Question

5. Stationary perturbation theory Consider two one-dimensional harmonic oscillators coupled by an interaction term, ie, -E mo? N?) + x,x): Calculate the ground-state energy up to second order in ..

Answer 1

Unpordiabed Hamiltonian How to CP 2+ 3) + moco? cx + x3) puodu balion Im H = X X1 X2 Two indepen dend has, monin 09 Now From anihilation and Qucation operatorii XI - [at+ a1] X2 = [ ]"? [ agt + mre] Figen State of Ho Ang ima, na) Energy Eming = (mit mint to Caround state 100) Energy Foo= te cu Now we know atin) = Vn+ilm+i) alm) svim-1) Ist ordea corection : Art L0019X1X2100) = 0

Since <mi xin) = {m} cat+03|n) enero =0 Second oo dese colocation DE 5 Kmy mal 2X1X2100)12 muto Eoo - Emiña m20 Now mw 2 me Now 1001XX,X2]m, me) b^2oo|(a+a)(c+d)|m) tel. (001[atast + cyat tartaz az imma) - they can cool aj az Imma) - tone vaims Coolmu-l, ma-1) - te snima Som- Som- Nom zero value only form = 1,m=1 So, 02. 121118X1X2100)12 Eoo - El t2x2/4/2002 tw-ho tex = 2mw 1 8 mus

So the ground state energy uplo end ogden E = Eoo + DEU!+DEC) ztew to - ha 8m203