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Hermite Polynomial

Hermite polynomials are a set of orthogonal polynomials that arise in probability, combinatorics, and physics, particularly in the context of quantum mechanics and the solution of differential equations. They are defined by the recurrence relation:

Hn(x)=2xHn−1(x)−2(n−1)Hn−2(x)H_n(x) = 2xH_{n-1}(x) - 2(n-1)H_{n-2}(x)Hn​(x)=2xHn−1​(x)−2(n−1)Hn−2​(x)

with the initial conditions H0(x)=1H_0(x) = 1H0​(x)=1 and H1(x)=2xH_1(x) = 2xH1​(x)=2x. The nnn-th Hermite polynomial can also be expressed in terms of the exponential function and is given by:

Hn(x)=(−1)nex2/2dndxne−x2/2H_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}Hn​(x)=(−1)nex2/2dxndn​e−x2/2

These polynomials are orthogonal with respect to the weight function w(x)=e−x2w(x) = e^{-x^2}w(x)=e−x2 on the interval (−∞,∞)(- \infty, \infty)(−∞,∞), meaning that:

∫−∞∞Hm(x)Hn(x)e−x2 dx=0for m≠n\int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = 0 \quad \text{for } m \neq n∫−∞∞​Hm​(x)Hn​(x)e−x2dx=0for m=n

Hermite polynomials play a crucial role in the formulation of the quantum harmonic oscillator and in the study of Gaussian integrals, making them significant in both theoretical and applied

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Hamiltonian System

A Hamiltonian system is a mathematical framework used to describe the evolution of a physical system in classical mechanics. It is characterized by the Hamiltonian function H(q,p,t)H(q, p, t)H(q,p,t), which represents the total energy of the system, where qqq denotes the generalized coordinates and ppp the generalized momenta. The dynamics of the system are governed by Hamilton's equations, which are given as:

dqdt=∂H∂p,dpdt=−∂H∂q\frac{dq}{dt} = \frac{\partial H}{\partial p}, \quad \frac{dp}{dt} = -\frac{\partial H}{\partial q}dtdq​=∂p∂H​,dtdp​=−∂q∂H​

These equations describe how the position and momentum of a system change over time. One of the key features of Hamiltonian systems is their ability to conserve quantities such as energy and momentum, leading to predictable and stable behavior. Furthermore, Hamiltonian mechanics provides a powerful framework for transitioning to quantum mechanics, making it a fundamental concept in both classical and modern physics.

Transfer Function

A transfer function is a mathematical representation that describes the relationship between the input and output of a linear time-invariant (LTI) system in the frequency domain. It is commonly denoted as H(s)H(s)H(s), where sss is a complex frequency variable. The transfer function is defined as the ratio of the Laplace transform of the output Y(s)Y(s)Y(s) to the Laplace transform of the input X(s)X(s)X(s):

H(s)=Y(s)X(s)H(s) = \frac{Y(s)}{X(s)}H(s)=X(s)Y(s)​

This function helps in analyzing the system's stability, frequency response, and time response. The poles and zeros of the transfer function provide critical insights into the system's behavior, such as resonance and damping characteristics. By using transfer functions, engineers can design and optimize control systems effectively, ensuring desired performance criteria are met.

Kaluza-Klein Theory

The Kaluza-Klein theory is a groundbreaking approach in theoretical physics that attempts to unify general relativity and electromagnetism by introducing additional spatial dimensions. Originally proposed by Theodor Kaluza in 1921 and later extended by Oskar Klein, the theory posits that our universe consists of not just the familiar four dimensions (three spatial dimensions and one time dimension) but also an extra compact dimension that is not directly observable. This extra dimension is theorized to be curled up or compactified, making it imperceptible at everyday scales.

In mathematical terms, the theory modifies the Einstein field equations to accommodate this additional dimension, leading to a geometric interpretation of electromagnetic phenomena. The resulting equations suggest that the electromagnetic field can be derived from the geometry of the higher-dimensional space, effectively merging gravity and electromagnetism into a single framework. The Kaluza-Klein theory laid the groundwork for later developments in string theory and higher-dimensional theories, demonstrating the potential of extra dimensions in explaining fundamental forces in nature.

Graphene Oxide Chemical Reduction

Graphene oxide (GO) is a derivative of graphene that contains various oxygen-containing functional groups such as hydroxyl, epoxide, and carboxyl groups. The chemical reduction of graphene oxide involves removing these oxygen groups to restore the electrical conductivity and structural integrity of graphene. This process can be achieved using various reducing agents, including hydrazine, sodium borohydride, or even green reducing agents like ascorbic acid. The reduction process not only enhances the electrical properties of graphene but also improves its mechanical strength and thermal conductivity. The overall reaction can be represented as:

GO+Reducing Agent→Reduced Graphene Oxide (rGO)+By-products\text{GO} + \text{Reducing Agent} \rightarrow \text{Reduced Graphene Oxide (rGO)} + \text{By-products}GO+Reducing Agent→Reduced Graphene Oxide (rGO)+By-products

Ultimately, the degree of reduction can be controlled to tailor the properties of the resulting material for specific applications in electronics, energy storage, and composite materials.

Proteome Informatics

Proteome Informatics is a specialized field that focuses on the analysis and interpretation of proteomic data, which encompasses the entire set of proteins expressed by an organism at a given time. This discipline integrates various computational techniques and tools to manage and analyze large datasets generated by high-throughput technologies such as mass spectrometry and protein microarrays. Key components of Proteome Informatics include:

  • Protein Identification: Determining the identity of proteins in a sample.
  • Quantification: Measuring the abundance of proteins to understand their functional roles.
  • Data Integration: Combining proteomic data with genomic and transcriptomic information for a holistic view of biological processes.

By employing sophisticated algorithms and databases, Proteome Informatics enables researchers to uncover insights into disease mechanisms, drug responses, and metabolic pathways, thereby facilitating advancements in personalized medicine and biotechnology.

Cpt Symmetry And Violations

CPT symmetry refers to the combined symmetry of Charge conjugation (C), Parity transformation (P), and Time reversal (T). In essence, CPT symmetry states that the laws of physics should remain invariant when all three transformations are applied simultaneously. This principle is fundamental to quantum field theory and underlies many conservation laws in particle physics. However, certain experiments, particularly those involving neutrinos, suggest potential violations of this symmetry. Such violations could imply new physics beyond the Standard Model, leading to significant implications for our understanding of the universe's fundamental interactions. The exploration of CPT violations challenges our current models and opens avenues for further research in theoretical physics.