Question

(6) Here is the Hamiltonian for two coupled harmonic oscillators oriented along ti same axis: 22 22 2mozz + 3.2) + -k(zỉ + 2) + Ariza, (a) What are the energy levels for the two oscillators when there is no coupling, or i = 0 (b) Now include the coupling. What change of variables can be made to solve this syste exactly? (C) What are the energy levels for the system? (d) How would one construct the raising and lowering operators for this system?

Answer 1

x a) H= -12 22 4 22 ) + tk (4,24 422) +AX, X2 do Ha – t2 2 2 + 2 + Ik (2,26222) 2-D Harmonie Oscillator. - Enne = (n + 2) to +(na + 2) to ...to (no +12 + 1) m., uz > eigen values, and they are - tue integers. los do. Let alia 9,442 x = 0, y z He £ 2 (2,2 4 X22) = 2 kly, 4 ) also f, = 2 + H2 134,42 = $14,2-4,3) Here w . Ik som om o Bay, Port lag, 27) (3,4%) (3, + 8y.) + ( 694 + 3324, 24) loto - boy @yogy) - Halog | - - - ) + $ (k+1)4,2 4 [(4-1) .2 - - - - -

c) : E = o(nit I) with + (M2' + t ) wa ti E= (n + £) w, t + (ná + 2) Wah v a ti V 25 mwi (55, + 1 x the os non Anatay sa Roy, fox, thanks ang kanila kay mwi â' - ma con lã,- pp + mva (forse - Foxx)) ât se poate año te, mw, (four + ford)) QI = muros lâu, -, - mis (Faxe - box)) fou a it 2 px2=-it o by Barbon ý - Home ww2