Question

Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the oscillator frequency: 2 K22 H = H. +V, (1) 2 K2 V = - K +K > 0. Of course, this problem (1), (2) is trivially solved exactly yielding the oscillator solutions with a new frequency. Show that corrections to the nth energy level as calculated within the perturbation theory indeed reproduce the exact result, restricting yourselves to terms up to the second order in the perturbation. That is, compare exact and perturbative results with the accuracy ~K

Answer 1

2 amw na K1 22 , wat En°= 10+5de 1943) =0,1,2, 3... ght order perturbation E = 4 Tv / Yn") = " <n/32")") = kx (934) i n Canti) s & Ind order correction En 5 min372 man EmEm 5 Ikmatam s kment y afectuatelyt mo K 5 Kmailm) + (hayu)+midats L Unt){n+2) (m/n+2) + Inn-1) { min-g) (H+) (min) + h min) 2. o ano niny) & ert1m) - Inti Inti) ki? 5 )4(n+1)(n+2) {n+2]n+22 non-)) {n-12]n-2)! it (nt) 1000+ nxol ki? lt 2 n²+ 3n+2 + neh y lamwl ato 2 hw - K2/h²x I an 3h 2 + n-nl 4 lam ghn - operate the x - 4|2n +1)

P. No: E2 - - K? 1 + 2/ont) Are 4tw lamw - 27 - 16 mw3 lanti) - - x+x1) Energy m En = {n+ 2) 29 , n = k + k. ut (k+k - 11 + 1 g 2 & walem - w/1t K]} - W[1+ y + 5531 13222-- Kamw 4 wt wk Wk2 OK2 En - Int L.)NA= 129,44% 23 t. wok. W - 1393!!w + kit wow ( 90+1) - (anti) tw k, 2 1662 -how - Enot Kh (2n+ 1)-(ant) k, 2 h 16 m²w3 En - En + En & En 12 e Note , od to lat + a) vamw

X2 = 5 lat? + aut + atq+012) al әрі юс -ARUN'S PAGE NO. DATE / / smw enlx/n) = t (n) at² + aat + atat 92/n) hujani artinst center aty n) + K mi arta] 1) + <n)824937 t o Lo + Jebel) (ne) <016) + Shin <nims+o] - [ #] + n] = m (3h+) dmw
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