Question

Answer all of the questions:

(25 pts) Consider a Harmonic Oscillator for a particle of mass m in a potential V = kx?, and with an added perturbation H' = Ex4. a. Find the first order correction to the energy of the nth state. b. Are there non-zero 2nd order corrections to the energy? Explain. c. Are there values for n where the first-order approximation breaks down? Explain physically why this makes intuitive sense. d. Are there any similarities of the answers to parts a) and c) to problem #3? Are the physical reasons the same? Comment.

Reference: (Problem 3, don't need to answer this)

Answer 1

given V(x)= + Kx² Perturbed folential & H')= EXA state of wa Now of we know Harmonic the nth oscillator e In exit) = (mw ) 1 1 g Hn (&) i El Entla Tak on where En = {n+ Y₂hhu. A HACE)= Hermite polynomial &&= mw.x & Kamwa. (9). We have to find lot order correction to the energy of the nith stake. = <414187 (6) 'Em - Keye 4:71? man to En.

this integral value in Eth depends on the nature of Uncat). of enclit) depends on Hermite polynomial, ie it is even function for even n & gt is old function for old n. re Incrit = even function, na even a odd function na odd. The term of an is zero when &m & en is different nature as I'm is even & an is odd. (C) for any value of n lnt-order approximation doesn't break down It depends on the Bequare of the Hermite polynomial & finally the integral of even tunction in rymmetric interval will never zero. - (2) Yeu actually both queation are arking about lat order calculation, part (as is for calculation & peut (b) is for existence,