy, the lower the probability of tunneling.
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Chaos-assisted tunnelling oscillations between two regular tori embedded in a chaotic sea, seen in phase space
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Cooper, W. G. (June 1993). "Roles of
Evolution, Quantum Mechanics and Point Mutations in Origins of Cancer".
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reviewed briefly the phenomenon of zero-time tunnelling. More recently, experimental tunnelling time data of
2617:) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive
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Quantum tunneling oscillations of probability in an integrable double well of potential, seen in phase space
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to satisfy the real part of the equation; for a good classical limit starting with the highest power of the
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to the other side, thus crossing the barrier. The reason for this difference comes from treating matter as
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923:, should not be passable due to the object not having sufficient energy to pass or surmount the barrier.
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Hong, Jooyoo; Vilenkin, Alexander; Winitzki, Serge (2003). "Particle creation in a tunneling universe".
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Vyas, P. B.; Naquin, C.; Edwards, H.; Lee, M.; Vandenberghe, W. G.; Fischetti, M. V. (23 January 2017).
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6874:"Theoretical simulation of negative differential transconductance in lateral quantum well nMOS devices"
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5268:= 0), there is a very high, but narrow potential barrier. A significant tunnelling effect can be seen.
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4676:{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots }
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developed a modified treatment of
Arrhenius kinetics that is commonly used to model this phenomenon.
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Some physicists have claimed that it is possible for spin-zero particles to travel faster than the
1572:). This was the first application of quantum tunnelling. Radioactive decay is a relevant issue for
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7042:
Wild, Robert; Nötzold, Markus; Simpson, Malcolm; Tran, Thuy Dung; Wester, Roland (16 March 2023).
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To solve this equation using the semiclassical approximation, each function must be expanded as a
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Wild, Robert; Nötzold, Markus; Simpson, Malcolm; Tran, Thuy Dung; Wester, Roland (1 March 2023).
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1891:{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=E\Psi (x)}
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level. Binnig and Rohrer were awarded the Nobel Prize in
Physics in 1986 for their discovery.
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Niels Bohr
Library & Archives, American Institute of Physics, College Park, Maryland, USA
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1349:. These are tunnel junctions, the study of which requires understanding quantum tunnelling.
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5712:, well established mathematical theorems imply that the process is completely subluminal.
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Friedrich Hund ... was the first to make use of quantum mechanical barrier penetration ...
6175:. International series in pure and applied physics. New York, NY: McGraw Hill. p. 2.
5757:
5477:{\displaystyle T(E)=e^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left}}},}
1621:â H + HD, has been measured experimentally in an ion trap. The deuterium was placed in an
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Gurney, R. W.; Condon, E. U. (1929). "Quantum
Mechanics and Radioactive Disintegration".
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Animation, applications and research linked to tunnel effect and other quantum phenomena
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Mandelstam, L.; Leontowitsch, M. (1928). "Zur
Theorie der Schrödingerschen Gleichung".
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The solutions of the Schrödinger equation take different forms for different values of
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7814:"Time delay of evanescent electromagnetic waves and the analogy to particle tunneling"
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5701:, seems to indicate that particles could tunnel at apparent speeds faster than light.
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6587:. Internet Archive. Cambridge, UK ; New York : Cambridge University Press.
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2827:{\displaystyle \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).}
2658:
2392:) is constant and positive, then the Schrödinger equation can be written in the form
2185:) is constant and negative, then the Schrödinger equation can be written in the form
1641:
1562:
1541:
Quantum tunnelling is an essential phenomenon for nuclear fusion. The temperature in
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5645:{\displaystyle T(E)=e^{-2{\sqrt {{\frac {2m}{\hbar ^{2}}}(V_{0}-E)}}(x_{2}-x_{1})}.}
3800:
which corresponds to classical motion. Resolving the next order of expansion yields
3085:{\displaystyle A'(x)+A(x)^{2}-B(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).}
1387:
device, based on the phenomenon of quantum tunnelling through the potential barriers
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6819:(2011). "Tunnel field-effect transistors as energy-efficient electronic switches".
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1721:) as well as enzymatic catalysis. Proton tunnelling is a key factor in spontaneous
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6017:
But quantum particles are able to tunnel into the classically forbidden region ...
5248:{\displaystyle C_{+}={\frac {1}{2}}C\cos {\left(\theta -{\frac {\pi }{4}}\right)}}
2581:
The solutions of this equation are rising and falling exponentials in the form of
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Several phenomena have the same behavior as quantum tunnelling. Two examples are
4106:
which corresponds to tunneling. Resolving the next order of the expansion yields
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7565:
5694:
3491:{\displaystyle B(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}B_{k}(x),}
2376:
The solutions of this equation represent travelling waves, with phase-constant +
1674:
1210:
entered the language in 1932 when it was used by Yakov
Frenkel in his textbook.
1083:, the time evolution of a known wave function can be deduced. The square of the
8934:
8903:
8893:
8515:
8495:
8324:
8143:
7949:
7165:
7067:
6475:
Feinberg, E. L. (2002). "The forefather (about Leonid Isaakovich Mandelstam)".
5709:
5705:
5678:
5666:
5260:
3396:{\displaystyle A(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}A_{k}(x)}
1714:
1666:
1536:
1436:
1324:
1160:
1120:
963:
792:
752:
732:
702:
682:
632:
598:
448:
438:
231:
7506:
7409:
6634:
6609:
5343:
With the coefficients found, the global solution can be found. Therefore, the
1345:
A simple barrier can be created by separating two conductors with a very thin
1174:
A great success of the tunnelling theory was the mathematical explanation for
9032:
8853:
8706:
8597:
8500:
8416:
8386:
8339:
8186:
8151:
7957:
7845:
7837:
7790:
7686:
7612:
7173:
7075:
7006:
6905:
6643:
6551:
6420:
6012:
5816:
4934:
1320:
1218:
1187:
1168:
1072:
931:
852:
847:
777:
747:
717:
588:
534:
261:
236:
114:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
8102:
7725:
7356:
10.1002/(SICI)1521-3889(199812)7:7/8<660::AID-ANDP660>3.0.CO;2-0
5824:
3277:. From the equations, the power series must start with at least an order of
8736:
8334:
8329:
7965:
7903:
7782:
7432:"Quantum-tunnelling time is measured using ultracold atoms â Physics World"
7238:
7181:
7116:
7083:
6961:
6850:
6559:
6394:
5910:
5859:
4783:{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)=v_{1}(x-x_{1}).}
3254:
1690:
1588:
1573:
1542:
1440:
1416:
1374:
1298:
1226:
1217:
demonstrated tunneling of electrons over a few nanometer wide barrier in a
1179:
983:
975:
915:
phenomenon in which an object such as an electron or atom passes through a
842:
837:
772:
757:
722:
216:
7853:
7733:
7319:
1280:
Tunnelling is the cause of some important macroscopic physical phenomena.
8774:
8200:
8062:
Introduction To Quantum Mechanics: Schrodinger Equation And Path Integral
6442:
Friedlander, Gerhart; Kennedy, Joseph E.; Miller, Julian Malcolm (1964).
5987:
5970:
5746:
1686:
1654:
1626:
experimental data that collisions happened one in every hundred billion.
1510:
1353:
take advantage of quantum tunnelling and superconductivity to create the
1253:
1242:
1175:
1123:
in a series of articles published in 1927. He studied the solutions of a
967:
943:
927:
807:
762:
697:
652:
7282:
Quantum Mechanics: In Physics and Chemistry with Applications to Bioloty
6989:
6842:
2112:
represents distance measured in the direction of motion of the particle,
1737:
was the first to develop this theory of spontaneous mutation within the
1139:
discovered tunneling independently and published their results in 1928.
6816:
6408:
6302:
5920:
5677:
then exists in which the particle arrives before it has left. In 1998,
4924:{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=v_{1}(x-x_{1})\Psi (x).}
1701:
Quantum tunnelling is among the central non-trivial quantum effects in
1035:
995:
797:
767:
687:
662:
657:
642:
8126:
6897:
6257:
2125:
of the particle (measured relative to any convenient reference level),
1106:" offer approximate solutions that are easier to compute, such as the
8628:
8309:
7765:
7604:
6350:
6325:
5820:
5690:
5333:{\displaystyle C_{-}=-C\sin {\left(\theta -{\frac {\pi }{4}}\right)}}
3172:
1767:
1749:
1614:
1316:
1214:
1191:
288:
8221:
8211:
7747:
Brodier, Olivier; Schlagheck, Peter; Ullmo, Denis (25 August 2002).
7333:
Low, F. E. (1998). "Comments on apparent superluminal propagation".
2131:
is the energy of the particle that is associated with motion in the
7548:
7148:
1730:
1622:
1610:
1402:
flow in one direction more than the other. The device depends on a
947:
667:
7869:"Unpredictable Tunneling of a Classical Wave-Particle Association"
7392:
7255:
Quantum Biochemistry: Electronic Structure and Biological Activity
1357:. This has applications in precision measurements of voltages and
5973:(6 May 2004). "Quantum mechanics and the equivalence principle".
5686:
5682:
2663:
The wave function is expressed as the exponential of a function:
1693:. Tunnelling of molecular hydrogen has been observed in the lab.
1645:
1545:
is generally insufficient to allow atomic nuclei to overcome the
1265:
8064:(2nd ed.). Singapore: World Scientific Publishing Company.
5524:
are the two classical turning points for the potential barrier.
1587:
Quantum tunnelling may be one of the mechanisms of hypothetical
962:
Tunneling plays an essential role in physical phenomena such as
5855:
5347:
for a particle tunneling through a single potential barrier is
3600:{\displaystyle A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-E\right)}
994:
because electrons tunnel readily through insulating layers and
951:
1023:
Animation showing the tunnel effect and its application to an
942:
describe their behavior. The probability of transmission of a
5812:
5742:
1678:
1391:
1379:
1222:
1062:
7377:
Nimtz, G. (2011). "Tunneling Confronts Special Relativity".
4099:{\displaystyle A_{0}(x)=\pm {\sqrt {2m\left(V(x)-E\right)}}}
3793:{\displaystyle B_{0}(x)=\pm {\sqrt {2m\left(E-V(x)\right)}}}
4499:{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)}
1463:
1269:
8212:
Animated illustration of quantum tunneling in a RTD device
7867:
Eddi, A.; Fort, E.; Moisy, F.; Couder, Y. (16 June 2009).
7626:
Keshavamurthy, Srihari; Schlagheck, Peter (9 March 2011).
7132:"Tunnelling measured in a very slow ionâmolecule reaction"
7044:"Tunnelling measured in a very slow ionâmolecule reaction"
6450:(2nd ed.). New York: John Wiley & Sons. pp.
5669:
when tunnelling. This appears to violate the principle of
5527:
For a rectangular barrier, this expression simplifies to:
3498:
with the following constraints on the lowest order terms,
1709:. Electron tunnelling is a key factor in many biochemical
6792:
Physics for Scientists and Engineers: With Modern Physics
6610:"Scanning tunneling microscopyâfrom birth to adolescence"
5741:, where bounded classical trajectories are confined onto
3984:
If the phase varies slowly as compared to the amplitude,
1722:
1682:
1229:
showed experimentally that tunnelling also took place in
1225:
based on tunnel effect. In 1960, following Esaki's work,
7205:"Quantum Tunnelling to the Origin and Evolution of Life"
6928:"Quantum tunnelling to the origin and evolution of life"
6441:
3678:
If the amplitude varies slowly as compared to the phase
1660:
7625:
6062:. Vol. 2 (Eighth ed.). Belmont: Brooks/Cole.
