How to Apply Hermite Polynomial for Physics Problems

[Alaguraja]

I have doubt since a long time, that is How we apply the Hermite polynomial for a physics problem. And I don't know weather everyone known about how the analyze a physics problem and how do they apply a correct mathematical methods?

 

[John Dalton]

This will clear your "ancient" doubt.
There are several ways that Hermite polynomials can be defined, but the one used by physicists is this: the Hermite polynomial of degree

is defined as

At first glance, this doesn’t look like a polynomial at all, since it contains only exponentials. But if we calculate the first few, we can see that we get a sequence of polynomials:

 

[Alaguraja]

Thank you Mr. John

 

[John Dalton]

The pleasure was all mine.

 

[marcusl]

You can find a short discussion of Hermite polynomials in a book on mathematical methods. I have the one written by Arfken, but I'm sure others (those by Boas or Riley) will cover it, too. An in-depth treatment is in Lebedev, Special Functions and Their Applications, which also has excellent coverage of the other important functions (polynomials, Bessel functions, spherical harmonics, etc.) with many physics applications. It's a Dover book so it's inexpensive.

Finally, Hermite polynomials are famous as the solution to the one dimensional quantum-mechanical harmonic oscillator. You can find this physics application in all quantum mechanics books. For an undergrad QM text, see any of the standards like Griffith, Shankar, Liboff, or an inexpensive used copy of E. Anderson.

As to the general question of how to solve physics problems, I think you need to start with a course or a basic physics text. It is traditional to start with mechanics.

 

[Alaguraja]

Thank you Mr.Marcus