The theory of quantum tunneling was first proposed by George Gamow in the late 1920s. Gamow was a physicist who worked on developing the theory of alpha decay in atomic nuclei, which is the process by which a nucleus emits an alpha particle (two protons and two neutrons bound together) and transforms into a new nucleus with a lower atomic number.
Gamow realized that the energy required for an alpha particle to escape the strong nuclear force holding it in the nucleus was much greater than it's kinetic energy. According to classical physics, the alpha particle would not have enough energy to escape, and therefore, alpha decay should not occur. However, Gamow proposed that there was a finite probability that the alpha particle could tunnel through the potential energy barrier and escape the nucleus, even though it did not have enough energy to do so classically.
The concept of quantum tunneling was further developed and verified by other physicists, including Richard Feynman and Freeman Dyson. Feynman used the theory of quantum electrodynamics (QED) to explain the quantum tunneling of electrons in the early 1950s, which led to a deeper understanding of the phenomenon.
Experimental verification of quantum tunneling has been carried out in various systems, including alpha decay, nuclear fusion, and tunneling in semiconductor devices. One of the most famous experiments demonstrating quantum tunneling is the scanning tunneling microscope (STM), which was invented by Gerd Binnig and Heinrich Rohrer in the 1980s. The STM uses quantum tunneling of electrons to image the surface of materials at the atomic scale, providing insights into the structure and properties of matter.
The mathematical theory of quantum tunneling is based on the Schrödinger equation, which describes the wave function of a particle in a potential energy well. The wave function gives the probability of finding the particle at a particular position and time, and it also predicts the probability of the particle tunneling through the potential barrier. The mathematical theory of quantum mechanics has been extensively tested and verified by numerous experiments, and it has become a cornerstone of modern physics.
Quantum tunneling occurs when a particle passes through a potential energy barrier that it does not have enough classical energy to surmount. The barrier can be of different types, depending on the system and the context.
In semiconductor devices, quantum tunneling occurs across potential energy barriers that arise due to the presence of thin insulating layers between different regions of the device. For example, in a tunnel diode, a thin insulating layer separates two regions of a semiconductor material with different doping levels. The potential energy barrier created by the insulating layer prevents the electrons from flowing freely between the two regions.
However, due to the wave-like nature of electrons in quantum mechanics, there is a finite probability that some electrons can tunnel through the barrier and flow from one region to another. This phenomenon is exploited in tunnel diodes to create high-speed switching circuits and other applications.
In nuclear physics, quantum tunneling occurs when alpha particles tunnel through the potential energy barrier created by the strong nuclear force that holds the nucleus together. This phenomenon explains the occurrence of alpha decay in which an alpha particle is emitted from the nucleus of an atom.
In general, the barrier can be of any potential energy, including an electric potential energy barrier, a gravitational potential energy barrier, or a potential energy barrier arising from the geometry or structure of the system. The size and mass of the particle do not matter in principle, as long as the potential energy barrier is of the right magnitude to prevent classical energy from being sufficient for surmounting it.
However, the probability of tunneling decreases exponentially with the thickness and height of the potential energy barrier, and with the mass of the particle. Hence, the probability of quantum tunneling for a macroscopic object is negligible, while for microscopic particles such as electrons, tunneling can be a significant effect.
The probability of quantum tunneling for a given particle depends on various factors, including the height and width of the potential energy barrier, the energy of the particle, and its mass. The probability of tunneling generally decreases with increasing mass and increasing height and width of the potential energy barrier.
The probability of quantum tunneling is given by the transmission coefficient, which is the ratio of the transmitted wave function to the incident wave function. The transmission coefficient can be calculated using quantum mechanical methods such as the WKB approximation or the scattering matrix approach.
It is difficult to give a general formula for the transmission coefficient as it depends on the specific system and the properties of the barrier. However, in general, the transmission coefficient decreases with increasing mass, as the wave nature of the particle becomes less significant and the classical behavior becomes more dominant. For example, the transmission coefficient for alpha particles is typically higher than that for protons or neutrons, which have higher masses.
Therefore a graph of the percentage of particles that tunnel versus their mass, would depend on the specific system and the properties of the potential energy barrier. In general, for a given barrier, the percentage of particles that tunnel would be higher for particles with lower masses and lower energies. However, the exact relationship would be complex and would depend on the details of the system.
A)That being said we do know that particles with mass are also related to energy.
Why do we assume that due to the effect of quantum tunneling , a proton actually tunnels through a barrier and the total energy is always equal to that of an original proton and not another particle with less energy like a pion or even an electron? This possibility should be examined in more detail.
The relationship between mass and energy is given by the famous equation E=mc^2, where E is the energy, m is the mass, and c is the speed of light. This equation shows that mass and energy are equivalent and interchangeable, and that a particle with mass has an associated energy even when it is at rest.
In quantum mechanics, the energy of a particle is quantized and can only take certain discrete values, which depend on the properties of the system. The wave function of a particle describes it's probability distribution over these energy states, and the probability of tunneling through a potential energy barrier depends on the energy of the particle relative to the height and width of the barrier.
