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Quantum tunnelling or tunneling (US) is the quantum mechanical phenomenon where a wavefunction can propagate through a potential barrier.
The transmission through the barrier can be finite and depends exponentially on the barrier height and barrier width. The wavefunction may disappear on one side and reappear on the other side. The wavefunction and its first derivative are continuous. In steady-state, the probability flux in the forward direction is spatially uniform. No particle or wave is lost. Tunneling occurs with barriers of thickness around 1–3 nm and smaller.
Some authors also identify the mere penetration of the wavefunction into the barrier, without transmission on the other side as a tunneling effect. Quantum tunneling is not predicted by the laws of classical mechanics where surmounting a potential barrier requires potential energy.
Quantum tunneling plays an essential role in physical phenomena, such as nuclear fusion.It has applications in the tunnel diode, quantum computing, and in the scanning tunneling microscope.
The effect was predicted in the early 20th century. Its acceptance as a general physical phenomenon came mid-century.
Quantum tunneling is projected to create physical limits to the size of the transistors used in microelectronics, due to electrons being able to tunnel past transistors that are too small.
Tunneling may be explained in terms of the Heisenberg uncertainty principle in that a quantum object can be known as a wave or as a particle in general. In other words, the uncertainty in the exact location of light particles allows these particles to break rules of classical mechanics and move in space without passing over the potential energy barrier.
Quantum tunnelling may be one of the mechanisms of proton decay.
Quantum tunneling was developed from the study of radioactivity,which was discovered in 1896 by Henri Becquerel. Radioactivity was examined further by Marie Curie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903. Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically by Friedrich Kohlrausch. The idea of half-life and the possibility of predicting decay was created from their work.
In 1901, Robert Francis Earhart discovered an unexpected conduction regime while investigating the conduction of gases between closely spaced electrodes using the Michelson interferometer. J. J. Thomson commented that the finding warranted further investigation. In 1911 and then 1914, then-graduate student Franz Rother directly measured steady field emission currents. He employed Earhart's method for controlling and measuring the electrode separation, but with a sensitive platform galvanometer. In 1926, Rother measured the field emission currents in a "hard" vacuum between closely spaced electrodes.
Quantum tunneling was first noticed in 1927 by Friedrich Hund while he was calculating the ground state of the double-well potentialLeonid Mandelstam and Mikhail Leontovich discovered it independently in the same year. They were analyzing the implications of the then new Schrödinger wave equation.
Its first application was a mathematical explanation for alpha decay, which was developed in 1928 by George Gamow (who was aware of Mandelstam and Leontovich's findings) and independently by Ronald Gurney and Edward Condon. The latter researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depended directly on the mathematical probability of tunneling.
After attending a Gamow seminar, Max Born recognised the generality of tunneling. He realised that it was not restricted to nuclear physics, but was a general result of quantum mechanics that applied to many different systems.Shortly thereafter, both groups considered the case of particles tunneling into the nucleus. The study of semiconductors and the development of transistors and diodes led to the acceptance of electron tunneling in solids by 1957. Leo Esaki, Ivar Giaever and Brian Josephson predicted the tunneling of superconducting Cooper pairs, for which they received the Nobel Prize in Physics in 1973. In 2016, the quantum tunneling of water was discovered.
Quantum tunneling falls under the domain of quantum mechanics: the study of what happens at the quantum scale. Tunneling cannot be directly perceived. Much of its understanding is shaped by the microscopic world, which classical mechanics cannot explain. 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. 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. A ball that lacks the energy to penetrate a wall bounces back. Alternatively, the ball might become part of the wall (absorption).
In quantum mechanics, these particles can, with a small probability, tunnel to the other side, thus crossing the barrier. The ball, in a sense, borrows energy from its surroundings to cross the wall. It then repays the energy by making the reflected electrons[ clarification needed ] more energetic than they otherwise would have been.
The reason for this difference comes from treating matter as having properties of waves and particles. One interpretation of this duality involves the Heisenberg uncertainty principle, which defines a limit on how precisely the position and the momentum of a particle can be simultaneously known.This implies that no solutions have a probability of exactly zero (or one), though it may approach infinity. If, for example, the calculation for its position was taken as a probability of 1, its speed, would have to be infinity (an impossibility). Hence, the probability of a given particle's existence on the opposite side of an intervening barrier is non-zero, and such particles will appear on the 'other' (a semantically difficult word in this instance) side in proportion to this probability.
The wave function of a particle summarizes everything that can be known about a physical system.Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the wave function can be deduced. The square of the absolute value of this wavefunction is directly related to the probability distribution of the particle's position, which describes the probability that the particle is at any given place. The wider the barrier and the higher the barrier energy, the lower the probability of tunneling.
