A 5.0-eV electron impacts on a barrier of with 0.60 nm. Find the probability of the electron to tunnel through the barrier if the barrier height is (a) \(7.0 \mathrm{eV} ;\) (b) \(9.0 \mathrm{eV} ;\) and (c) \(13.0 \mathrm{eV}\)

The transmission coefficient, denoted as T, is a critical figure in the realm of quantum mechanics, particularly when discussing phenomena such as quantum tunneling. It quantifies the likelihood that a particle will tunnel through and overcome a potential energy barrier, rather than reflecting off it. Imagine rolling a ball up a hill; if the ball doesn't have enough energy to get over the hill, it rolls back. However, quantum particles can sometimes 'cheat' and tunnel through the barrier, even if they apparently lack the requisite energy.

In mathematical terms, the transmission coefficient can be represented by the equation \[T = \exp(-2\alpha L)\], where \(\alpha\) is the inverse of the decay length of the particle’s wave function inside the barrier, and L is the barrier's width. The decay length is a measure of how quickly the wave function of a particle diminishes inside the barrier. A higher T signifies a higher tunneling probability. Understanding this concept is essential for interpreting tunneling behavior in various physical systems, from electron flow in semiconductors to nuclear fusion in stars.

When studying quantum mechanics and particle physics, you'll often encounter the term electron volts (eV), a unit of energy. It specifically refers to the amount of kinetic energy gained by a single electron when it’s accelerated through an electric potential difference of one volt. The conversion between electron volts and the more familiar joules (J) is vital to bridge the gap between the small scale of particle physics and the larger, everyday measurements.

The conversion factor is \[1 \text{eV} = 1.602 \times 10^{-19} \text{J}\]. To convert electron volts to joules, you multiply the energy in eV by this constant. For instance, an energy level of 5.0 eV corresponds to \(5.0 \times 1.602 \times 10^{-19}\) joules. This conversion is necessary for calculations like the transmission coefficient where energies must be in joules to use standard physics formulas that are based on SI units.

Quantum tunneling is an extraordinary concept where particles penetrate through a potential energy barrier that they seemingly do not have enough energy to surpass classically. The barrier penetration probability is the chance of this occurring and is determined by the transmission coefficient discussed earlier.

As quantum particles, like electrons, approach a barrier, they do not simply stop or bounce back—they have a nonzero probability of 'appearing' on the other side. Remarkably, this is due precisely to their wave-like nature and is described by the quantum mechanical wave function. The transmission coefficient T reflects this property and, with it, you can calculate the barrier penetration probability. For instance, when an electron with a particular energy faces a barrier of a certain height and width, you can apply the formulas provided to determine the values of T\(_1\), T\(_2\), and T\(_3\), which represent the respective probabilities of tunneling through barriers of different energies.