Question

H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥ, = îl + Ĥ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... 1. Write down explicitly the expression for the eigenvalues En,m in terms of 01 and W2 2. Prove that the resonant frequency of the first oscillator is independent of the states of the second oscillator: En+1m - Enm = ħwy for any m. We introduce an interaction between the oscillators described by the perturbation term û = ħB (âßt + atb), where ß is the coupling constant. 3. Assume w1 # w2 and use non-degenerate perturbation theory to calculate the first- order energy corrections Enm. Use the equation Enm = (n, mlû|n, m). -(2) 4. Find the second-order energy corrections Use the equation (2) En'un m'#m |(n' m'\V\n,m) Enm Enm = Enm-En'mi

Answer 1

1 » Å lm, m) = ( , + f₂) im, m) En,mln,m) Ĥi 19, m m, m) + , 11,4) [tw, (n+1) + ħ Hz (m+ + y)] in, m) {t(nw, + mws) + $(w,+w,))in, ) En,m. Entim = t [lon +1W, * ww2] + {W, 4w 2) 2.) > Enti, m-En, me t , for any 3) ant ab ta 0) Enn At 1n,m) + tß Knıml <n, ml û In, m) â @"]m,m) (nıml nti <nt, mln, m+) Imts + In Im {n-1,ml n, w-l) tiß o

12) enti, mln, mi) (n-1,min, m-1) =0 |<n; m' l ll ln,m m)? n'in, m't m En,m - Enim 4.) E vi'u (nim' n, m) = Jen'+1)(m+1) <n't, mil ,m +1) + Inism <n'-1, milm, m-1) 2 (2) E Enim = tips '+1) (mt) (n'+1, "\ n, m+1) +frism <1-1,4mment n'in, m'tm En,m - Enim' (nim,wowy <n'ti, m'ln, uti) (n-s, m'|m,m-1) = и iff n'tl=n १ ma i=mtl n'an-1 a m'am-1 = 1 n-1=n =) n =hti 22 2. Enim u = tiß E (n'+1) (m+i) | n't yoxm, m+12) N'En, wit m Enim - Eni,ems! + n'mkn't, m'ln, m-1), En, m - Enim' [ 6-1+1) (m+1) (n+1)m En, m - Eni, mti En,m-Entium-i 22 tiß + 1) 9

Enim-En-1,miti ħ(nw, +mW 2) + t (w, +W2) -to [(-1)w, +(m+1) wz] - Ę (w, twe) - ħW, - ħWz En, ne-Enti,mars = -tw, + ħ Wz 2,2 2 Enm h p2 + In(m+1) th(w,-W2) (n+1) m t(W₂_W,) 2 (N-m) t 8² han+n-mom (W, W2) Wi-wz