Solve Hermite Equation & Find Its Use in Math Physics

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In summary, the conversation discusses the Hermite equation and its use in physics, specifically in the context of quantum mechanics. The Hermite equation produces the Hermite polynomials, which are solutions to the Schrodinger Equation for a particle in a harmonic potential. The conversation also mentions finding a series solution for the Hermite equation and its relationship to the Schrodinger Equation.
  • #1
darkone1687
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I have a problem for a Math Physics course, that I was suppose to find the Hermite equation, find solutions to it, plug them into see if they work, and lastly find a use for the Hermite equation. I've done everything but I can't find a use for the equation I looked almost everywhere I would really applicate the help if someone could give me a use for the equation, it would be best if it was used in physics but if not it's alright. THANK YOU for the help!
 
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  • #2
Hermite Equation as in the differential equation producing the Hermite polynomials?

For the case of a particle in a harmonic potential ([itex]\frac{1}{2}m\omega^2x^2[/itex]) in quantum mechanics, the Schrodinger Equation (once rescaled to be in terms of nondimensional variables) has solutions which are proportional to [itex]\exp[-x^2/2][/itex] times the Hermite polynomials.

http://en.wikipedia.org/wiki/Hermite_polynomials
 
  • #3
I can't find the series solution to this equation can someone post it here please, also I was wondering how it fit in with shrodingers equation I'm suppose to somehow use Hermite to get to Schrodinger, or vice versa?
 
  • #4

What is the Hermite equation?

The Hermite equation is a type of differential equation named after mathematician Charles Hermite. It is a second-order linear differential equation that has applications in physics and engineering, particularly in the study of quantum mechanics and harmonic oscillator systems.

How is the Hermite equation solved?

The Hermite equation can be solved using various methods such as power series, Frobenius method, and recurrence relations. These methods involve manipulating the equation to reduce it to a form that can be solved using standard techniques, such as integration or differentiation.

What is the significance of solving the Hermite equation?

Solving the Hermite equation allows us to understand and predict the behavior of physical systems, such as the motion of a particle in a harmonic potential or the dynamics of a simple pendulum. It also has applications in other branches of mathematics, such as probability theory and statistics.

How is the Hermite equation used in mathematical physics?

The Hermite equation is used to describe the wave function of a quantum mechanical system, specifically for the quantum harmonic oscillator. It is also used in the study of Brownian motion and random walks, as well as in the solution of certain partial differential equations.

What are some real-world applications of the Hermite equation?

The Hermite equation has applications in fields such as quantum mechanics, statistical mechanics, and signal processing. It is used to model the behavior of systems in these fields, such as the energy levels of a quantum particle or the fluctuations in a random process. It also has applications in engineering, such as in the design of electronic circuits and control systems.

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