In order for a thermonuclear fusion reaction of two deuterons \(\left(_{1}^{2} \mathrm{H}^{+}\right)\) to take place, the deuterons must collide with each deuteron traveling at \(1 \times 10^{6} \mathrm{m} / \mathrm{s}\) . Find the wavelength of such a deuteron.

Thermonuclear fusion is a process where atomic nuclei merge to form a heavier nucleus, releasing massive amounts of energy. This reaction is fundamental to the operation of stars, including our Sun. For fusion to occur, conditions of extremely high temperature and pressure are needed to overcome the electrostatic repulsion between nuclei. In stars, this is achieved by their immense gravitational forces. On Earth, achieving the necessary conditions for fusion is challenging, and is usually attempted with either inertial confinement (using lasers) or magnetic confinement (using tokamaks or similar devices).

The energy released in a fusion reaction comes from the small loss of mass that occurs when nuclei combine, as predicted by Einstein's mass-energy equivalence principle, E=mc^2. This energy can be harnessed for power generation, which is the goal of thermonuclear reactors. However, to date, controlled thermonuclear fusion as a practical power source remains experimental and not yet commercially viable.

A deuteron is the nucleus of an isotope of hydrogen known as deuterium, symbolized as \( ^{2}H \). It consists of one proton and one neutron. Deuteron collision is a key step in the fusion process, as deuterons must come into very close contact for the strong nuclear force to overcome their electrostatic repulsion and bind them together. In the exercise, a scenario was described where two deuterons must be moving at speeds of \(1 \times 10^{6} \mathrm{m} / \mathrm{s}\) for fusion to occur.

This high velocity imparts the necessary kinetic energy for the deuterons to surmount the Coulomb barrier, which is the natural repulsion between the positively charged nuclei. The likelihood of collision and subsequent fusion increases with the temperature and pressure—akin to providing the deuterons with sufficient speed—and is also influenced by quantum mechanical phenomena, such as tunneling, which allows particles to pass through energy barriers under certain conditions.

In physics, momentum is a measure of the motion of an object and is directly proportional to both its mass and velocity. It is a vector quantity, which means that it has both magnitude and direction. For a particle moving at a constant speed, the momentum \(p\) can be calculated simply by multiplying the mass \(m\) of the particle by its velocity \(v\), given by the equation \(p = m \times v\).

In the context of the exercise, the momentum of a deuteron is crucial for determining its de Broglie wavelength. According to de Broglie's hypothesis, every moving particle exhibits wave-like properties, and its wavelength \(\lambda\) can be found by the equation \(\lambda = \frac{h}{p}\), where \(h\) is the Planck constant. The calculation of a particle's momentum is therefore the first step in finding the de Broglie wavelength, which helps us to understand the quantum mechanical behaviour of particles, especially at the subatomic level where both their particle-like and wave-like properties become significant.