1. The problem statement, all variables and given/known data "Show that the Hermite polynomials generated in the Taylor series expansion e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞) are the same as generated in 7.58*." 2. Relevant equations *7.58 is an equation in the book "Introductory Quantum Mechanics" by Richard Liboff. (ξ-(∂/∂ξ))n × e-(ξ2/2) = Hn(ξ) × e-(ξ2/2) ∑ƒn(a)/n! × (x-a)n 3. The attempt at a solution To check if I am doing things correctly, I chose n=2 and according to the book I should get A2(4ξ2 – 2)e-(ξ2)/2 where A2 is a normalization constant. I am told to Taylor Expand e(2ξt - t2) Now the Right Hand Side tells me that Hn is a function of ξ so I believe I am supposed to to apply Hn to e(2ξt-t2) with respect to ξ. So from the equation 7.58 from the book, if I choose n=2 then I get H2 (ξ-(∂/∂ξ))2 which equals (ξ2 - ξ(∂/∂ξ) - (∂/∂ξ)(ξ) - (∂2/∂ξ2)) so if I now perform the operation [(ξ2 - ξ(∂/∂ξ) - (∂/∂ξ)(ξ) - (∂2/∂ξ2)) × e(2ξt - t2)](t2/2!) I should get A2(4ξ2 – 2) × e-(ξ2)/2 but with e(2ξt - t2) in place of e-(ξ2)/2 correct? Well I do not. Instead I get (ξ2e(2ξt - t2) - ξ(2t)e(2ξt - t2) - e(2ξt - t2) - ξ(2t)e(2ξt - t2) + (2t)2e(2ξt - t2)) × t2/2 I can factor out e(2ξt - t2) but it doesn't do anything that would lead me to an answer. I mean, it's obvious this is incorrect since I have the variable t but it doesn't show up anywhere in the table that the Taylor expansion is supposed to correspond to. I have looked up videos and checked textbooks for performing a Taylor expansion but they just show me how to evaluate a polynomial at a point a on the function ƒ(x) but I am explicitly given the Taylor expansion to have tn instead of (t-a)n so I don't think I should just pick a random point. I also tried using (t-(∂/∂t))n to see if maybe I was supposed to infer a change of variable to t instead of ξ but that just ended in a huge long equation that did not seem to simplify. Is my approach wrong? Please let me know. Also, please, please, please, please.....do not be rude. To say I love physics is a complete understatement. I have been and will continue to put in the work to learn as much as possible. I do not like to take shortcuts, but my calculus knowledge is lacking because I didn't realize that physics was what I wanted to do until a little bit later in life. So I ask anyone who may help me to not treat me like someone that doesn't value the process of learning by saying something along the lines of "Just look at the definition! Did you even read the book??" or one that I have seen here many times "why would you even do that?!" I would not like to get berated by those whom I automatically respect because of their knowledge of physics. Thank you in advance.