Quantum tunnelling

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 does not 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.[1]

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.[2] It has applications in the tunnel diode,[3] 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.[4]

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.[5][6]

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.

History

Quantum tunneling was developed from the study of radioactivity,[4] which was discovered in 1896 by Henri Becquerel.[7] Radioactivity was examined further by Marie Curie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903.[7] 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.[4]

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.[8]

Quantum tunneling was first noticed in 1927 by Friedrich Hund while he was calculating the ground state of the double-well potential[7] Leonid Mandelstam and Mikhail Leontovich discovered it independently in the same year. They were analyzing the implications of the then new Schrödinger wave equation.[9]

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[10]) and independently by Ronald Gurney and Edward Condon.[11][12][13][14] 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.[4] 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.[4] In 2016, the quantum tunneling of water was discovered.[15]
Introduction to the concept
File:Quantum tunnel effect and its application to the scanning tunneling microscope.ogvPlay media
Animation showing the tunnel effect and its application to an STM

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 more energetic than they otherwise would have been.[16]

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.[7] 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 tunneling problem
A simulation of a wave packet incident on a potential barrier. In relative units, the barrier energy is 20, greater than the mean wave packet energy of 14. A portion of the wave packet passes through the barrier.

The wave function of a particle summarizes everything that can be known about a physical system.[17] 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.
An electron wavepacket directed at a potential barrier. Note the dim spot on the right that represents tunneling electrons.

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.
Dynamical tunneling
Quantum tunneling oscillations of probability in an integrable double well of potential, seen in phase space.

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[18][19].
Tunneling in phase space

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.[20]
Chaos-assisted tunneling
Chaos-assisted tunneling oscillations between two regular tori embedded in a chaotic sea, seen in phase space

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.[21] and is characterized by sharp resonances of the tunneling rate when varying any system parameter.
Resonance-assisted tunneling

When ℏ {\displaystyle \hbar } \hbar 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 [22].
Related phenomena

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,[23] 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.
Applications

Tunneling is the cause of some important macroscopic physical phenomena.
Electronics

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.[24] tunneling is a fundamental technique used to program the floating gates of flash memory.
Cold emission
Main article: Semiconductor devices

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.[25] These materials are important for flash memory, vacuum tubes, as well as some electron microscopes.
Tunnel junction
Main article: Tunnel junction

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.[26] 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,[25] as well as the multijunction solar cell.
Quantum-dot cellular automata

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.[27]
Tunnel diode
Main article: Tunnel diode
A working mechanism of a resonant tunneling diode device, based on the phenomenon of quantum tunneling through the potential barriers.

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.[28]

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.[28]

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.[29]
Tunnel field-effect transistors
Main article: Tunnel field-effect transistor

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.[30]
Nuclear fusion
Main article: Nuclear fusion

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–a precondition for the evolution of life in insolation habitable zones.[31]
Radioactive decay
Main article: Radioactive decay

Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the 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.[31]
Astrochemistry in interstellar clouds

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.[31]
Quantum biology

Quantum tunneling is among the central non-trivial quantum effects in quantum biology. Here it is important both as electron tunneling and proton tunneling[32] . 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.[31]

Spontaneous mutation occurs when normal DNA replication takes place after a particularly significant proton has tunnelled.[33] 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.[34] 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.[35]
Quantum conductivity

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.[25] 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.[25]
Scanning tunneling microscope
Main article: Scanning tunnelling microscope

The scanning tunneling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, may allow imaging of individual atoms on the surface of a material.[25] 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.[25] STMs are accurate to 0.001 nm, or about 1% of atomic diameter.[29]
Kinetic isotope effect
Main article: Kinetic isotope effect

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.[36]
Faster than light
See also: Faster-than-light

Some physicists have claimed that it is possible for spin-zero particles to travel faster than the speed of light when tunneling.[4] 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.[37] More recently, experimental tunneling time data of phonons, photons, and electrons was published by Günter Nimtz.[38]

Other physicists, such as Herbert Winful,[39] 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.
Mathematical discussion
Quantum tunneling through a barrier. The energy of the tunnelled particle is the same but the probability amplitude is decreased.
The Schrödinger equation

The time-independent Schrödinger equation for one particle in one dimension can be written as

\( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x) \) or

\( \frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} \left( V(x) - E \right) \Psi(x) \equiv \frac{2m}{\hbar^2} M(x) \Psi(x) , \)

where

\( \hbar \) is the reduced Planck's constant,
m is the particle mass,
x represents distance measured in the direction of motion of the particle,
Ψ is the Schrödinger wave function,
V is the potential energy of the particle (measured relative to any convenient reference level),
E is the energy of the particle that is associated with motion in the x-axis (measured relative to V),
M(x) is a quantity defined by V(x) – E which has no accepted name in physics.