1705:. Here it is important both as electron tunnelling and
1264:, which is based on tunnelling and is used for imaging
990:. Tunneling limits the minimum size of devices used in
8026:(3. ed.). Upper Saddle River, NJ: Prentice Hall.
7746:
7129:
7041:
6280:
3171:
in atomic units (a.u.). The solid lines represent the
1509:
The scanning tunnelling microscope (STM), invented by
1233:. The tunnelling spectrum gave direct evidence of the
1206:
was used in 1931 by Walter Schottky. The English term
7021:
5780:
5533:
5490:
5353:
5276:
5264:
Quantum tunnelling through a barrier. At the origin (
5184:
5144:
5118:
4943:
4829:
4798:
4691:
4685:
Keeping only the first order term ensures linearity:
4539:
4512:
4445:
4418:
4380:
4112:
4032:
3990:
3806:
3726:
3684:
3613:
3504:
3409:
3317:
3283:
3263:
3184:
3119:
2964:
2873:
2839:
2723:
2669:
2601:) determines the nature of the medium, with negative
2398:
2191:
2082:
1904:
1776:
1473:
52:
8109:
6974:
6871:
2636:
The mathematics of dealing with the situation where
1599:
1495:, and that impurities in the metal will disrupt it.
8217:
Interactive Solution of Schrodinger Tunnel Equation
7866:
3670:At this point two extreme cases can be considered.
8080:
7579:Davis, Michael J.; Heller, Eric J. (1 July 1981).
7438:
6443:
6326:"Quantum Mechanics and Radioactive Disintegration"
6139:
5786:
5644:
5516:
5476:
5332:
5247:
5170:
5130:
5098:
4923:
4811:
4782:
4675:
4525:
4498:
4431:
4401:
4364:
4098:
4018:
3969:
3792:
3712:
3660:
3599:
3490:
3395:
3299:
3269:
3241:
3163:
3084:
2929:
2859:
2826:
2709:
2571:
2366:
2088:
2065:
1890:
1466:, they would improve the performance per power of
1446:
113:
7532:"Subluminality of relativistic quantum tunneling"
7529:
2867:is then separated into real and imaginary parts:
9030:
7986:(3. ed.). Great Malvern: Cappella Archive.
7530:Gavassino, L.; Disconzi, M. M. (13 March 2023).
5769:
2625:) is sandwiched between two regions of negative
1498:
1249:for their works on quantum tunneling in solids.
1098:Some models of a tunneling barrier, such as the
8045:(4th ed.). San Francisco: Addison-Wesley.
8003:JWKB Approximation: Contributions to the Theory
7932:Bush, John W M; Oza, Anand U (1 January 2021).
6435:
1245:. Esaki, Giaever and Josephson shared the 1973
8059:
7811:
7700:Tomsovic, Steven; Ullmo, Denis (1 July 1994).
6921:
6919:
6917:
6915:
6046:Tunneling into the barrier (wall) is possible.
1198:In the early days of quantum theory, the term
1004:
8237:
8169:Principles of electron tunneling spectroscopy
7981:
7812:Martin, Th.; Landauer, R. (1 February 1992).
7699:
7581:"Quantum dynamical tunneling in bound states"
6584:Nobel laureates and twentieth-century physics
1419:is such that the energy level of the p and n
1293:Tunnelling is a source of current leakage in
1030:Quantum tunnelling falls under the domain of
880:
8000:
6608:Binnig, Gerd; Rohrer, Heinrich (July 1987).
6607:
6419:(Interview). Interviewed by Charles Weiner;
6372:
6323:
6171:Bjorken, James D.; Drell, Sidney D. (1964).
6170:
6057:
5843:effects provided that the wave equation has
5752:
5732:
4792:Using this approximation, the equation near
3113:for tunneling through the potential barrier
108:
82:
7578:
6912:
6814:
6721:
6699:Modern Physics for Scientists and Engineers
6085:Modern Physics for Scientists and Engineers
5938:
5830:These effects are modeled similarly to the
1673:can be explained, such as the synthesis of
1260:developed a new type of microscope, called
1241:predicted the tunneling of superconducting
934:describes the state of a particle or other
8244:
8230:
8207:Animated illustration of quantum tunneling
7629:Dynamical Tunneling: Theory and Experiment
6231:
5963:
1744:
1038:, particles attempting to travel across a
887:
873:
8125:
8019:
7764:
7652:
7547:
7391:
7228:
7147:
6988:
6951:
6785:
6783:
6762:. New York: John Wiley and Sons. p.
6692:
6690:
6688:
6686:
6684:
6682:
6633:
6349:
5986:
2609:) corresponding to medium A and positive
1629:
1457:A European research project demonstrated
8001:Fröman, Nanny; Fröman, Per Olof (1965).
7655:"Tunnelling between tori in phase space"
7278:
6751:
6749:
6728:(2nd ed.). New York: VCH. pp.
6474:
6234:"The Early History of Quantum Tunneling"
6202:. North Holland, John Wiley & Sons.
6133:
6131:
6129:
6031:"Particle in a Finite Box and Tunneling"
5756:
5719:
5259:
3093:
2161:, which has no accepted name in physics.
1748:
1580:where insolation would not be possible (
1378:
1061:
1052:having properties of waves and particles
1008:
7931:
7653:Wilkinson, Michael (1 September 1986).
7494:
7202:
6925:
6195:
5945:(2nd ed.). New York: VCH. p.
3311:possible is preferable, which leads to
2633:), hence creating a potential barrier.
1757:
998:that are thinner than about 1 nm.
14:
9031:
8078:
8040:
7982:Binney, James; Skinner, David (2010).
7458:
7305:
6789:
6780:
6696:
6679:
6189:
6137:
6082:
6028:
5969:
5715:
2710:{\displaystyle \Psi (x)=e^{\Phi (x)},}
1283:
8251:
8225:
8060:Muller-kirsten, Harald J. W. (2012).
7376:
7251:
6755:
6746:
6580:
6517:
6407:
6324:Gurney, R. W.; Condon, E. U. (1928).
6227:
6225:
6223:
6221:
6219:
6126:
6109:"Quantum Effects At 7/5nm And Beyond"
4412:To start, a classical turning point,
2930:{\displaystyle \Phi '(x)=A(x)+iB(x),}
1764:time-independent Schrödinger equation
1665:By including quantum tunnelling, the
1661:Astrochemistry in interstellar clouds
1091:that the probability the particle is
1057:
8166:
8087:. River Edge, NJ: World Scientific.
7196:
7096:
5798:
5654:
4506:is expanded in a power series about
3164:{\displaystyle U(x)=8e^{-0.25x^{2}}}
2652:
1556:
7332:
6794:. Pearson Education. p. 1311.
6029:Fowler, Michael (5 December 2019).
3661:{\displaystyle A_{0}(x)B_{0}(x)=0.}
3242:{\displaystyle H(x,p)=p^{2}/2+U(x)}
2115:Κ is the Schrödinger wave function,
1150:, published a paper that discussed
24:
7975:
6216:
4944:
4906:
4859:
4113:
3807:
3451:
3359:
2875:
2841:
2745:
2725:
2690:
2670:
2504:
2477:
2428:
2298:
2270:
2221:
2048:
1999:
1934:
1876:
1858:
1831:
1696:
1669:syntheses of various molecules in
1525:
1474:Conductivity of crystalline solids
1334:
926:Tunneling is a consequence of the
419:Sum-over-histories (path integral)
105:
79:
35:Part of a series of articles about
25:
9060:
8194:
6232:Merzbacher, Eugen (August 2002).
5697:. Another experiment overseen by
3430:
3338:
1600:Energetically forbidden reactions
1530:
1178:, which was developed in 1928 by
971:
938:, and wave equations such as the
56:
9013:
9012:
7984:The physics of quantum mechanics
2177:) is positive or negative. When
1304:
8023:Introduction to electrodynamics
7925:
7860:
7805:
7740:
7693:
7646:
7619:
7585:The Journal of Chemical Physics
7572:
7523:
7488:
7452:
7424:
7370:
7326:
7299:
7272:
7245:
7123:
7090:
7035:
6865:
6808:
6715:
6650:
6601:
6574:
6511:
6497:10.1070/PU2002v045n01ABEH001126
6468:
6401:
6366:
6317:
6274:
5906:Superconducting tunnel junction
5847:solutions in medium A but real
2525:
2519:
2319:
2313:
1447:Tunnel field-effect transistors
1368:
1275:
8962:Relativistic quantum mechanics
8043:Introductory quantum mechanics
7938:Reports on Progress in Physics
7934:"Hydrodynamic quantum analogs"
7896:10.1103/PhysRevLett.102.240401
7749:"Resonance-Assisted Tunneling"
7659:Physica D: Nonlinear Phenomena
7308:Cancer Biochemistry Biophysics
7099:The tunnel effect in chemistry
6701:. Prentice Hall. p. 479.
6173:Relativistic quantum mechanics
6164:
6101:
6087:. Prentice Hall. p. 234.
6076:
6051:
6022:
5932:
5634:
5608:
5603:
5584:
5543:
5537:
5453:
5447:
5363:
5357:
5088:
5069:
5018:
4999:
4953:
4947:
4915:
4909:
4903:
4884:
4868:
4862:
4774:
4755:
4728:
4722:
4658:
4638:
4622:
4603:
4576:
4570:
4482:
4476:
4396:
4390:
4339:
4333:
4282:
4276:
4196:
4190:
4122:
4116:
4080:
4074:
4049:
4043:
4007:
4001:
3950:
3944:
3888:
3882:
3816:
3810:
3780:
3774:
3743:
3737:
3701:
3695:
3649:
3643:
3630:
3624:
3583:
3577:
3551:
3544:
3522:
3515:
3482:
3476:
3419:
3413:
3390:
3384:
3327:
3321:
3236:
3230:
3200:
3188:
3129:
3123:
3065:
3059:
3017:
3010:
2995:
2988:
2979:
2973:
2921:
2915:
2903:
2897:
2888:
2882:
2854:
2848:
2807:
2801:
2759:
2752:
2738:
2732:
2699:
2693:
2679:
2673:
2513:
2507:
2486:
2480:
2474:
2468:
2437:
2431:
2307:
2301:
2279:
2273:
2267:
2261:
2230:
2224:
2057:
2051:
2045:
2039:
2008:
2002:
1985:
1979:
1943:
1937:
1885:
1879:
1867:
1861:
1855:
1849:
1840:
1834:
1644:, the substitution of a light
1505:Scanning tunnelling microscope
1453:Tunnel field-effect transistor
1288:
1204:wellenmechanische Tunneleffekt
1171:that agreed with experiments.
569:Relativistic quantum mechanics
101:
94:
75:
13:
1:
8940:Quantum statistical mechanics
8717:Quantum differential calculus
8639:Delayed-choice quantum eraser
8407:Symmetry in quantum mechanics
7482:10.1016/j.physrep.2006.09.002
6544:10.1126/science.183.4130.1149
6520:"Long Journey into Tunneling"
6146:. World Scientific. pp.
5975:Classical and Quantum Gravity
5926:
5832:rectangular potential barrier
5815:) and the application of the
5770:Resonance-assisted tunnelling
2953:) are real-valued functions.