When we describe the mass or energy of a proton, we are referring to it's rest mass and its associated energy when it is at rest. However, in quantum mechanics, a particle is not necessarily at rest and can have a range of energies and momenta depending on the system and the context. Therefore, the energy that tunnels through a barrier is not always equal to the rest energy of a proton, but depends on the energy of the particle in the specific quantum state that is involved in the tunneling process.
In general, the probability of tunneling is higher for particles with lower masses and lower energies, as these particles have a more wave-like behavior and are more likely to penetrate through the potential energy barrier. However, the specific energy and mass of the particle involved in tunneling depend on the details of the system and the quantum state of the particle.
In some cases, a particle may interact with the barrier in such a way that it loses energy or momentum, and this can result in a change in the particle's mass or energy. However, this is not a general feature of quantum tunneling and would depend on the specific system and the properties of the barrier.
In general, the probability of tunneling depends on a variety of factors, including the properties of the barrier, the energy and momentum of the particle, and the quantum state of the particle. The relationship between mass and energy is an important consideration in understanding the behavior of particles in quantum mechanics, but it is not the only factor that determines the probability of tunneling.
B) Quantum tunneling through barriers of energy that gradually increase in size or increased resistance to tunneling, decreasing the chances of quantum tunneling.
The phenomenon of gradually increasing energy barriers is known as a potential energy ramp or a sloping barrier. This is a type of potential energy barrier where the energy of the barrier increases gradually over a certain distance.
In the context of quantum tunneling, a sloping barrier can have interesting effects on the transmission probability of particles. At low energies, particles with a low mass are more likely to tunnel through a sloping barrier compared to particles with higher mass. However, as the energy of the particles increases, the transmission probability decreases for all particles, regardless of their mass.
One interesting feature of sloping barriers is that they can lead to resonant tunneling. This occurs when the sloping barrier is accompanied by a narrow potential energy well, and the energy of the particle is such that it matches the energy of a bound state within the well. In this case, the transmission probability of the particle can be greatly enhanced due to resonant tunneling.
In general, the behavior of particles tunneling through sloping barriers is complex and depends on the specific properties of the barrier and the quantum state of the particle. However, sloping barriers are an interesting example of the diverse range of potential energy barriers that can be encountered in quantum mechanics, and they can have important implications for the behavior of particles in various physical systems.
The behavior of particles tunneling through sloping barriers can be described mathematically using quantum mechanical models. One way to approach this problem is to use the WKB (Wentzel-Kramers-Brillouin) approximation, which is a semiclassical method for calculating the transmission probability of particles through potential energy barriers.
The WKB approximation involves solving the Schrödinger equation for the particle in the presence of the sloping barrier, and then using this solution to calculate the transmission probability of the particle. The method takes into account the gradual increase in the energy barrier over a certain distance, as well as the wave-like behavior of the particle.
The behavior of a particle such as a proton, tunneling through a sloping barrier can be affected by many factors, including the temperature of the system. At higher temperatures, the particle may have a higher kinetic energy, which can increase the likelihood of tunneling through the barrier. However, the behavior of the particle will also depend on the specific properties of the barrier, such as it's height, width, and slope.
The behavior of particles tunneling through sloping barriers can be quite complex, and may require numerical or computational methods to accurately describe. However, the WKB approximation provides a useful starting point for understanding the transmission probability of particles through sloping barriers, and can be used to explore the behavior of particles in a range of physical systems.
The increase in temperature can certainly affect the shape and properties of a potential energy barrier, including a sloping barrier. However, the effect of temperature on the barrier will depend on the specific system and the nature of the barrier.
An increase in temperature can lead to thermal expansion of the system, which can cause the barrier to increase in size or change shape. This effect may be more pronounced for certain types of barriers, such as those composed of materials with temperature-dependent properties.
In addition to temperature, an increase in pressure can also affect the shape and properties of a potential energy barrier. An increase in pressure can lead to changes in the interatomic distances and bonding between atoms, which can alter the shape of the barrier.
However, the effect of pressure on the barrier will also depend on the specific system and the nature of the barrier. For example, a sloping barrier may be less sensitive to changes in pressure than a sharp barrier due to it's gradual increase in energy over a certain distance.
Both temperature and pressure can affect the shape and properties of a potential energy barrier, including a sloping barrier. However, the effect of these factors will depend on the specific system and the nature of the barrier, and may require further analysis to fully understand.
The effect of entropy on a sloping barrier can be complex and depends on the specific system and the nature of the barrier. However, in general, an increase in entropy can lead to an increase in the thermal fluctuations and disorder of the system, which can affect the shape and properties of the potential energy barrier.
In some cases, an increase in entropy can lead to a decrease in the height of the sloping barrier, making it easier for particles to tunnel through. This effect may be more pronounced for barriers composed of materials with temperature-dependent properties, as thermal fluctuations can cause changes in the interatomic distances and bonding between atoms that make up the barrier.
On the other hand, an increase in entropy can also lead to an increase in the width of the barrier or the number of energy levels available to the particle. This effect can make it more difficult for particles to tunnel through, as there are more energy states available for the particle to occupy before it can tunnel through the barrier.