A simple model of a tunneling barrier, such as the rectangular barrier, can be analysed and solved algebraically. In canonical field theory, the tunneling is described by a wave function which has a non-zero amplitude inside the tunnel; but the current is zero there because the relative phase of the amplitude of the conjugate wave function (the time derivative) is orthogonal to it.
The simulation shows one such system.
The 2nd illustration shows the uncertainty principle at work. A wave impinges on the barrier; the barrier forces it to become taller and narrower. The wave becomes much more de-localized–it is now on both sides of the barrier, it is wider on each side and lower in maximum amplitude but equal in total amplitude. In both illustrations, the localization of the wave in space causes the localization of the action of the barrier in time, thus scattering the energy/momentum of the wave.
Problems in real life often do not have one, so "semiclassical" or "quasiclassical" methods have been developed to offer approximate solutions, such as the WKB approximation. Probabilities may be derived with arbitrary precision, as constrained by computational resources, via Feynman's path integral method. Such precision is seldom required in engineering practice.[ citation needed ]
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 tunneling.
The concept of dynamical tunneling is particularly suited to address the problem of quantum tunneling in high dimensions (d>1). In the case of an integrable system, where bounded classical trajectories are confined onto tori in phase space, tunneling can be understood as the quantum transport between semi-classical states built on two distinct but symmetric tori.
In real life, most system 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 is classically allowed, between the two symmetric tori then assists the quantum tunneling between them. This phenomenon is referred as chaos-assisted tunneling.and is characterized by sharp resonances of the tunneling rate when varying any system parameter.
When is small in front of the size of the regular islands, the fine structure of the classical phase space plays a key role in tunneling. In particular the two symmetric tori are coupled "via a succession of classically forbidden transitions across nonlinear resonances" surrounding the two islands.
Several phenomena have the same behavior as quantum tunneling, and can be accurately described by tunneling. Examples include the tunneling of a classical wave-particle association,evanescent wave coupling (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". Evanescent wave coupling, until recently, was only called "tunneling" in quantum mechanics; now it is used in other contexts.
These effects are modeled 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.
In optics, 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 total energy is greater than its potential energy 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.
Tunneling is the cause of some important macroscopic physical phenomena. Quantum tunnelling has important implications on functioning of nanotechnology.
Tunneling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in the substantial power drain and heating effects that plague such devices. It is considered the lower limit on how microelectronic device elements can be made.Tunneling is a fundamental technique used to program the floating gates of flash memory.
Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission, 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, as well as some electron microscopes.
A simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions, the study of which requires understanding quantum tunneling.Josephson junctions take advantage of quantum tunneling and the superconductivity of some semiconductors to create the Josephson effect. This has applications in precision measurements of voltages and magnetic fields, as well as the multijunction solar cell.
QCA is a molecular binary logic synthesis technology that operates by the inter-island electron tunneling system. This is a very low power and fast device that can operate at a maximum frequency of 15 PHz.
Diodes are electrical semiconductor devices that allow electric current flow in one direction more than the other. The device depends on a depletion layer between N-type and P-type semiconductors to serve its purpose. When these are heavily doped the depletion layer can be thin enough for tunneling. When a small forward bias is applied, the current due to tunneling is significant. This has a maximum at the point where the voltage bias is such that the energy level of the p and n conduction bands are the same. As the voltage bias is increased, the two conduction bands no longer line up and the diode acts typically.
Because the tunneling 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 tunneling probability changes as rapidly as the bias voltage.
The resonant tunneling diode makes use of quantum tunneling in a very different manner to achieve a similar result. This diode has a resonant voltage for which a lot of current favors a particular voltage, achieved by placing two thin layers with a high energy conductance band near each other. This creates a quantum potential well that has a discrete lowest energy level. When this energy level is higher than that of the electrons, no tunneling 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, tunneling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable.
A European research project demonstrated field effect transistors in which the gate (channel) is controlled via quantum tunneling rather than by thermal injection, reducing gate voltage from ≈1 volt to 0.2 volts and reducing power consumption by up to 100×. If these transistors can be scaled up into VLSI chips, they would improve the performance per power of integrated circuits.
Quantum tunneling is an essential phenomenon for nuclear fusion. The temperature in stars' cores is generally insufficient to allow atomic nuclei to overcome the Coulomb barrier and achieve thermonuclear fusion. Quantum tunneling 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.
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 tunneling of a particle out of the nucleus (an electron tunneling into the nucleus is electron capture). This was the first application of quantum tunneling. Radioactive decay is a relevant issue for astrobiology as this consequence of quantum tunneling 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.
By including quantum tunneling, 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.
Quantum tunneling is among the central non-trivial quantum effects in quantum biology. Here it is important both as electron tunneling and proton tunneling.Electron tunneling is a key factor in many biochemical redox reactions (photosynthesis, cellular respiration) as well as enzymatic catalysis. Proton tunneling is a key factor in spontaneous DNA mutation.