1499:Scanning tunneling microscope
1427:rapidly as the bias voltage.
1262:scanning tunneling microscope
988:scanning tunneling microscope
609:Quantum statistical mechanics
27:Quantum mechanical phenomenon
8020:Griffiths, David J. (2004).
8005:. Amsterdam: North-Holland.
7679:10.1016/0167-2789(86)90009-6
7221:10.2174/13852728113179990083
7101:. London: Chapman and Hall.
6944:10.2174/13852728113179990083
6518:Esaki, Leo (22 March 1974).
5881:Dielectric barrier discharge
5817:non-dispersive wave-equation
2135:-axis (measured relative to
1594:
1578:circumstellar habitable zone
1295:very-large-scale integration
18:Quantum mechanical tunneling
7:
8742:Quantum stochastic calculus
8732:Quantum measurement problem
8654:MachâZehnder interferometer
8083:Quantum theory of tunneling
8041:Liboff, Richard L. (2002).
7566:10.1103/PhysRevA.107.032209
7097:Bell, Ronald Percy (1980).
6142:Quantum Theory of Tunneling
6005:10.1088/0264-9381/21/11/017
5874:
5517:{\displaystyle x_{1},x_{2}}
5171:{\displaystyle C_{+},C_{-}}
3300:{\displaystyle \hbar ^{-1}}
2149:) is a quantity defined by
1439:that has a discrete lowest
1005:Introduction to the concept
579:Quantum information science
10:
9065:
8144:10.1103/PhysRevD.68.023520
7702:"Chaos-assisted tunneling"
7166:10.1038/s41586-023-05727-z
7068:10.1038/s41586-023-05727-z
6878:Journal of Applied Physics
6581:Dardo, M. (Mauro) (2004).
6446:Nuclear and Radiochemistry
5704:Other physicists, such as
5658:
5109:solution to connect them.
4019:{\displaystyle B_{0}(x)=0}
3713:{\displaystyle A_{0}(x)=0}
2656:
1633:
1604:Chemical reactions in the
1560:
1534:
1502:
1450:
1372:
1338:
1327:physics. It is similar to
1308:
1235:superconducting energy gap
1221:structure and developed a
1113:
9008:
8970:
8922:
8802:Quantum complexity theory
8780:Quantum cellular automata
8755:
8687:
8621:
8534:
8483:
8470:Path integral formulation
8437:
8302:
8259:
7507:10.1007/978-3-662-02753-0
7410:10.1007/s10701-011-9539-2
7252:Matta, Cherif F. (2014).
7209:Current Organic Chemistry
6932:Current Organic Chemistry
6635:10.1103/RevModPhys.59.615
6614:Reviews of Modern Physics
6113:Semiconductor Engineering
5753:Chaos-assisted tunnelling
5733:Tunnelling in phase space
5131:{\displaystyle C,\theta }
4933:This can be solved using
3105:Quantum tunneling in the
2860:{\displaystyle \Phi '(x)}
1432:resonant tunnelling diode
1385:resonant tunnelling diode
1383:A working mechanism of a
8869:Quantum machine learning
8849:Quantum key distribution
8839:Quantum image processing
8829:Quantum error correction
8679:Wheeler's delayed choice
7950:10.1088/1361-6633/abc22c
7838:10.1103/PhysRevA.45.2611
7007:10.1103/PhysRevC.58.3280
6413:"Hans Bethe â Session I"
6196:Messiah, Albert (1966).
5805:evanescent wave coupling
5345:transmission coefficient
1766:for one particle in one
1459:field effect transistors
1363:multijunction solar cell
1159:published two papers on
917:potential energy barrier
614:Quantum machine learning
367:Wheeler's delayed-choice
9049:Solid state engineering
8785:Quantum finite automata
8079:Razavy, Mohsen (2003).
7876:Physical Review Letters
7726:10.1103/PhysRevE.50.145
7495:Thaller, Bernd (1992).
7279:Majumdar, Rabi (2011).
7258:. Weinheim: Wiley-VCH.
6756:Krane, Kenneth (1983).
6725:Encyclopedia of Physics
6138:Razavy, Mohsen (2003).
6058:Serway; Vuille (2008).
5942:Encyclopedia of Physics
5891:HolsteinâHerring method
5886:Field electron emission
5851:solutions in medium B.
5809:Maxwell's wave-equation
3107:phase space formulation
2169:, depending on whether
2098:reduced Planck constant
1745:Mathematical discussion
1651:transition state theory
1484:electrical conductivity
1311:Field electron emission
968:alpha radioactive decay
324:LeggettâGarg inequality
8889:Quantum neural network
8203:(Université Paris Sud)
7783:10.1006/aphy.2002.6281
7285:. Newi: PHI Learning.
6790:Knight, R. D. (2004).
6722:Lerner; Trigg (1991).
6395:10.1103/PhysRev.33.127
6283:Zeitschrift fĂŒr Physik
5939:Lerner; Trigg (1991).
5834:. In these cases, one
5788:
5787:{\displaystyle \hbar }
5762:
5725:
5646:
5518:
5478:
5334:
5269:
5249:
5172:
5132:
5100:
4925:
4813:
4784:
4677:
4527:
4500:
4433:
4403:
4402:{\displaystyle E=V(x)}
4366:
4100:
4020:
3971:
3794:
3714:
3662:
3601:
3492:
3455:
3397:
3363:
3301:
3271:
3270:{\displaystyle \hbar }
3250:
3243:
3165:
3109:of quantum mechanics.
3086:
2931:
2861:
2828:
2711:
2573:
2368:
2090:
2089:{\displaystyle \hbar }
2067:
1892:
1754:
1636:Kinetic isotope effect
1630:Kinetic isotope effect
1388:
1247:Nobel Prize in Physics
1068:
1027:
972:Tunneling applications
115:
8914:Quantum teleportation
8427:Waveâparticle duality
5789:
5760:
5723:
5647:
5519:
5479:
5335:
5263:
5250:
5173:
5133:
5101:
4926:
4821:differential equation
4814:
4812:{\displaystyle x_{1}}
4785:
4678:
4528:
4526:{\displaystyle x_{1}}
4501:
4434:
4432:{\displaystyle x_{1}}
4404:
4367:
4101:
4021:
3972:
3795:
3715:
3663:
3602:
3493:
3435:
3398:
3343:
3302:
3272:
3244:
3166:
3104:
3087:
2932:
2862:
2829:
2712:
2574:
2369:
2106:is the particle mass,
2091:
2068:
1893:
1752:
1731:tautomeric transition
1412:P-type semiconductors
1396:semiconductor devices
1382:
1182:and independently by
1157:J. Robert Oppenheimer
1125:double-well potential
1104:Semiclassical methods
1065:
1022:
957:finite potential well
928:wave nature of matter
309:Elitzur–Vaidman
299:Davisson–Germer
116:
8945:Quantum field theory
8874:Quantum metamaterial
8819:Quantum cryptography
8549:Consistent histories
8167:Wolf, E. L. (2012).
7203:Trixler, F. (2013).
7029:"adsabs.harvard.edu"
6926:Trixler, F. (2013).
6815:Ionescu, Adrian M.;
6423:. Cornell University
6035:LibreTexts Chemistry
5862:is greater than its
5807:(the application of
5778:
5531:
5488:
5351:
5274:
5182:
5142:
5116:
4941:
4827:
4796:
4689:
4537:
4510:
4443:
4416:
4378:
4110:
4030:
3988:
3804:
3724:
3682:
3611:
3502:
3407:
3315:
3281:
3261:
3182:
3117:
2962:
2871:
2837:
2721:
2667:
2396:
2384:. Alternatively, if
2189:
2080:
1902:
1774:
1758:Schrödinger equation
1719:cellular respiration
1551:thermonuclear fusion
1100:rectangular barriers
1081:Schrödinger equation
1034:. To understand the
940:Schrödinger equation
930:, where the quantum
574:Quantum field theory
486:Consistent histories
123:Schrödinger equation
50:
8930:Quantum fluctuation
8899:Quantum programming
8859:Quantum logic gates
8844:Quantum information
8824:Quantum electronics
8284:Classical mechanics
8136:2003PhRvD..68b3520H
7888:2009PhRvL.102x0401E
7830:1992PhRvA..45.2611M
7775:2002AnPhy.300...88B
7718:1994PhRvE..50..145T
7671:1986PhyD...21..341W
7597:1981JChPh..75..246D
7558:2023PhRvA.107c2209G
7474:2006PhR...436....1W
7402:2011FoPh...41.1193N
7348:1998AnP...510..660L
7158:2023Natur.615..425W
7060:2023Natur.615..425W
6999:1998PhRvC..58.3280T
6890:2017JAP...121d4501V
6843:10.1038/nature10679
6835:2011Natur.479..329I
6697:Taylor, J. (2004).
6666:psi.phys.wits.ac.za
6626:1987RvMP...59..615B
6536:1974Sci...183.1149E
6530:(4130): 1149â1155.
6489:2002PhyU...45...81F
6411:(27 October 1966).
6387:1929PhRv...33..127G
6342:1928Natur.122..439G
6295:1928ZPhy...47..131M
6250:2002PhT....55h..44M
6083:Taylor, J. (2004).
5997:2004CQGra..21.2761D
5836:transmission medium
5716:Dynamical tunneling
5408:
1671:interstellar clouds
1606:interstellar medium
1480:Drude-Lorentz model
1468:integrated circuits
1351:Josephson junctions
1329:thermionic emission
1284:Solid-state physics
1152:thermionic emission
921:classical mechanics
919:that, according to
905:barrier penetration
362:Stern–Gerlach
159:Classical mechanics
8983:in popular culture
8765:Quantum algorithms
8613:Von NeumannâWigner
8593:Objective collapse
8289:Old quantum theory
7498:The Dirac Equation
7448:. 20 October 2020.
6303:10.1007/BF01391061
5838:through which the
5825:"waves on strings"
5784:
5763:
5726:
5675:frame of reference
5642:
5514:
5474:
5380:
5330:
5270:
5245:
5168:
5128:
5096:
4921:
4809:
4780:
4673:
4523:
4496:
4429:
4399:
4362:
4096:
4016:
3967:
3790:
3710:
3658:
3597:
3488:
3393:
3297:
3267:
3251:
3239:
3161:
3082:
2927:
2857:
2824:
2707:
2569:
2364:
2086:
2063:
1888:
1770:can be written as
1755:
1675:molecular hydrogen
1519:piezoelectric rods
1389:
1137:Mikhail Leontovich
1069:
1058:Tunnelling problem
1028:
970:of atomic nuclei.
913:quantum mechanical
901:quantum tunnelling
550:Von NeumannâWigner
530:Objective-collapse
329:Mach–Zehnder
319:Leggett inequality
314:Franck–Hertz
164:Old quantum theory
111:
9044:Quantum mechanics
9026:
9025:
9000:Quantum mysticism
8978:Schrödinger's cat
8909:Quantum simulator
8879:Quantum metrology
8807:Quantum computing
8770:Quantum amplifier
8747:Quantum spacetime
8712:Quantum cosmology
8702:Quantum chemistry
8402:Scattering theory
8350:Zero-point energy
8345:Degenerate levels
8253:Quantum mechanics
8178:978-0-19-958949-4
8113:Physical Review D
8094:978-981-238-019-7
8071:978-981-4397-76-6
8052:978-0-8053-8714-8
8033:978-0-13-805326-0
8012:978-0-7204-0085-4
7993:978-1-902918-51-8
7818:Physical Review A
7753:Annals of Physics
7706:Physical Review E
7639:978-1-4398-1666-0
7536:Physical Review A
7516:978-3-642-08134-7
7446:"Quanta Magazine"
7265:978-3-527-62922-0
7215:(16): 1758â1770.