The effect of entropy on the chances of tunneling through a sloping barrier will depend on the specific system and the nature of the barrier. However, in general, an increase in entropy can make it more difficult for particles to tunnel through a barrier, particularly if the barrier becomes wider or if there are more energy levels available to the particle.
C)How does resonance of particles affect the sloping barrier and chances or probability of tunneling through the barrier under static pressure .
Resonance can play an important role in the dynamics of tunneling particles and can affect the shape and properties of a sloping barrier. When a particle has a resonant energy state that is close to the energy of the barrier, it can increase the probability of tunneling through the barrier by effectively "tuning" itself to the energy of the barrier.
In the case of a sloping barrier, resonance can lead to an increase in the transmission probability of the particle through the barrier. This effect is more pronounced for particles with wave-like behavior, such as electrons,which can exhibit wave-like interference patterns when they encounter a sloping barrier.
Under static pressure, resonance can also affect the dynamics of the system and the properties of the barrier. For example, if the pressure causes a change in the interatomic distances and bonding between atoms that make up the barrier, this can alter the resonant energy states of the particle and affect its probability of tunneling through the barrier.
Overall, the effect of resonance on the probability of tunneling through a sloping barrier will depend on the specific system and the nature of the barrier. However, in general, resonance can increase the probability of tunneling through the barrier by effectively "matching" the energy of the particle to the energy of the barrier, leading to an increase in the transmission probability of the particle.
It is possible to study a system and manipulate it's resonant properties, although the exact methods and techniques used will depend on the specific system and the properties of the barrier.
In some cases, it may be possible to adjust the parameters of the system, such as temperature or pressure, to control the resonant properties of the particle and increase the probability of tunneling through the barrier. For example, if the resonant energy states of the particle shift in response to changes in temperature or pressure, it may be possible to tune the system to increase the probability of tunneling through a barrier with a larger width.
Other methods for manipulating resonance may involve the use of external fields or interactions to alter the properties of the barrier and the resonant properties of the particle. For example, the application of an external magnetic field can cause the resonant energy states of an electron to shift, potentially increasing it's probability of tunneling through a barrier.
Lastly the likelihood of quantum tunneling for a particle depends on the specific properties of the system, such as the energy of the particle, the properties of the barrier, and the surrounding medium.
In general, a particle in a medium may have a higher probability of tunneling than a particle in free space, due to the interactions between the particle and the surrounding atoms or molecules. These interactions can lead to the formation of bound states or resonant energy levels that may increase the probability of tunneling through a barrier.
However, the exact probability of tunneling will depend on the specific properties of the system, and it is possible for a particle in free space to have a higher probability of tunneling than a particle in a medium, depending on the specific parameters of the system.
Further more resonant properties of particles with different masses can be different, which can affect the probability of quantum tunneling through a barrier. By controlling the resonant properties of a system, it may be possible to enhance the probability of tunneling for particles with certain masses or properties, while suppressing the tunneling of particles with other masses or properties.
For example, in semiconductor devices, the design of the barrier and the choice of materials can be used to tune the resonant properties of electrons and holes, allowing for selective control over the tunneling of different types of carriers. This can be important for the operation of semiconductor devices such as transistors and diodes.
D) Quantum tunneling in the Sun.
Proton-to-proton tunneling is a key process that powers the Sun, as well as other stars, through the process of nuclear fusion. In the Sun, protons are heated and compressed in the core to the point where they can overcome their electrostatic repulsion and tunnel through the Coulomb barrier to fuse together and form a helium nucleus. This process releases a large amount of energy in the form of gamma rays and other particles, which helps to maintain the Sun's high temperature and pressure.
On Earth, researchers are working to develop practical nuclear fusion reactors that can provide a source of clean, renewable energy. These reactors use a variety of different fusion reactions, but one of the most promising is the fusion of deuterium and tritium (D-T) nuclei. In these reactors, the fusion fuel is heated and compressed to high temperatures and pressures using powerful lasers or magnetic fields, which can help to overcome the Coulomb barrier and initiate fusion.
The proton-to-proton tunneling process in the Sun and in fusion reactors on Earth is similar in many respects, as both involve the tunneling of charged particles through the Coulomb barrier to initiate nuclear fusion. However, there are also important differences in the specific properties of the systems, such as the temperature, pressure, and density of the fusion fuel, which can affect the efficiency and stability of the fusion reaction.
In summary even though the coulomb barrier is over come by composite particles/baryons called protons in the sun ,it's still unclear how composite particles can do it just as effectively as electrons or gamma rays ,it's also not known if the high temperature and entropy, favor quantum tunneling or it's the resonance of the composite particles that actually enables sufficient quantum tunneling in the core of the sun .
There other factors that we haven't examined in relation to the sun such as the presence of other mediums, such as atoms, neutrons, magnetic fields and their geometric shapes. The possiblity that the coulomb barrier actually decreases due to certain factors or actually increases but some unknown factors still enable quantum tunneling.
However the above discussion could perhaps in some ways lead to more efficient and intelligent semiconductor designs.
1/5/2023.
Image of fusion reactor.