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 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 tautomeric transition. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised, causing a mutation. Per-Olov Lowdin was the first to develop this theory of spontaneous mutation within the double helix. Other instances of quantum tunneling-induced mutations in biology are believed to be a cause of ageing and cancer.
While the Drude model of electrical conductivity makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunneling to explain the nature of the electron's collisions.When a free electron wave packet encounters a long array of uniformly spaced barriers, 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 conductance, and that impurities in the metal will disrupt it significantly.
The scanning tunneling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, may allow imaging of individual atoms on the surface of a material. nm, or about 1% of atomic diameter.It operates by taking advantage of the relationship between quantum tunneling 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 tunneling between the needle and the surface reveals the distance between the needle and the surface. By using piezoelectric rods that change in size when voltage is applied, the height of the tip can be adjusted to keep the tunneling 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
In chemical kinetics, the substitution of a light isotope 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 transition state theory. However, in certain cases, large isotope effects are observed that cannot be accounted for by a semi-classical treatment, and quantum tunneling is required. R. P. Bell developed a modified treatment of Arrhenius kinetics that is commonly used to model this phenomenon.
Some physicists have claimed that it is possible for spin-zero particles to travel faster than the speed of light when tunneling.This apparently violates the principle of causality, since a frame of reference then exists in which the particle arrives before it has left. In 1998, Francis E. Low reviewed briefly the phenomenon of zero-time tunneling. More recently, experimental tunneling time data of phonons, photons, and electrons was published by Günter Nimtz.
Other physicists, such as Herbert Winful,disputed these claims. Winful argued that the wavepacket of a tunneling 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 wavepacket does not measure its speed, but is related to the amount of time the wavepacket is stored in the barrier. But the problem remains that the wave function still rises inside the barrier at all points at the same time. In other words, in any region that is inaccessible to measurement, non-local propagation is still mathematically certain.
An experiment done in 2020, overseen by Aephraim Steinberg, showed that particles should be able to tunnel at apparent speeds faster than light.
The time-independent Schrödinger equation for one particle in one dimension can be written as
The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M(x) is positive or negative. When M(x) is constant and negative, then the Schrödinger equation can be written in the form
The solutions of this equation represent travelling waves, with phase-constant +k or -k. Alternatively, if M(x) is constant and positive, then the Schrödinger equation can be written in the form
The solutions of this equation are rising and falling exponentials in the form of evanescent waves. When M(x) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of M(x) determines the nature of the medium, with negative M(x) corresponding to medium A and positive M(x) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M(x) is sandwiched between two regions of negative M(x), hence creating a potential barrier.
The mathematics of dealing with the situation where M(x) varies with x 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.
The wave function is expressed as the exponential of a function:
is then separated into real and imaginary parts:
Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:
To solve this equation using the semiclassical approximation, each function must be expanded as a power series in . From the equations, the power series must start with at least an order of to satisfy the real part of the equation; for a good classical limit starting with the highest power of Planck's constant possible is preferable, which leads to
with the following constraints on the lowest order terms,
At this point two extreme cases can be considered.
Case 1 If the amplitude varies slowly as compared to the phase and
In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points . 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.
To start, a classical turning point, is chosen and is expanded in a power series about :
Keeping only the first order term ensures linearity:
Using this approximation, the equation near becomes a differential equation:
This can be solved using Airy functions as solutions.
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 solution to connect them.
Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between and are
With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particle tunneling through a single potential barrier is
where are the two classical turning points for the potential barrier.
For a rectangular barrier, this expression simplifies to:
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow, quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.
A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986. STM senses the surface by using an extremely sharp conducting tip that can distinguish features smaller than 0.1 nm with a 0.01 nm depth resolution. This means that individual atoms can routinely be imaged and manipulated. Most microscopes are built for use in ultra-high vacuum at temperatures approaching zero kelvin, but variants exist for studies in air, water and other environments, and for temperatures over 1000 °C.
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ.
The Klein–Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.
In theoretical physics, the pilot wave theory, also known as Bohmian mechanics, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets quantum mechanics as a deterministic theory, avoiding troublesome notions such as wave–particle duality, instantaneous wave function collapse, and the paradox of Schrödinger's cat. To solve these problems, the theory is inherently nonlocal.
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space.
The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the mass of that particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons.
In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is approximately −9.284764×10−24 J/T. The electron magnetic moment has been measured to an accuracy of 7.6 parts in 1013.
A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the Hamiltonian. This corresponds to a state with a single definite energy. It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket. It is very similar to the concept of atomic orbital and molecular orbital in chemistry, with some slight differences explained below.
In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves. The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension. Typically, the potential is modelled as a Heaviside step function.
In quantum mechanics, the rectangularpotential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left.
In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.
The quantum potential or quantum potentiality is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952.
The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin (1999) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term fractional Schrödinger equation was coined by Nick Laskin.
In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry.
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
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