7142:(7952): 425â429.
7054:(7952): 425â429.
6977:Physical Review C
6938:(16): 1758â1770.
6898:10.1063/1.4974469
6829:(7373): 329â337.
6801:978-0-321-22369-2
6773:978-0-471-07963-7
6739:978-0-89573-752-6
6708:978-0-13-805715-2
6658:"Applications of
6594:978-0-521-83247-2
6461:978-0-471-86255-0
6258:10.1063/1.1510281
6199:Quantum Mechanics
6182:978-0-07-005493-6
6094:978-0-13-805715-2
6069:978-0-495-55475-2
5981:(11): 2761â2772.
5956:978-0-89573-752-6
5799:Related phenomena
5739:integrable system
5693:was published by
5661:Faster-than-light
5655:Faster than light
5606:
5582:
5467:
5437:
5322:
5237:
5206:
5067:
4997:
4857:
4712:
4560:
4466:
4360:
4359:
4323:
4296:
4266:
4210:
4180:
4094:
3965:
3964:
3928:
3896:
3866:
3788:
3433:
3341:
3102:
3049:
2791:
2659:WKB approximation
2653:WKB approximation
2561:
2523:
2463:
2426:
2356:
2317:
2256:
2219:
2034:
1969:
1932:
1829:
1800:
1707:proton tunnelling
1642:chemical kinetics
1582:subsurface oceans
1563:Radioactive decay
1557:Radioactive decay
1522:atomic diameter.
1361:, as well as the
1315:Cold emission of
1133:Leonid Mandelstam
1129:molecular spectra
1108:WKB approximation
1040:potential barrier
1032:quantum mechanics
1020:
980:quantum computing
897:
896:
604:Scattering theory
584:Quantum computing
357:Schrödinger's cat
289:Bell's inequality
97:
72:
41:Quantum mechanics
16:(Redirected from
9056:
9039:Particle physics
9016:
9015:
8727:Quantum geometry
8722:Quantum dynamics
8579:Superdeterminism
8511:RaritaâSchwinger
8460:Matrix mechanics
8315:Braâket notation
8246:
8239:
8232:
8223:
8222:
8190:
8163:
8129:
8106:
8086:
8075:
8056:
8037:
8016:
7997:
7970:
7969:
7929:
7923:
7922:
7920:
7918:
7913:on 10 March 2016
7912:
7906:. Archived from
7873:
7864:
7858:
7857:
7824:(4): 2611â2617.
7809:
7803:
7802:
7768:
7744:
7738:
7737:
7697:
7691:
7690:
7650:
7644:
7643:
7623:
7617:
7616:
7605:10.1063/1.441832
7576:
7570:
7569:
7551:
7527:
7521:
7520:
7492:
7486:
7485:
7456:
7450:
7449:
7442:
7436:
7435:
7428:
7422:
7421:
7395:
7386:(7): 1193â1199.
7374:
7368:
7367:
7342:(7â8): 660â661.
7330:
7324:
7323:
7303:
7297:
7296:
7276:
7270:
7269:
7249:
7243:
7242:
7232:
7200:
7194:
7193:
7151:
7127:
7121:
7120:
7094:
7088:
7087:
7039:
7033:
7032:
7025:
7019:
7018:
6992:
6983:(6): 3280â3285.
6972:
6966:
6965:
6955:
6923:
6910:
6909:
6869:
6863:
6862:
6812:
6806:
6805:
6787:
6778:
6777:
6753:
6744:
6743:
6719:
6713:
6712:
6694:
6677:
6676:
6674:
6672:
6654:
6648:
6647:
6637:
6605:
6599:
6598:
6578:
6572:
6571:
6515:
6509:
6508:
6472:
6466:
6465:
6449:
6439:
6433:
6432:
6430:
6428:
6405:
6399:
6398:
6370:
6364:
6363:
6353:
6351:10.1038/122439a0
6321:
6315:
6314:
6289:(1â2): 131â136.
6278:
6272:
6271:
6266:
6264:
6229:
6214:
6213:
6193:
6187:
6186:
6168:
6162:
6161:
6145:
6135:
6124:
6123:
6121:
6119:
6105:
6099:
6098:
6080:
6074:
6073:
6055:
6049:
6048:
6043:
6041:
6026:
6020:
6019:
5990:
5988:quant-ph/0403027
5967:
5961:
5960:
5936:
5896:Proton tunneling
5864:potential energy
5793:
5791:
5790:
5785:
5651:
5649:
5648:
5643:
5638:
5637:
5633:
5632:
5620:
5619:
5607:
5596:
5595:
5583:
5581:
5580:
5571:
5563:
5561:
5523:
5521:
5520:
5515:
5513:
5512:
5500:
5499:
5483:
5481:
5480:
5475:
5470:
5469:
5468:
5466:
5462:
5438:
5436:
5435:
5426:
5418:
5416:
5407:
5406:
5405:
5395:
5394:
5393:
5339:
5337:
5336:
5331:
5329:
5328:
5324:
5323:
5315:
5286:
5285:
5254:
5252:
5251:
5246:
5244:
5243:
5239:
5238:
5230:
5207:
5199:
5194:
5193:
5177:
5175:
5174:
5169:
5167:
5166:
5154:
5153:
5137:
5135:
5134:
5129:
5105:
5103:
5102:
5097:
5095:
5091:
5087:
5086:
5068:
5066:
5061:
5060:
5051:
5038:
5037:
5025:
5021:
5017:
5016:
4998:
4996:
4991:
4990:
4981:
4968:
4967:
4930:
4928:
4927:
4922:
4902:
4901:
4883:
4882:
4858:
4856:
4855:
4854:
4841:
4840:
4831:
4818:
4816:
4815:
4810:
4808:
4807:
4789:
4787:
4786:
4781:
4773:
4772:
4754:
4753:
4741:
4737:
4713:
4711:
4710:
4701:
4693:
4682:
4680:
4679:
4674:
4666:
4665:
4656:
4655:
4637:
4636:
4621:
4620:
4602:
4601:
4589:
4585:
4561:
4559:
4558:
4549:
4541:
4532:
4530:
4529:
4524:
4522:
4521:
4505:
4503:
4502:
4497:
4495:
4491:
4467:
4465:
4464:
4455:
4447:
4438:
4436:
4435:
4430:
4428:
4427:
4408:
4406:
4405:
4400:
4371:
4369:
4368:
4363:
4361:
4358:
4353:
4352:
4348:
4324:
4322:
4321:
4312:
4304:
4301:
4300:
4299:
4298:
4297:
4295:
4291:
4267:
4265:
4264:
4255:
4247:
4245:
4226:
4225:
4213:
4212:
4211:
4209:
4205:
4181:
4179:
4178:
4169:
4161:
4159:
4140:
4139:
4129:
4105:
4103:
4102:
4097:
4095:
4093:
4089:
4059:
4042:
4041:
4025:
4023:
4022:
4017:
4000:
3999:
3976:
3974:
3973:
3968:
3966:
3963:
3958:
3957:
3953:
3929:
3927:
3926:
3917:
3909:
3906:
3905:
3904:
3897:
3895:
3891:
3867:
3865:
3864:
3855:
3847:
3845:
3826:
3799:
3797:
3796:
3791:
3789:
3787:
3783:
3753:
3736:
3735:
3719:
3717:
3716:
3711:
3694:
3693:
3667:
3665:
3664:
3659:
3642:
3641:
3623:
3622:
3606:
3604:
3603:
3598:
3596:
3592:
3559:
3558:
3543:
3542:
3530:
3529:
3514:
3513:
3497:
3495:
3494:
3489:
3475:
3474:
3465:
3464:
3454:
3449:
3434:
3426:
3402:
3400:
3399:
3394:
3383:
3382:
3373:
3372:
3362:
3357:
3342:
3334:
3306:
3304:
3303:
3298:
3296:
3295:
3276:
3274:
3273:
3268:
3248:
3246:
3245:
3240:
3220:
3215:
3214:
3170:
3168:
3167:
3162:
3160:
3159:
3158:
3157:
3103:
3091:
3089:
3088:
3083:
3078:
3074:
3050:
3048:
3047:
3038:
3030:
3025:
3024:
3003:
3002:
2972:
2936:
2934:
2933:
2928:
2881:
2866:
2864:
2863:
2858:
2847:
2833:
2831:
2830:
2825:
2820:
2816:
2792:
2790:
2789:
2780:
2772:
2767:
2766:
2751:
2731:
2716:
2714:
2713:
2708:
2703:
2702:
2583:evanescent waves
2578:
2576:
2575:
2570:
2562:
2560:
2559:
2550:
2542:
2537:
2536:
2531:
2524:
2521:
2503:
2502:
2497:
2464:
2462:
2461:
2452:
2444:
2427:
2425:
2424:
2423:
2410:
2409:
2400:
2373:
2371:
2370:
2365:
2357:
2355:
2354:
2345:
2337:
2329:
2328:
2318:
2315:
2297:
2296:
2257:
2255:
2254:
2245:
2237:
2220:
2218:
2217:
2216:
2203:
2202:
2193:
2123:potential energy
2095:
2093:
2092:
2087:
2072:
2070:
2069:
2064:
2035:
2033:
2032:
2023:
2015:
1998:
1994:
1970:
1968:
1967:
1958:
1950:
1933:
1931:
1930:
1929:
1916:
1915:
1906:
1897:
1895:
1894:
1889:
1830:
1828:
1827:
1826:
1813:
1812:
1803:
1801:
1799:
1791:
1790:
1781:
1584:) or effective.
1570:electron capture
1421:conduction bands
1400:electric current
1355:Josephson effect
1021:
992:microelectronics
889:
882:
875:
516:Superdeterminism
169:Braâket notation
120:
118:
117:
112:
104:
99:
98:
90:
78:
73:
71:
60:
32:
31:
21:
9064:
9063:
9059:
9058:
9057:
9055:
9054:
9053:
9029:
9028:
9027:
9022:
9004:
8990:Wigner's friend
8966:
8957:Quantum gravity
8918:
8904:Quantum sensing
8884:Quantum network
8864:Quantum machine
8834:Quantum imaging
8797:Quantum circuit
8792:Quantum channel
8751:
8697:Quantum biology
8683:
8659:ElitzurâVaidman
8634:DavissonâGermer
8617:
8569:Hidden-variable
8559:de BroglieâBohm
8536:Interpretations
8530:
8479:
8433:
8320:Complementarity
8298:
8255:
8250:
8197:
8179:
8095:
8072:
8053:
8034:
8013:
7994:
7978:
7976:Further reading
7973:
7930:
7926:
7916:
7914:
7910:
7871:
7865:
7861:
7810:
7806:
7745:
7741:
7698:
7694:
7651:
7647:
7640:
7624:
7620:
7577:
7573:
7528:
7524:
7517:
7493:
7489:
7457:
7453:
7444:
7443:
7439:
7434:. 22 July 2020.
