Discussion Overview
The discussion revolves around finding a general formula for the integral of the product of Hermite polynomials and a power of the variable in the context of the one-dimensional harmonic oscillator. Participants explore the mathematical properties and references related to this integral, particularly in relation to quantum mechanics.
Discussion Character
- Mathematical reasoning
- Technical explanation
- Debate/contested
Main Points Raised
- One participant seeks a general formula for the integral \(\int_{-\infty}^{\infty} \xi^p H_n(\xi) H_k(\xi) d\xi\) for integer \(p \geq 0\), noting they have found results for \(p=0\) and \(p=1\) but require further hints for general \(p\).
- Another participant provides a link to a resource but does not offer further hints, suggesting that the integral might be found in the referenced document.
- A participant mentions that the integral is more comprehensive than what is presented in Abramowitz and Stegun's handbook.
- There is a request for clarification regarding the reference to Abramowitz and Stegun, which is then explained as a well-known mathematical functions handbook.
- Another participant suggests that the integral in question may not be what is typically calculated in quantum mechanics, where a Gaussian weighting function is usually involved, and recommends integrating by parts as a potential method.
Areas of Agreement / Disagreement
Participants express differing views on the integral's complexity and relevance to quantum mechanics, indicating that there is no consensus on the approach to take or the integral's significance in this context.
Contextual Notes
Some participants reference specific mathematical handbooks and resources, indicating a reliance on established literature for integrals involving Hermite polynomials. There is also mention of the potential need for integrating by parts, suggesting that the discussion may involve unresolved mathematical steps or assumptions.