Question

Quantum mechanics

Consider a two-dimensional harmonic oscillator $\small H_{o}=\frac{P_{x}^{2}+P_{y}^{2}}{2m}+\frac{1}{2}m\omega^2(x^2+y^2)$ . If $\small H=H_{o}+\epsilon xy$ find the energy of the base state until second order in theory of disturbances and the energies of the first level excited to first order in $\small \epsilon$ .

Answer 1

A) E degree_n_1 n_1 = epsilonvariant_n_1 + epsilonavaraint_n_2 = (n_1 + 1/2) planckconstantovertwopi w + (n_2 + 1/2) planckconstantovertwopi w E degree_n_1 n_2 = (n_1 + n_2 + 1) planckconstantovertwopi w (d) = n_1 + n_2 + 1 (i) ground state, = n_1 = n_2 = 0 E degree_00 = planckconstantovertwopi w & d = 0 + 0 + 1 = 1 (ii) I^st state, n_1 = 0 n_2 = 1, or n_2 = 0, n_1 = 1 E degree_01 = E degree_10 = 2 planckconstantovertwopi w, d = 0 + 1 + 1 = 2 (ii) II^nd state n_1 = 0 n_2 = 2, n_1 = 2, n_2 = 0 n_1 = 1, n_2 = 1 E degree_02 = E degree_20 = E degree_11 = 3 planckconstantovertwopi w, d = 3 (iv) III^rd state d = 0 + 3 + 1 = 4 E degree_03
please note in hamiltomian E is replaced by a .if you have any querry let me know.thank you