7430:
7429:
7425:
7375:
7371:
7331:
7327:
7304:
7300:
7293:
7277:
7273:
7266:
7250:
7246:
7201:
7197:
7128:
7124:
7109:
7095:
7091:
7040:
7036:
7027:
7026:
7022:
6990:nucl-th/9809006
6973:
6969:
6924:
6913:
6870:
6866:
6813:
6809:
6802:
6788:
6781:
6774:
6754:
6747:
6740:
6720:
6716:
6709:
6695:
6680:
6670:
6668:
6656:
6655:
6651:
6606:
6602:
6595:
6579:
6575:
6516:
6512:
6477:Physics-Uspekhi
6473:
6469:
6462:
6440:
6436:
6426:
6424:
6406:
6402:
6375:Physical Review
6371:
6367:
6322:
6318:
6279:
6275:
6262:
6260:
6230:
6217:
6210:
6194:
6190:
6183:
6169:
6165:
6158:
6136:
6127:
6117:
6115:
6107:
6106:
6102:
6095:
6081:
6077:
6070:
6060:College Physics
6056:
6052:
6039:
6037:
6027:
6023:
5968:
5964:
5957:
5937:
5933:
5929:
5916:Tunnel junction
5901:Quantum cloning
5877:
5845:travelling wave
5840:wave propagates
5801:
5779:
5776:
5775:
5772:
5755:
5735:
5718:
5699:A. M. Steinberg
5663:
5657:
5628:
5624:
5615:
5611:
5591:
5587:
5576:
5572:
5564:
5562:
5560:
5553:
5549:
5532:
5529:
5528:
5508:
5504:
5495:
5491:
5489:
5486:
5485:
5443:
5439:
5431:
5427:
5419:
5417:
5415:
5401:
5397:
5396:
5389:
5385:
5384:
5373:
5369:
5352:
5349:
5348:
5314:
5307:
5303:
5302:
5281:
5277:
5275:
5272:
5271:
5229:
5222:
5218:
5217:
5198:
5189:
5185:
5183:
5180:
5179:
5162:
5158:
5149:
5145:
5143:
5140:
5139:
5117:
5114:
5113:
5082:
5078:
5062:
5056:
5052:
5050:
5049:
5045:
5033:
5029:
5012:
5008:
4992:
4986:
4982:
4980:
4979:
4975:
4963:
4959:
4942:
4939:
4938:
4897:
4893:
4878:
4874:
4850:
4846:
4842:
4836:
4832:
4830:
4828:
4825:
4824:
4803:
4799:
4797:
4794:
4793:
4768:
4764:
4749:
4745:
4718:
4714:
4706:
4702:
4694:
4692:
4690:
4687:
4686:
4661:
4657:
4651:
4647:
4632:
4628:
4616:
4612:
4597:
4593:
4566:
4562:
4554:
4550:
4542:
4540:
4538:
4535:
4534:
4517:
4513:
4511:
4508:
4507:
4472:
4468:
4460:
4456:
4448:
4446:
4444:
4441:
4440:
4423:
4419:
4417:
4414:
4413:
4379:
4376:
4375:
4354:
4329:
4325:
4317:
4313:
4305:
4303:
4302:
4272:
4268:
4260:
4256:
4248:
4246:
4244:
4231:
4227:
4221:
4217:
4186:
4182:
4174:
4170:
4162:
4160:
4158:
4145:
4141:
4135:
4131:
4130:
4128:
4111:
4108:
4107:
4070:
4066:
4058:
4037:
4033:
4031:
4028:
4027:
3995:
3991:
3989:
3986:
3985:
3959:
3934:
3930:
3922:
3918:
3910:
3908:
3907:
3872:
3868:
3860:
3856:
3848:
3846:
3844:
3831:
3827:
3825:
3805:
3802:
3801:
3764:
3760:
3752:
3731:
3727:
3725:
3722:
3721:
3689:
3685:
3683:
3680:
3679:
3637:
3633:
3618:
3614:
3612:
3609:
3608:
3573:
3569:
3554:
3550:
3538:
3534:
3525:
3521:
3509:
3505:
3503:
3500:
3499:
3470:
3466:
3460:
3456:
3450:
3439:
3425:
3408:
3405:
3404:
3378:
3374:
3368:
3364:
3358:
3347:
3333:
3316:
3313:
3312:
3309:Planck constant
3288:
3284:
3282:
3279:
3278:
3262:
3259:
3258:
3216:
3210:
3206:
3183:
3180:
3179:
3153:
3149:
3142:
3138:
3118:
3115:
3114:
3111:Wigner function
3094:
3055:
3051:
3043:
3039:
3031:
3029:
3020:
3016:
2998:
2994:
2965:
2963:
2960:
2959:
2874:
2872:
2869:
2868:
2840:
2838:
2835:
2834:
2797:
2793:
2785:
2781:
2773:
2771:
2762:
2758:
2744:
2724:
2722:
2719:
2718:
2689:
2685:
2668:
2665:
2664:
2661:
2655:
2555:
2551:
2543:
2541:
2532:
2527:
2526:
2520:
2498:
2493:
2492:
2457:
2453:
2445:
2443:
2419:
2415:
2411:
2405:
2401:
2399:
2397:
2394:
2393:
2350:
2346:
2338:
2336:
2324:
2320:
2314:
2292:
2288:
2250:
2246:
2238:
2236:
2212:
2208:
2204:
2198:
2194:
2192:
2190:
2187:
2186:
2081:
2078:
2077:
2028:
2024:
2016:
2014:
1975:
1971:
1963:
1959:
1951:
1949:
1925:
1921:
1917:
1911:
1907:
1905:
1903:
1900:
1899:
1822:
1818:
1814:
1808:
1804:
1802:
1792:
1786:
1782:
1780:
1775:
1772:
1771:
1760:
1747:
1735:Per-Olov Lowdin
1711:redox reactions
1703:quantum biology
1699:
1697:Quantum biology
1663:
1638:
1632:
1620:
1602:
1597:
1565:
1559:
1547:Coulomb barrier
1539:
1533:
1528:
1526:Nuclear physics
1515:Heinrich Rohrer
1507:
1501:
1476:
1455:
1449:
1404:depletion layer
1394:are electrical
1377:
1371:
1359:magnetic fields
1343:
1341:Tunnel junction
1337:
1335:Tunnel junction
1319:is relevant to
1313:
1307:
1291:
1286:
1278:
1258:Heinrich Rohrer
1239:Brian Josephson
1231:superconductors
1144:Lothar Nordheim
1116:
1077:physical system
1060:
1009:
1007:
936:physical system
893:
864:
863:
862:
627:
619:
618:
564:
563:Advanced topics
556:
555:
554:
506:Hidden-variable
496:de BroglieâBohm
475:
473:Interpretations
465:
464:
463:
433:
425:
424:
423:
381:
373:
372:
371:
338:
294:CHSH inequality
283:
275:
274:
273:
202:Complementarity
196:
188:
187:
186:
154:
125:
100:
89:
88:
74:
64:
59:
51:
48:
47:
28:
23:
22:
15:
12:
11:
5:
9062:
9052:
9051:
9046:
9041:
9024:
9023:
9021:
9020:
9009:
9006:
9005:
9003:
9002:
8997:
8992:
8987:
8986:
8985:
8974:
8972:
8968:
8967:
8965:
8964:
8959:
8954:
8953:
8952:
8942:
8937:
8935:Casimir effect
8932:
8926:
8924:
8920:
8919:
8917:
8916:
8911:
8906:
8901:
8896:
8894:Quantum optics
8891:
8886:
8881:
8876:
8871:
8866:
8861:
8856:
8851:
8846:
8841:
8836:
8831:
8826:
8821:
8816:
8815:
8814:
8804:
8799:
8794:
8789:
8788:
8787:
8777:
8772:
8767:
8761:
8759:
8753:
8752:
8750:
8749:
8744:
8739:
8734:
8729:
8724:
8719:
8714:
8709:
8704:
8699:
8693:
8691:
8685:
8684:
8682:
8681:
8676:
8671:
8669:Quantum eraser
8666:
8661:
8656:
8651:
8646:
8641:
8636:
8631:
8625:
8623:
8619:
8618:
8616:
8615:
8610:
8605:
8600:
8595:
8590:
8585:
8584:
8583:
8582:
8581:
8566:
8561:
8556:
8551:
8546:
8540:
8538:
8532:
8531:
8529:
8528:
8523:
8518:
8513:
8508:
8503:
8498:
8493:
8487:
8485:
8481:
8480:
8478:
8477:
8472:
8467:
8462:
8457:
8452:
8447:
8441:
8439:
8435:
8434:
8432:
8431:
8430:
8429:
8424:
8414:
8409:
8404:
8399:
8394:
8389:
8384:
8379:
8374:
8369:
8364:
8359:
8354:
8353:
8352:
8347:
8342:
8337:
8327:
8325:Density matrix
8322:
8317:
8312:
8306:
8304:
8300:
8299:
8297:
8296:
8291:
8286:
8281:
8280:
8279:
8269:
8263:
8261:
8257:
8256:
8249:
8248:
8241:
8234:
8226:
8220:
8219:
8214:
8209:
8204:
8196:
8195:External links
8193:
8192:
8191:
8177:
8164:
8107:
8093:
8076:
8070:
8057:
8051:
8038:
8032:
8017:
8011:
7998:
7992:
7977:
7974:
7972:
7971:
7924:
7882:(24): 240401.
7859:
7804:
7739:
7712:(1): 145â162.
7692:
7665:(2): 341â354.
7645:
7638:
7618:
7591:(1): 246â254.
7571:
7522:
7515:
7487:
7451:
7437:
7423:
7369:
7325:
7314:(3): 147â170.
7298:
7291:
7271:
7264:
7244:
7195:
7122:
7107:
7089:
7034:
7020:
6967:
6911:
6864:
6807:
6800:
6779:
6772:
6759:Modern Physics
6745:
6738:
6714:
6707:
6678:
6649:
6620:(3): 615â625.
6600:
6593:
6573:
6510:
6467:
6460:
6434:
6400:
6381:(2): 127â140.
6365:
6316:
6273:
6215:
6208:
6188:
6181:
6163:
6157:978-9812564887
6156:
6125:
6100:
6093:
6075:
6068:
6050:
6021:
5962:
5955:
5930:
5928:
5925:
5924:
5923:
5918:
5913:
5908:
5903:
5898:
5893:
5888:
5883:
5876:
5873:
5800:
5797:
5783:
5771:
5768:
5754:
5751:
5734:
5731:
5717:
5714:
5710:Dirac equation
5706:Herbert Winful
5679:Francis E. Low
5667:speed of light
5656:
5653:
5641:
5636:
5631:
5627:
5623:
5618:
5614:
5610:
5605:
5602:
5599:
5594:
5590:
5586:
5579:
5575:
5570:
5567:
5559:
5556:
5552:
5548:
5545:
5542:
5539:
5536:
5511:
5507:
5503:
5498:
5494:
5473:
5465:
5461:
5458:
5455:
5452:
5449:
5446:
5442:
5434:
5430:
5425:
5422:
5414:
5411:
5404:
5400:
5392:
5388:
5383:
5379:
5376:
5372:
5368:
5365:
5362:
5359:
5356:
5341:
5340:
5327:
5321:
5318:
5313:
5310:
5306:
5301:
5298:
5295:
5292:
5289:
5284:
5280:
5242:
5236:
5233:
5228:
5225:
5221:
5216:
5213:
5210:
5205:
5202:
5197:
5192:
5188:
5165:
5161:
5157:
5152:
5148:
5127:
5124:
5121:
5094:
5090:
5085:
5081:
5077:
5074:
5071:
5065:
5059:
5055:
5048:
5044:
5041:
5036:
5032:
5028:
5024:
5020:
5015:
5011:
5007:
5004:
5001:
4995:
4989:
4985:
4978:
4974:
4971:
4966:
4962:
4958:
4955:
4952:
4949:
4946:
4937:as solutions.
