A simple model of a radioactive nuclear decay assumes that \(\alpha\) -particles are trapped inside a well of nuclear potential that walls are the barriers of a finite width \(2.0 \mathrm{fm}\) and height \(30.0 \mathrm{MeV}\). Find the tunneling probability across the potential barrier of the wall for \(\alpha\) -particles having kinetic energy (a) \(29.0 \mathrm{MeV}\) and \((\mathrm{b}) 20.0 \mathrm{MeV} .\) The mass of the \(\alpha\) -particle is \(m=6.64 \times 10^{-27} \mathrm{kg}\)

Quantum mechanics, a fundamental theory in physics, describes the physical properties of nature at the scale of atoms and subatomic particles. It introduces concepts that may seem at odds with classical physics, such as the superposition of states and the behavior of particles as both waves and particles.

One of the astonishing phenomena explained by quantum mechanics is 'quantum tunneling'. It's a quantum phenomenon where particles move through a barrier that they classically shouldn't be able to pass. This is due to the wave-like properties of particles in quantum mechanics, as explained by the wave-particle duality. According to classical physics, if a particle doesn't possess enough energy to overcome a barrier, it should not be able to pass through it. However, quantum mechanics tells us that there is a probability, albeit sometimes exceedingly small, that the particle will appear on the other side of the barrier.

In a simplified model of radioactive nuclear decay, like the one in the exercise, an alpha particle trapped inside a nuclear well can somehow 'tunnel' through the potential barrier, and escape, leading to alpha decay. This tunneling probability is calculated using the wave numbers inside and outside the barrier, and the potential barrier height and width. The lower the particle's kinetic energy relative to the barrier height, the less likely it is to tunnel through. The wave number, which describes the 'waviness' of a particle, is crucial in determining how readily a particle can exhibit such tunneling behavior (e.g., the difference between the particle's energy and the barrier's potential).

Alpha decay is one of the types of radioactive decay in which an unstable nucleus emits an alpha particle—consisting of two protons and two neutrons, identical to a helium-4 nucleus—to form a new nucleus. This decay is typical in heavy elements where the strong nuclear force struggles to hold all the protons and neutrons together, leading to the ejection of an alpha particle to achieve a more stable state.

During this process, the well-known concept of quantum tunneling comes into play. Despite lacking the requisite energy to overcome the nuclear potential barrier, the alpha particle can still 'leak out' due to the probability prescribed by its wavefunction. The potential well and barrier in the exercise represent a simplistic view of the forces at work within the nucleus that retain or release an alpha particle. In quantum mechanics, the likelihood of alpha decay is quantified by a tunneling probability, which can be determined mathematically as was displayed in the step by step solution in the exercise. The kinetic energy of the alpha particle, the potential barrier's height and width all factor into this probability, which diminishes exponentially with increased barrier width and height.

Nuclear physics is the branch of physics that studies the constituents and interactions of atomic nuclei. The primary forces within an atomic nucleus include the strong nuclear force, which holds the nucleus together, and the electrostatic force, which causes protons to repel each other. Nuclear physics aims to understand the behavior of these forces and how they contribute to nuclear stability and phenomena such as radioactivity.

In the context of the textbook exercise, the nuclear potential well represents the strong nuclear force's grip on the alpha particle. When an alpha particle has enough energy to overcome the potential barrier hypothetically, it can be emitted from the nucleus naturally—this is called alpha decay. However, as shown in the calculation of tunneling probability, even when the alpha particle's energy isn't sufficient, there is still a non-zero chance that the particle can 'escape' the potential well through quantum tunneling, which is a purely quantum mechanical effect. This explains how, in real-world observations, we can detect alpha particles emitted from radioactive substances even though the alpha particles do not have the classically required energy to climb over the potential barrier preventing their escape.