Science and math

Hilbert space

Jovan Stosic July 11, 2023

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space.

The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of “dropping the altitude” of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space.

https://en.wikipedia.org/wiki/Hilbert_space

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Jovan Stosic July 1, 2023

Hey guys! For all the self learners out there, there is a really good YouTube channel called eigenchris. He teachers Tensor Analysis (from the beginner level up to complex level) and is starting a series on special and general relativity. He’s doing these videos for free, so please support him!
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Electron configuration

Jovan Stosic April 18, 2023

Electron atomic and molecular orbitals

A Bohr diagram of lithium

In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom is 1s2 2s2 2p6, meaning that the 1s, 2s and 2p subshells are occupied by 2, 2 and 6 electrons respectively.

Electronic configurations describe each electron as moving independently in an orbital, in an average field created by all other orbitals. Mathematically, configurations are described by Slater determinants or configuration state functions.

According to the laws of quantum mechanics, for systems with only one electron, a level of energy is associated with each electron configuration and in certain conditions, electrons are able to move from one configuration to another by the emission or absorption of a quantum of energy, in the form of a photon.

Knowledge of the electron configuration of different atoms is useful in understanding the structure of the periodic table of elements. This is also useful for describing the chemical bonds that hold atoms together. In bulk materials, this same idea helps explain the peculiar properties of lasers and semiconductors.

Shells and subshells

Main article: Electron shells (l = 0)p (l = 1)m = 0m = 0m = ±1spzpxpyn = 1n = 2

Electron configuration was first conceived under the Bohr model of the atom, and it is still common to speak of shells and subshells despite the advances in understanding of the quantum-mechanical nature of electrons.

An electron shell is the set of allowed states that share the same principal quantum number, n (the number before the letter in the orbital label), that electrons may occupy. An atom’s nth electron shell can accommodate 2n2 electrons. For example, the first shell can accommodate 2 electrons, the second shell 8 electrons, the third shell 18 electrons and so on. The factor of two arises because the allowed states are doubled due to electron spin—each atomic orbital admits up to two otherwise identical electrons with opposite spin, one with a spin +1⁄2 (usually denoted by an up-arrow) and one with a spin of −1⁄2 (with a down-arrow).

A subshell is the set of states defined by a common azimuthal quantum number, l, within a shell. The value of l is in the range from 0 to n − 1. The values l = 0, 1, 2, 3 correspond to the s, p, d, and f labels, respectively. For example, the 3d subshell has n = 3 and l = 2. The maximum number of electrons that can be placed in a subshell is given by 2(2l + 1). This gives two electrons in an s subshell, six electrons in a p subshell, ten electrons in a d subshell and fourteen electrons in an f subshell.

The numbers of electrons that can occupy each shell and each subshell arise from the equations of quantum mechanics, in particular the Pauli exclusion principle, which states that no two electrons in the same atom can have the same values of the four quantum numbers.

https://en.wikipedia.org/wiki/Electron_configuration