4935:Airy functions
4920:
4917:
4914:
4911:
4908:
4905:
4900:
4896:
4892:
4889:
4886:
4881:
4877:
4873:
4870:
4867:
4864:
4861:
4853:
4849:
4845:
4839:
4835:
4806:
4802:
4779:
4776:
4771:
4767:
4763:
4760:
4757:
4752:
4748:
4744:
4740:
4736:
4733:
4730:
4727:
4724:
4721:
4717:
4709:
4705:
4700:
4697:
4672:
4669:
4664:
4660:
4654:
4650:
4646:
4643:
4640:
4635:
4631:
4627:
4624:
4619:
4615:
4611:
4608:
4605:
4600:
4596:
4592:
4588:
4584:
4581:
4578:
4575:
4572:
4569:
4565:
4557:
4553:
4548:
4545:
4520:
4516:
4494:
4490:
4487:
4484:
4481:
4478:
4475:
4471:
4463:
4459:
4454:
4451:
4439:is chosen and
4426:
4422:
4398:
4395:
4392:
4389:
4386:
4383:
4357:
4351:
4347:
4344:
4341:
4338:
4335:
4332:
4328:
4320:
4316:
4311:
4308:
4294:
4290:
4287:
4284:
4281:
4278:
4275:
4271:
4263:
4259:
4254:
4251:
4243:
4240:
4237:
4234:
4230:
4224:
4220:
4216:
4208:
4204:
4201:
4198:
4195:
4192:
4189:
4185:
4177:
4173:
4168:
4165:
4157:
4154:
4151:
4148:
4144:
4138:
4134:
4127:
4124:
4121:
4118:
4115:
4092:
4088:
4085:
4082:
4079:
4076:
4073:
4069:
4065:
4062:
4057:
4054:
4051:
4048:
4045:
4040:
4036:
4015:
4012:
4009:
4006:
4003:
3998:
3994:
3962:
3956:
3952:
3949:
3946:
3943:
3940:
3937:
3933:
3925:
3921:
3916:
3913:
3903:
3900:
3894:
3890:
3887:
3884:
3881:
3878:
3875:
3871:
3863:
3859:
3854:
3851:
3843:
3840:
3837:
3834:
3830:
3824:
3821:
3818:
3815:
3812:
3809:
3786:
3782:
3779:
3776:
3773:
3770:
3767:
3763:
3759:
3756:
3751:
3748:
3745:
3742:
3739:
3734:
3730:
3709:
3706:
3703:
3700:
3697:
3692:
3688:
3657:
3654:
3651:
3648:
3645:
3640:
3636:
3632:
3629:
3626:
3621:
3617:
3595:
3591:
3588:
3585:
3582:
3579:
3576:
3572:
3568:
3565:
3562:
3557:
3553:
3549:
3546:
3541:
3537:
3533:
3528:
3524:
3520:
3517:
3512:
3508:
3487:
3484:
3481:
3478:
3473:
3469:
3463:
3459:
3453:
3448:
3445:
3442:
3438:
3432:
3429:
3424:
3421:
3418:
3415:
3412:
3392:
3389:
3386:
3381:
3377:
3371:
3367:
3361:
3356:
3353:
3350:
3346:
3340:
3337:
3332:
3329:
3326:
3323:
3320:
3294:
3291:
3287:
3266:
3238:
3235:
3232:
3229:
3226:
3223:
3219:
3213:
3209:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3156:
3152:
3148:
3145:
3141:
3137:
3134:
3131:
3128:
3125:
3122:
3081:
3077:
3073:
3070:
3067:
3064:
3061:
3058:
3054:
3046:
3042:
3037:
3034:
3028:
3023:
3019:
3015:
3012:
3009:
3006:
3001:
2997:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2971:
2968:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2880:
2877:
2856:
2853:
2850:
2846:
2843:
2823:
2819:
2815:
2812:
2809:
2806:
2803:
2800:
2796:
2788:
2784:
2779:
2776:
2770:
2765:
2761:
2757:
2754:
2750:
2747:
2743:
2740:
2737:
2734:
2730:
2727:
2706:
2701:
2698:
2695:
2692:
2688:
2684:
2681:
2678:
2675:
2672:
2657:Main article:
2654:
2651:
2644:) varies with
2568:
2565:
2558:
2554:
2549:
2546:
2540:
2535:
2530:
2518:
2515:
2512:
2509:
2506:
2501:
2496:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2460:
2456:
2451:
2448:
2442:
2439:
2436:
2433:
2430:
2422:
2418:
2414:
2408:
2404:
2363:
2360:
2353:
2349:
2344:
2341:
2335:
2332:
2327:
2323:
2312:
2309:
2306:
2303:
2300:
2295:
2291:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2253:
2249:
2244:
2241:
2235:
2232:
2229:
2226:
2223:
2215:
2211:
2207:
2201:
2197:
2163:
2162:
2140:
2126:
2116:
2113:
2107:
2101:
2085:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2031:
2027:
2022:
2019:
2013:
2010:
2007:
2004:
2001:
1997:
1993:
1990:
1987:
1984:
1981:
1978:
1974:
1966:
1962:
1957:
1954:
1948:
1945:
1942:
1939:
1936:
1928:
1924:
1920:
1914:
1910:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1825:
1821:
1817:
1811:
1807:
1798:
1795:
1789:
1785:
1779:
1759:
1756:
1746:
1743:
1715:photosynthesis
1698:
1695:
1662:
1659:
1634:Main article:
1631:
1628:
1618:
1601:
1598:
1596:
1593:
1561:Main article:
1558:
1555:
1537:Nuclear fusion
1535:Main article:
1532:
1531:Nuclear fusion
1529:
1527:
1524:
1503:Main article:
1500:
1497:
1475:
1472:
1451:Main article:
1448:
1445:
1437:potential well
1373:Main article:
1370:
1367:
1339:Main article:
1336:
1333:
1325:superconductor
1321:semiconductors
1309:Main article:
1306:
1303:
1290:
1287:
1285:
1282:
1277:
1274:
1169:work functions
1161:field emission
1146:, assisted by
1127:and discussed
1121:Friedrich Hund
1115:
1112:
1085:absolute value
1059:
1056:
1006:
1003:
964:nuclear fusion
895:
894:
892:
891:
884:
877:
869:
866:
865:
861:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
629:
628:
625:
624:
621:
620:
617:
616:
611:
606:
601:
599:Density matrix
596:
591:
586:
581:
576:
571:
565:
562:
561:
558:
557:
553:
552:
547:
542:
537:
532:
527:
522:
521:
520:
519:
518:
503:
498:
493:
488:
483:
477:
476:
471:
470:
467:
466:
462:
461:
456:
451:
446:
441:
435:
434:
431:
430:
427:
426:
422:
421:
416:
411:
406:
401:
396:
390:
389:
388:
382:
379:
378:
375:
374:
370:
369:
364:
359:
353:
352:
351:
350:
349:
347:Delayed-choice
342:Quantum eraser
337:
336:
331:
326:
321:
316:
311:
306:
301:
296:
291:
285:
284:
281:
280:
277:
276:
272:
271:
270:
269:
259:
254:
249:
244:
239:
234:
232:Quantum number
229:
224:
219:
214:
209:
204:
198:
197:
194:
193:
190:
189:
185:
184:
179:
173:
172:
171:
166:
161:
155:
152:
151:
148:
147:
146:
145:
140:
135:
127:
126:
121:
110:
107:
103:
96:
93:
87:
84:
81:
77:
70:
67:
63:
58:
55:
44:
43:
37:
36:
26:
9:
6:
4:
3:
2:
9061:
9050:
9047:
9045:
9042:
9040:
9037:
9036:
9034:
9019:
9011:
9010:
9007:
9001:
8998:
8996:
8993:
8991:
8988:
8984:
8981:
8980:
8979:
8976:
8975:
8973:
8969:
8963:
8960:
8958:
8955:
8951:
8948:
8947:
8946:
8943:
8941:
8938:
8936:
8933:
8931:
8928:
8927:
8925:
8921:
8915:
8912:
8910:
8907:
8905:
8902:
8900:
8897:
8895:
8892:
8890:
8887:
8885:
8882:
8880:
8877:
8875:
8872:
8870:
8867:
8865:
8862:
8860:
8857:
8855:
8854:Quantum logic
8852:
8850:
8847:
8845:
8842:
8840:
8837:
8835:
8832:
8830:
8827:
8825:
8822:
8820:
8817:
8813:
8810:
8809:
8808:
8805:
8803:
8800:
8798:
8795:
8793:
8790:
8786:
8783:
8782:
8781:
8778:
8776:
8773:
8771:
8768:
8766:
8763:
8762:
8760:
8758:
8754:
8748:
8745:
8743:
8740:
8738:
8735:
8733:
8730:
8728:
8725:
8723:
8720:
8718:
8715:
8713:
8710:
8708:
8707:Quantum chaos
8705:
8703:
8700:
8698:
8695:
8694:
8692:
8690:
8686:
8680:
8677:
8675:
8674:SternâGerlach
8672:
8670:
8667:
8665:
8662:
8660:
8657:
8655:
8652:
8650:
8647:
8645:
8642:
8640:
8637:
8635:
8632:
8630:
8627:
8626:
8624:
8620:
8614:
8611:
8609:
8608:Transactional
8606:
8604:
8601:
8599:
8598:Quantum logic
8596:
8594:
8591:
8589:
8586:
8580:
8577:
8576:
8575:
8572:
8571:
8570:
8567:
8565:
8562:
8560:
8557:
8555:
8552:
8550:
8547:
8545:
8542:
8541:
8539:
8537:
8533:
8527:
8524:
8522:
8519:
8517:
8514:
8512:
8509:
8507:
8504:
8502:
8499:
8497:
8494:
8492:
8489:
8488:
8486:
8482:
8476:
8473:
8471:
8468:
8466:
8463:
8461:
8458:
8456:
8453:
8451:
8448:
8446:
8443:
8442:
8440:
8436:
8428:
8425:
8423:
8420:
8419:
8418:
8417:Wave function
8415:
8413:
8410:
8408:
8405:
8403:
8400:
8398:
8395:
8393:
8392:Superposition
8390:
8388:
8387:Quantum state
8385:
8383:
8380:
8378:
8375:
8373:
8370:
8368:
8365:
8363:
8360:
8358:
8355:
8351:
8348:
8346:
8343:
8341:
8340:Excited state
8338:
8336:
8333:
8332:
8331:
8328:
8326:
8323:
8321:
8318:
8316:
8313:
8311:
8308:
8307:
8305:
8301:
8295:
8292:
8290:
8287:
8285:
8282:
8278:
8275:
8274:
8273:
8270:
8268:
8265:
8264:
8262:
8258:
8254:
8247:
8242:
8240:
8235:
8233:
8228:
8227:
8224:
8218:
8215:
8213:
8210:
8208:
8205:
8202:
8199:
8198:
8188:
8184:
8180:
8174:
8170:
8165:
8161:
8157:
8153:
8149:
8145:
8141:
8137:
8133:
8128:
8127:gr-qc/0210034
8123:
8120:(2): 023520.
