The linear potential well is supposed to demonstrate the quantization of energies. The photoelectric effect showed that light behaves like a particle. The wave-particle-duality states that quantum particles, such as electrons, exhibit both wavelike and particle-like properties. 

The potential well helps to visualize how particles, like electrons, can only absorb a certain amount of energy, allowing them to jump between discrete energy levels.

The linear potential well

An electron is confined in a pot of width L with infinitely high walls, meaning it would require infinite energy to escape. The electron can only move in x-direction (1D) and possesses kinetic energy along this axis. Its motion is described by a wave function which provides the probability of the particle's location. 

The electron forms a stationary wave, created when the wave reflects off the fixed walls of the well. A stationary wave is characterized by the superposition of two progressive waves of same frequency and amplitude traveling in opposite directions. At the walls of the potential well, the superposition results in a zero-probability location for the particle. 

The following illustrates a wave function inside the potential well for different energy levels n, as well as the corresponding probability distribution:

The wave function Y describes the probability amplitude, and the square of its magnitude gives the probability of finding the electron at a certain position.

Waves inside the potential well

Inside the well, the wavelength λ must satisfy the condition that exactly half-wavelengths, or multiples of half-wavelengths, fit within the well. In general, L=n×(λ/2), where n is a positive integer. By rearranging this relationship and substituting it into the total energy equation (total energy= kinetic energy + potential energy), we see that the total energy of the electron depends on n, meaning it can only have discrete values. Think of it like steps on a staircase- electros can only stand on specific steps, not in between. Notably, the lowest energy state (the ground state) can never be zero due to Heisenbergs uncertainty principle, as the momentum cannot be zero.

Inside the sun

This quantization may seem unrelated to daily life, but it is crucial for understanding why the sun provides us with energy. The sun fuses hydrogen nuclei to form helium, releasing immense amounts of energy in the process (nuclear fusion). 

However, for the hydrogen nuclei (protons) to fuse, they need to overcome the repulsive force caused by their positive charges - the Coulomb force. At a distance of approximately 10 femtometer (1÷10¹⁵m) from the proton, the strong force becomes stronger than the Coulomb force, enabling the protons to attract and fuse. 

The Coulomb barrier represents the minimal distance two protons must overcome to attract each other due to the strong force and for fusion to occur.

Overcoming the Coulomb barrier

One might assume that the extreme pressure in the sun forces protons close enough to overcome the Coulomb barrier. However, the 15million Kelvin inside the sun cause too little pressure for the Coulomb barriers to be overcome. 

Quantum tunneling provides the solution: imagine a proton trapped in a potential well where its position is described by a probability distribution rather than a precise point. The probability looks like this in quantum physics:

In quantum mechanics, there is a small but finite probability of the proton existing in a region it would not be allowed to occupy according to classical mechanics-outside the potential well. 

This is like a ball rolling up a hill. Classically, if the ball does not have enough energy, it would stop. In quantum mechanics, however, there is a small chance the ball tunnels through the hill to the other side.

In the sun, protons are not close enough together for their Coulomb barrier to overlap, but quantum tunneling allows them to leak through their potential well. The wave function of each proton spreads slightly beyond the classical barrier, giving a small probability for tunneling. 

This makes it possible for the strong nuclear force to act and for fusion to occur.