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

\( \frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} M(x) \Psi(x) = -k^2 \Psi(x),\;\;\;\;\;\; \mathrm{where} \;\;\; k^2=- \frac{2m}{\hbar^2} M. \)

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

\( \frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} M(x) \Psi(x) = {\kappa}^2 \Psi(x), \;\;\;\;\;\; \mathrm{where} \;\;\; {\kappa}^2= \frac{2m}{\hbar^2} M. \)

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 WKB approximation
Main article: WKB approximation

The wave function is expressed as the exponential of a function:

\( {\displaystyle \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right).} \Phi''(x) + \Phi'(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right). \)

\( \Phi'(x) is then separated into real and imaginary parts:

\( \Phi'(x) = A(x) + i B(x) \), where A(x) and B(x) are real-valued functions.

Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:

\( A'(x) + A(x)^2 - B(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right) \).

Quantum tunneling in the phase space formulation of quantum mechanics. Wigner function for tunneling through the potential barrier \( U(x)=8e^{-0.25 x^2} \) in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian \( H(x,p) = p^2 / 2 + U(x) . \)

To solve this equation using the semiclassical approximation, each function must be expanded as a power series in ℏ {\displaystyle \hbar } \hbar . From the equations, the power series must start with at least an order of ℏ − 1 {\displaystyle \hbar ^{-1}} \hbar^{-1} 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

\( A(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k A_k(x) \)

and

\( B(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k B_k(x), \)

with the following constraints on the lowest order terms,

\( A_0(x)^2 - B_0(x)^2 = 2m \left( V(x) - E \right) \)

and

\( A_0(x) B_0(x) = 0. \)

At this point two extreme cases can be considered.

Case 1 If the amplitude varies slowly as compared to the phase \( A_0(x) = 0 \) and

\( B_0(x) = \pm \sqrt{ 2m \left( E - V(x) \right) } \)

which corresponds to classical motion. Resolving the next order of expansion yields

\( \Psi(x) \approx C \frac{ e^{i \int dx \sqrt{\frac{2m}{\hbar^2} \left( E - V(x) \right)} + \theta} }{\sqrt[4]{\frac{2m}{\hbar^2} \left( E - V(x) \right)}} \)

Case 2

If the phase varies slowly as compared to the amplitude, \( B_0(x) = 0 \) and

\( A_0(x) = \pm \sqrt{ 2m \left( V(x) - E \right) }

which corresponds to tunneling. Resolving the next order of the expansion yields

\( \Psi(x) \approx \frac{ C_{+} e^{+\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} + C_{-} e^{-\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}}{\sqrt[4]{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} \)

In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points \( E = V(x) \). 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, \( x_{1} \) is chosen and \( \frac{2m}{\hbar^2}\left(V(x)-E\right) \) is expanded in a power series about \( x_{1} \):

\( \frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1) + v_2 (x - x_1)^2 + \cdots \)

Keeping only the first order term ensures linearity:

\( \frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1). \)

Using this approximation, the equation near \( x_{1} \) becomes a differential equation:

\( \frac{d^2}{dx^2} \Psi(x) = v_1 (x - x_1) \Psi(x). \)

This can be solved using Airy functions as solutions.

\( \Psi(x) = C_A Ai\left( \sqrt[3]{v_1} (x - x_1) \right) + C_B Bi\left( \sqrt[3]{v_1} (x - x_1) \right) \)

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 \( C,\theta \) and \( C_{+},C_{-} \) are

\( C_{+} = \frac{1}{2} C \cos{\left(\theta - \frac{\pi}{4}\right)} \)

and

Quantum tunneling through a barrier. At the origin (x=0), there is a very high, but narrow potential barrier. A significant tunneling effect can be seen.
\( C_{-} = - C \sin{\left(\theta - \frac{\pi}{4}\right)} \)

With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particle tunneling through a single potential barrier is

\( T(E)=e^{{-2\int _{{x_{1}}}^{{x_{2}}}{\mathrm {d}}x{\sqrt {{\frac {2m}{\hbar ^{2}}}\left[V(x)-E\right]}}}}, \)

where \( x_1,x_2 \) are the two classical turning points for the potential barrier.

For a rectangular barrier, this expression simplifies to:

\( T(E)=e^{{-2{\sqrt {{\frac {2m}{\hbar ^{2}}}(V_{0}-E)}}(x_{2}-x_{1})}}={\tilde V}_{0}^{{-(x_{2}-x_{1})}}. \)