8119:
8115:
8114:
8108:
8104:
8100:
8096:
8090:
8085:
8084:
8077:
8073:
8067:
8063:
8058:
8054:
8048:
8044:
8039:
8035:
8029:
8025:
8024:
8018:
8014:
8008:
8004:
7999:
7995:
7989:
7985:
7980:
7979:
7967:
7963:
7959:
7955:
7951:
7947:
7944:(1): 017001.
7943:
7939:
7935:
7928:
7909:
7905:
7901:
7897:
7893:
7889:
7885:
7881:
7877:
7870:
7863:
7855:
7851:
7847:
7843:
7839:
7835:
7831:
7827:
7823:
7819:
7815:
7808:
7800:
7796:
7792:
7788:
7784:
7780:
7776:
7772:
7767:
7762:
7759:(1): 88â136.
7758:
7754:
7750:
7743:
7735:
7731:
7727:
7723:
7719:
7715:
7711:
7707:
7703:
7696:
7688:
7684:
7680:
7676:
7672:
7668:
7664:
7660:
7656:
7649:
7641:
7635:
7632:. CRC Press.
7631:
7630:
7622:
7614:
7610:
7606:
7602:
7598:
7594:
7590:
7586:
7582:
7575:
7567:
7563:
7559:
7555:
7550:
7545:
7542:(3): 032209.
7541:
7537:
7533:
7526:
7518:
7512:
7508:
7504:
7500:
7499:
7491:
7483:
7479:
7475:
7471:
7468:(1â2): 1â69.
7467:
7464:
7463:
7455:
7447:
7441:
7433:
7427:
7419:
7415:
7411:
7407:
7403:
7399:
7394:
7389:
7385:
7382:
7381:
7373:
7365:
7361:
7357:
7353:
7349:
7345:
7341:
7338:
7337:
7329:
7321:
7317:
7313:
7309:
7302:
7294:
7292:9788120343047
7288:
7284:
7283:
7275:
7267:
7261:
7257:
7256:
7248:
7240:
7236:
7231:
7226:
7222:
7218:
7214:
7210:
7206:
7199:
7191:
7187:
7183:
7179:
7175:
7171:
7167:
7163:
7159:
7155:
7150:
7145:
7141:
7137:
7133:
7126:
7118:
7114:
7110:
7104:
7100:
7093:
7085:
7081:
7077:
7073:
7069:
7065:
7061:
7057:
7053:
7049:
7045:
7038:
7030:
7024:
7016:
7012:
7008:
7004:
7000:
6996:
6991:
6986:
6982:
6978:
6971:
6963:
6959:
6954:
6949:
6945:
6941:
6937:
6933:
6929:
6922:
6920:
6918:
6916:
6907:
6903:
6899:
6895:
6891:
6887:
6884:(4): 044501.
6883:
6879:
6875:
6868:
6860:
6856:
6852:
6848:
6844:
6840:
6836:
6832:
6828:
6824:
6823:
6818:
6811:
6803:
6797:
6793:
6786:
6784:
6775:
6769:
6765:
6761:
6760:
6752:
6750:
6741:
6735:
6731:
6727:
6726:
6718:
6710:
6704:
6700:
6693:
6691:
6689:
6687:
6685:
6683:
6667:
6663:
6661:
6653:
6645:
6641:
6636:
6631:
6627:
6623:
6619:
6615:
6611:
6604:
6596:
6590:
6586:
6585:
6577:
6569:
6565:
6561:
6557:
6553:
6549:
6545:
6541:
6537:
6533:
6529:
6525:
6521:
6514:
6506:
6502:
6498:
6494:
6490:
6486:
6483:(1): 81â100.
6482:
6478:
6471:
6463:
6457:
6453:
6448:
6447:
6438:
6422:
6421:Jagdish Mehra
6418:
6414:
6410:
6404:
6396:
6392:
6388:
6384:
6380:
6376:
6369:
6361:
6357:
6352:
6347:
6343:
6339:
6336:(3073): 439.
6335:
6331:
6327:
6320:
6312:
6308:
6304:
6300:
6296:
6292:
6288:
6284:
6277:
6270:
6259:
6255:
6251:
6247:
6243:
6239:
6238:Physics Today
6235:
6228:
6226:
6224:
6222:
6220:
6211:
6205:
6201:
6200:
6192:
6184:
6178:
6174:
6167:
6159:
6153:
6149:
6144:
6143:
6134:
6132:
6130:
6114:
6110:
6104:
6096:
6090:
6086:
6079:
6071:
6065:
6061:
6054:
6047:
6036:
6032:
6025:
6018:
6014:
6010:
6006:
6002:
5998:
5994:
5989:
5984:
5980:
5976:
5972:
5971:Davies, P C W
5966:
5958:
5952:
5948:
5944:
5943:
5935:
5931:
5922:
5919:
5917:
5914:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5884:
5882:
5879:
5878:
5872:
5868:
5865:
5861:
5857:
5852:
5850:
5846:
5841:
5837:
5833:
5828:
5826:
5822:
5818:
5814:
5810:
5806:
5796:
5781:
5767:
5759:
5750:
5748:
5744:
5740:
5730:
5722:
5713:
5711:
5707:
5702:
5700:
5696:
5692:
5688:
5684:
5680:
5676:
5672:
5668:
5662:
5652:
5639:
5629:
5625:
5621:
5616:
5612:
5600:
5597:
5592:
5588:
5577:
5573:
5568:
5565:
5557:
5554:
5550:
5546:
5540:
5534:
5525:
5509:
5505:
5501:
5496:
5492:
5471:
5463:
5459:
5456:
5450:
5444:
5440:
5432:
5428:
5423:
5420:
5412:
5409:
5402:
5398:
5390:
5386:
5381:
5377:
5374:
5370:
5366:
5360:
5354:
5346:
5325:
5319:
5316:
5311:
5308:
5304:
5299:
5296:
5293:
5290:
5287:
5282:
5278:
5267:
5262:
5258:
5257:
5256:
5240:
5234:
5231:
5226:
5223:
5219:
5214:
5211:
5208:
5203:
5200:
5195:
5190:
5186:
5163:
5159:
5155:
5150:
5146:
5125:
5122:
5119:
5110:
5106:
5092:
5083:
5079:
5075:
5072:
5063:
5057:
5053:
5046:
5042:
5039:
5034:
5030:
5026:
5022:
5013:
5009:
5005:
5002:
4993:
4987:
4983:
4976:
4972:
4969:
4964:
4960:
4956:
4950:
4936:
4931:
4918:
4912:
4898:
4894:
4890:
4887:
4879:
4875:
4871:
4865:
4851:
4847:
4843:
4837:
4833:
4822:
4804:
4800:
4790:
4777:
4769:
4765:
4761:
4758:
4750:
4746:
4742:
4738:
4734:
4731:
4725:
4719:
4715:
4707:
4703:
4698:
4695:
4683:
4670:
4667:
4662:
4652:
4648:
4644:
4641:
4633:
4629:
4625:
4617:
4613:
4609:
4606:
4598:
4594:
4590:
4586:
4582:
4579:
4573:
4567:
4563:
4555:
4551:
4546:
4543:
4518:
4514:
4492:
4488:
4485:
4479:
4473:
4469:
4461:
4457:
4452:
4449:
4424:
4420:
4410:
4393:
4387:
4384:
4381:
4372:
4355:
4349:
4345:
4342:
4336:
4330:
4326:
4318:
4314:
4309:
4306:
4292:
4288:
4285:
4279:
4273:
4269:
4261:
4257:
4252:
4249:
4241:
4238:
4235:
4232:
4228:
4222:
4218:
4214:
4206:
4202:
4199:
4193:
4187:
4183:
4175:
4171:
4166:
4163:
4155:
4152:
4149:
4146:
4142:
4136:
4132:
4125:
4119:
4090:
4086:
4083:
4077:
4071:
4067:
4063:
4060:
4055:
4052:
4046:
4038:
4034:
4013:
4010:
4004:
3996:
3992:
3982:
3981:
3977:
3960:
3954:
3947:
3941:
3938:
3935:
3931:
3923:
3919:
3914:
3911:
3901:
3898:
3892:
3885:
3879:
3876:
3873:
3869:
3861:
3857:
3852:
3849:
3841:
3838:
3835:
3832:
3828:
3822:
3819:
3813:
3784:
3777:
3771:
3768:
3765:
3761:
3757:
3754:
3749:
3746:
3740:
3732:
3728:
3707:
3704:
3698:
3690:
3686:
3676:
3675:
3671:
3668:
3655:
3652:
3646:
3638:
3634:
3627:
3619:
3615:
3593:
3589:
3586:
3580:
3574:
3570:
3566:
3563:
3560:
3555:
3547:
3539:
3535:
3531:
3526:
3518:
3510:
3506:
3485:
3479:
3471:
3467:
3461:
3457:
3446:
3443:
3440:
3436:
3427:
3422:
3416:
3410:
3387:
3379:
3375:
3369:
3365:
3354:
3351:
3348:
3344:
3335:
3330:
3324:
3318:
3310:
3292:
3289:
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2020:
2017:
2011:
2005:
1995:
1991:
1988:
1982:
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1688:
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1680:
1676:
1672:
1668:
1667:astrochemical
1658:
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1647:
1643:
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1627:
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1616:
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1607:
1592:
1590:
1585:
1583:
1579:
1575:
1571:
1564:
1554:
1552:
1548:
1544:
1543:stellar cores
1538:
1523:
1520:
1516:
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1506:
1496:
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1471:
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1326:
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1318:
1312:
1305:Cold emission
1302:
1300:
1296:
1281:
1273:
1271:
1267:
1263:
1259:
1255:
1250:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1219:semiconductor
1216:
1211:
1209:
1208:tunnel effect
1205:
1201:
1200:tunnel effect
1196:
1193:
1189:
1188:Edward Condon
1185:
1184:Ronald Gurney
1181:
1177:
1172:
1170:
1166:
1162:
1158:
1153:
1149:
1145:
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1122:
1111:
1109:
1105:
1101:
1096:
1094:
1088:
1086:
1082:
1078:
1074:
1073:wave function
1064:
1055:
1053:
1049:
1043:
1041:
1037:
1033:
1026:
1002:
999:
997:
993:
989:
985:
981:
977:
973:
969:
965:
960:
958:
953:
949:
945:
941:
937:
933:
932:wave function
929:
924:
922:
918:
914:
910:
906:
902:
890:
885:
883:
878:
876:
871:
870:
868:
867:
859:
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854:
851:
849:
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844:
841:
839:
836:
834:
831:
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789:
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779:
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771:
769:
766:
764:
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759:
756:
754:
751:
749:
746:
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741:
739:
736:
734:
731:
729:
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724:
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631:
630:
623:
622:
615:
612:
610:
607:
605:
602:
600:
597:
595:
592:
590:
589:Quantum chaos
587:
585:
582:
580:
577:
575:
572:
570:
567:
566:
560:
559:
551:
548:
546:
545:Transactional
543:
541:
538:
536:
535:Quantum logic
533:
531:
528:
526:
523:
517:
514:
513:
512:
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302:
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287:
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279:
278:
268:
265:
264:
263:
262:Wave function
260:
258:
255:
253:
250:
248:
245:
243:
242:Superposition
240:
238:
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233:
230:
228:
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128:
124:
91:
85:
68:
65:
61:
53:
46:
45:
42:
39:
38:
34:
33:
30:
19:
8737:Quantum mind
8649:FranckâHertz
8491:KleinâGordon
8445:Formulations
8438:Formulations
8396:
8367:Interference
8357:Entanglement
8335:Ground state
8330:Energy level
8303:Fundamentals
8267:Introduction
8168:
8117:
8111:
8082:
8061:
8042:
8022:
8002:
7983:
7941:
7937:
7927:
7915:. Retrieved
7908:the original
7879:
7875:
7862:
7821:
7817:
7807:
7766:nlin/0205054
7756:
7752:
7742:
7709:
7705:
7695:
7662:
7658:
7648:
7628:
7621:
7588:
7584:
7574:
7539:
7535:
7525:
7497:
7490:
7465:
7460:
7454:
7440:
7426:
7383:
7380:Found. Phys.
