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Hermite polynomials: How these mathematical formulas hold key secrets in quantum physics.
Hermitian polynomials are a set of classic orthogonal polynomials. These mathematical structures not only occupy an important position in pure mathematics, but also play a huge role in many fields such as signal processing, probability theory, numerical analysis and physics. . They are particularly relevant to quantum physics because in the quantum harmonic oscillator model, the Hermite polynomials give exactly the energy eigenstates. What secrets are hidden in these seemingly abstract polynomial backgrounds?
Hermitian polynomials not only appear in probability and mathematical analysis, they also play a crucial role in the field of quantum mechanics in physics.
There are two common standard definitions of Hermite polynomials, known as the "probabilist's Hermite polynomials" and the "physicist's Hermite polynomials". These two different definitions reflect the applications of polynomials in different fields, which makes Hermite polynomials an example of research diversity and interactivity.
In physics, Hermite polynomials are connected to the quantum oscillator model. A quantum oscillator is an idealized quantum system in which particles can change between specific energy states. The Hermite polynomials are used to describe these energy states - that is, the wave functions of quantum states.
Hermitian polynomials are a mathematical tool in quantum physics that describe the energy eigenstates of a harmonic oscillator, giving us insight into the workings of the microscopic world.
Historically, the concept of Hermite polynomials was first proposed by Pierre-Simon Laplace in 1810, although in an imperfect form at that time. Subsequently, Russian mathematician Pavnuty Chebyshev conducted in-depth research in 1859. In 1864, French mathematician Charles Hermite finally completed their multidimensional definition and gave these polynomials their name, although this is not entirely correct, as Hermite's work built on Chebyshev's work. Above.
The definitions of Hermite polynomials can be arranged differently according to different starting points, which also reflects their flexibility and adaptability in mathematics. For example, the probabilist's Hermite polynomials are defined as:
He_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}
And the Hermite polynomials for physicists are:
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}
The connection between these two definitions is mutual, and there is a proportional relationship between them. This diversity makes its application range in scientific research more extensive.
The applications of Hermite polynomials are not limited to quantum physics; they are also used in many fields, such as random matrix theory, the heat equation, Gaussian noise treatment in system theory, and Gaussian numerical integration. In signal processing, Hermite polynomial-based Hermann wavelet can effectively perform wavelet transform analysis, showing the power of Hermite polynomials in extracting signal features.
The outstanding performance of Hermite polynomials makes them an indispensable tool in mathematics and physics, advancing our understanding of the universe.
Given the multifaceted nature of Hermite polynomials, studying these mathematical objects can help us gain a deeper understanding of many phenomena, especially physical processes in the microscopic world. In the future, as our technology and theory develop, Hermite polynomials are likely to demonstrate their potential again in new areas.
As an important building block in mathematics, Hermite polynomials reveal many key theoretical foundations in the study of quantum physics, which makes people wonder: what else is hidden in these seemingly simple mathematical formulas? What about secrets we haven't discovered yet?
