![]() ![]() Beyond that, random matrix theory is an important tool in computer science and applied mathematics. It has also had unexpected applications to quantum gravity. In physics, it is now understood as one of the basic tools in understanding quantum chaos. Random matrix theory has developed into a major topic with far-flung applications in physics and mathematics. The original goal was to give a statistical description of the energy levels of atomic nuclei. Starting in 1962, Dyson, along with Eugene Wigner and others, was largely responsible for developing what is now known as random matrix theory. (This problem was independently analyzed by Elliott Lieb and Walter Thirring.) Dyson made many subtle contributions regarding the “phases” of quantum and sometimes classical matter, generalizing the fact that water has solid, liquid, and gas phases. In 1966, with Andrew Lenard, Dyson gave the first rigorous proof that the Pauli exclusion principle between electrons is enough to ensure that matter is stable and does not undergo spontaneous collapse. The general goal of this subject is to understand the quantum-mechanical behavior of an assembly of many particles-for example, the electrons and atomic nuclei in a piece of metal. In the 1960s and afterwards, Dyson contributed extensively to quantum statistical mechanics. He retained a passion for number theory throughout his career and made multiple contributions, of which one will be mentioned shortly. Dyson made a fundamental contribution in a paper published in 1947, when he was 24 years old. In a more sophisticated modern version, one considers approximations by more general algebraic numbers. The simplest version of Diophantine approximation-which goes back to the ancient Greeks, as the name suggests-is to approximate a real number such as π by rational numbers. Remarkably, by this time he had already established a reputation in a completely different area: the field of mathematics known as Diophantine approximation. The initial breakthroughs were made in 19 Dyson turned 25 in 1948. This work was done at an astonishingly young age. ![]() As the position of the particle becomes more precise when the slit is narrowed, the direction, or therefore the momentum, of the particle becomes less known as seen by a wider horizontal distribution of the light.ġ.) The Heisenberg Uncertainty Principle discredits the aspect of the Bohr atom model that an electron is constrained to a one-dimensional orbit of a fixed radius around the nucleus.Ģ.In the 1950s, Dyson made multiple important contributions to continue developing the framework of quantum electrodynamics. Here is a video that demonstrates particles of light passing through a slit and as the slit becomes smaller, the final possible array of directions of the particles becomes wider. Therefore, the momentum is unknown, but the initial position of the particle is known. He had light passing through a slit, which causes an uncertainty of momentum because the light behaves like a particle and a wave as it passes through the slit. Einstein created a slit experiment to try and disprove the Uncertainty Principle. ![]() Several scientists have debated the Uncertainty Principle, including Einstein. Therefore, there is no way to find both the position and momentum of a particle simultaneously. Conversely, if we want a more precise momentum, we would add less wavelengths to the "wave packet" and then the position would become more uncertain. The more waves that are combined in the "wave packet", the more precise the position of the particle becomes and the more uncertain the momentum becomes because more wavelengths of varying momenta are added. An accumulation of waves of varying wavelengths can be combined to create an average wavelength through an interference pattern: this average wavelength is called the "wave packet". A "wave packet" can be used to demonstrate how either the momentum or position of a particle can be precisely calculated, but not both of them simultaneously. However, in quantum mechanics, the wave-particle duality of electrons does not allow us to accurately calculate both the momentum and position because the wave is not in one exact location but is spread out over space. ![]() It is hard for most people to accept the uncertainty principle, because in classical physics the velocity and position of an object can be calculated with certainty and accuracy. Understanding the Uncertainty Principle through Wave Packets and the Slit Experiment However, the more accurately momentum is known the less accurately position is known. Mathematically, this occurs because the smaller Δx becomes, the larger Δp must become in order to satisfy the inequality. What this equation reveals is that the more accurately a particle’s position is known, or the smaller Δx is, the less accurately the momentum of the particle Δp is known. ![]()
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