7378:
7372:
7339:
7334:
7328:
7311:
7307:
7301:
7281:
7274:
7254:
7247:
7212:
7208:
7198:
7139:
7135:
7125:
7098:
7092:
7051:
7047:
7037:
7023:
6980:
6976:
6970:
6935:
6931:
6881:
6877:
6867:
6826:
6820:
6810:
6791:
6758:
6724:
6717:
6698:
6669:. Retrieved
6665:
6659:
6652:
6617:
6613:
6603:
6583:
6576:
6527:
6523:
6513:
6480:
6476:
6470:
6445:
6437:
6425:. Retrieved
6416:
6403:
6378:
6374:
6368:
6333:
6329:
6319:
6286:
6282:
6276:
6268:
6261:. Retrieved
6244:(8): 44â49.
6241:
6237:
6198:
6191:
6172:
6166:
6141:
6116:. Retrieved
6112:
6103:
6084:
6078:
6059:
6053:
6045:
6038:. Retrieved
6034:
6024:
6016:
5978:
5974:
5965:
5941:
5934:
5911:Tunnel diode
5869:
5860:total energy
5853:
5829:
5802:
5773:
5764:
5736:
5727:
5703:
5695:GĂŒnter Nimtz
5664:
5526:
5342:
5265:
5111:
5107:
4932:
4791:
4684:
4411:
4373:
3983:
3979:
3978:
3677:
3673:
3672:
3669:
3255:power series
3252:
2958:
2955:
2950:
2946:
2942:
2938:
2662:
2645:
2641:
2637:
2635:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2598:
2594:
2590:
2586:
2580:
2389:
2385:
2381:
2377:
2375:
2182:
2178:
2174:
2170:
2166:
2164:
2158:
2154:
2150:
2146:
2142:
2136:
2132:
2128:
2118:
2109:
2103:
1761:
1739:double helix
1727:
1700:
1691:formaldehyde
1664:
1639:
1603:
1589:proton decay
1586:
1574:astrobiology
1566:
1549:and achieve
1540:
1508:
1477:
1456:
1441:energy level
1429:
1425:
1417:voltage bias
1390:
1375:Tunnel diode
1369:Tunnel diode
1344:
1314:
1299:flash memory
1292:
1279:
1276:Applications
1251:
1243:Cooper pairs
1227:Ivar Giaever
1212:
1207:
1203:
1199:
1197:
1180:George Gamow
1173:
1164:
1148:Ralph Fowler
1141:
1117:
1097:
1092:
1089:
1070:
1047:
1044:
1029:
1000:
984:flash memory
976:tunnel diode
974:include the
961:
925:
908:
907:, or simply
904:
900:
899:In physics,
898:
444:KleinâGordon
380:Formulations
251:
217:Energy level
212:Entanglement
195:Fundamentals
182:Interference
133:Introduction
29:
8995:EPR paradox
8775:Quantum bus
8644:Double-slit
8622:Experiments
8588:Many-worlds
8526:Schrödinger
8475:Phase space
8465:Schrödinger
8455:Interaction
8412:Uncertainty
8382:Nonlocality
8377:Measurement
8372:Decoherence
8362:Hamiltonian
6817:Riel, Heike
6409:Bethe, Hans
6040:4 September
5849:exponential
5823:applied to
5747:phase space
3177:Hamiltonian
1511:Gerd Binnig
1493:conductance
1398:that allow
1289:Electronics
1254:Gerd Binnig
1237:. In 1962,
1176:alpha decay
996:transistors
944:wave packet
833:von Neumann
818:Schrödinger
594:EPR paradox
525:Many-worlds
459:Schrödinger
414:Schrödinger
409:Phase-space
399:Interaction
304:Double-slit
282:Experiments
257:Uncertainty
227:Nonlocality
222:Measurement
207:Decoherence
177:Hamiltonian
9033:Categories
8923:Extensions
8757:Technology
8603:Relational
8554:Copenhagen
8450:Heisenberg
8397:Tunnelling
8260:Background
7549:2208.09742
7462:Phys. Rep.
7336:Ann. Phys.
7149:2303.14948
7108:0412213400
6209:0486409244
5927:References
5921:White hole
5673:, since a
5659:See also:
4819:becomes a
1725:mutation.
1689:important
1685:) and the
1655:R. P. Bell
1478:While the
1464:VLSI chips
1036:phenomenon
986:, and the
909:tunnelling
828:Sommerfeld
743:Heisenberg
738:Gutzwiller
678:de Broglie
626:Scientists
540:Relational
491:Copenhagen
394:Heisenberg
252:Tunnelling
153:Background
8629:Bell test
8484:Equations
8310:Born rule
8187:768067375
8160:118969589
8152:0556-2821
7958:0034-4885
7846:1050-2947
7791:0003-4916
7687:0167-2789
7613:0021-9606
7418:119249900
7393:1003.3944
7364:122717505
7190:257282176
7174:1476-4687
7076:0028-0836
7015:119075457
6906:0021-8979
6730:1308â1309
6660:tunneling
6644:0034-6861
6552:0036-8075
6505:250780246
6311:125101370
6263:17 August
6013:0264-9381
5821:acoustics
5782:ℏ
5691:electrons
5671:causality
5622:−
5598:−
5574:ℏ
5555:−
5457:−
5429:ℏ
5382:∫
5375:−
5317:π
5312:−
5309:θ
5300:
5291:−
5283:−
5232:π
5227:−
5224:θ
5215:
5164:−
5126:θ
5076:−
5006:−
4945:Ψ
4907:Ψ
4891:−
4860:Ψ
4762:−
4732:−
4704:ℏ
4671:⋯
4645:−
4610:−
4580:−
4552:ℏ
4486:−
4458:ℏ
4343:−
4315:ℏ
4286:−
4258:ℏ
4236:∫
4233:−
4223:−
4200:−
4172:ℏ
4150:∫
4126:≈
4114:Ψ
4084:−
4056:±
3939:−
3920:ℏ
3902:θ
3877:−
3858:ℏ
3836:∫
3820:≈
3808:Ψ
3769:−
3750:±
3587:−
3532:−
3458:ℏ
3452:∞
3437:∑
3431:ℏ
3366:ℏ
3360:∞
3345:∑
3339:ℏ
3290:−
3286:ℏ
3265:ℏ
3173:level set
3144:−
3069:−
3041:ℏ
3005:−
2876:Φ
2842:Φ
2811:−
2783:ℏ
2746:Φ
2726:Φ
2691:Φ
2671:Ψ
2553:ℏ
2529:κ
2505:Ψ
2495:κ
2478:Ψ
2455:ℏ
2429:Ψ
2348:ℏ
2334:−
2299:Ψ
2286:−
2271:Ψ
2248:ℏ
2222:Ψ
2084:ℏ
2049:Ψ
2026:ℏ
2012:≡
2000:Ψ
1989:−
1961:ℏ
1935:Ψ
1877:Ψ
1859:Ψ
1832:Ψ
1784:ℏ
1778:−
1768:dimension
1687:prebiotic
1615:deuterium
1595:Chemistry
1347:insulator
1317:electrons
1252:In 1981,
1215:Leo Esaki
1192:half-life
1142:In 1927,
1093:somewhere
948:electrons
858:Zeilinger
703:Ehrenfest
432:Equations
109:⟩
106:Ψ
95:^
83:⟩
80:Ψ
57:ℏ
9018:Category
8812:Timeline
8564:Ensemble
8544:Bayesian
8506:Majorana
8422:Collapse
8294:Glossary
8277:Timeline
8103:52498470
7966:33065567
7904:19658983
7799:51895893
7239:24039543
7182:36859549
7084:36859549
6962:24039543
6851:22094693
6671:30 April
6568:44642243
6560:17789212
5875:See also
2970:′
2879:′
2845:′
2749:′
2729:″
1623:ion trap
1613:isotope
1611:hydrogen
1489:barriers
1406:between
1266:surfaces
1213:In 1957
1067:barrier.
783:Millikan
708:Einstein
693:Davisson
648:Blackett
633:Aharonov
501:Ensemble
481:Bayesian
386:Overview
267:Collapse
247:Symmetry
138:Glossary
8971:Related
8950:History
8689:Science
8521:Rydberg
8272:History
8132:Bibcode
7884:Bibcode
7854:9907285
7826:Bibcode
7771:Bibcode
7734:9961952
7714:Bibcode
7667:Bibcode
7593:Bibcode
7554:Bibcode
7470:Bibcode
7398:Bibcode
7344:Bibcode
7320:8111728
7230:3768233
7154:Bibcode
7117:6854792
7056:Bibcode
6995:Bibcode
6953:3768233
6886:Bibcode
6859:4322368
6831:Bibcode
6622:Bibcode
6532:Bibcode
6524:Science
6485:Bibcode
6383:Bibcode
6360:4090561
6338:Bibcode
6291:Bibcode
6246:Bibcode
6150:, 462.
6118:15 July
5993:Bibcode
5687:photons
5683:phonons
3175:of the
2585:. When
2121:is the
2096:is the
1646:isotope
1617:, D + H
1268:at the
1114:History
952:protons
823:Simmons
813:Rydberg
778:Moseley
758:Kramers
748:Hilbert
733:Glauber
728:Feynman
713:Everett
683:Compton
454:Rydberg
143:History
8664:Popper
8185:
8175:
8158:
8150:
8101:
8091:
8068:
8049:
8030:
8009:
7990:
7964:
7956:
7902:
7852:
7844:
7797:
7789:
7732:
7685:
7636:
7611:
7513:
7416:
7362:
7318:
7289:
7262:
7237:
7227:
7188:
7180:
7172:
7136:Nature
7115:
7105:
7082:
7074:
7048:Nature
7013:
6960:
6950:
6904:
6857:
6849:
6822:Nature
6798:
6770:
6736:
6705:
6642:
6591:
6566:
6558:
6550:
6503:
6458:
6358:
6330:Nature
6309:
6206:
6179:
6154:
6091:
6066:
6011:
5953:
5856:optics
5689:, and
5484:where
3980:Case 2
3674:Case 1
2945:) and
2937:where
2717:where
2073:where
1408:N-type
1392:Diodes
1270:atomic
1048:tunnel
853:Zeeman
848:Wigner
798:Planck
768:Landau
753:Jordan
404:Matrix
334:Popper
8574:Local
8516:Pauli
8496:Dirac
8156:S2CID
8122:arXiv
7917:1 May
7911:(PDF)
7872:(PDF)
7795:S2CID
7761:arXiv
7544:arXiv
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