Question

is A system with an unper tubed Hamiltonian H Subjected to a perturbation HC Mon , 3 . • Il (1 1 . i eil 2 0 ; H = da o vz; 0 0 -2 0 II 131 l-ai vei o 2, where a is a real Constant with The dimesions of an energy anda is a dimension less real Constant. as show that the following vectors are the eigen vectors of M" and determine their associated eigenvalues. 14>*3) ; =(- 4- (1) - ****(:) JU, >- b) Find the eigenstates of H" that are stable under the perturbation. (0) c) Find the eigenvalues of 14 - Hº+H" to first order indl

Answer 1

o The unperturbed Hamiltonian of the system Ha) aſs 1 0 0 31 -za o The correspondig eigenvalue 40 Cos 49 lue> za = - 2 12 2a 142> The corresponding eigenvalue in za.. #lug) - 734 0 al 0 0 24 (0) --- () - eigen: value is - 2a. - The Corresponding 494) = 22 0 - za ma con le lua) The eigen we in za

The perturbed Hamiltonian HF HO HO 300 tai atsida H 2+2 al 0 - 2 i 1- dai o -2arxa 0 la-2aai darai o Barca H/4)= a o dar ozat da o - dai o Zarja labaai saaai io atraxi držai o 3 6*219, | IQ+ radi -datai This in not prertarbed Har eigen red for 19a - 2x aitzia Hu s o e 0 a 2atxao 0 -20+10 maai aai -vejai. 0 Bataa/ á -zare ratta This is not perturbed eigenreefor. V2 AA/ | 3a d ai 2aa o hai o Zatta a-zati retai 0 atrasi retai o Batera, lo Laai - 20.0tal This is not pertumbed eigen rector of Hamiltoniame HUA) sa o dai atrasi Lo2arta ao -vzati. 200 - 200 . a zati Vzdai o s atzaal 20 1 2a- radi . Praxi This in not perfurbed Hamitomin eigen Keetor. 29-2da -zadi.

que a Correction The first order Rigenvalue HD <**> CU,H UI sh L-21+2 ga (zieci + 2) = de ei +2) = 20 The corrected eigen value di = 9G HÅR. Lual Hºus) = 4( . ..) Eda looo) - ja Corrected eigenvalue A2 = 2atda <uz1H"143) = de oor 1 0 - Vzi E (44|H9|44) = sa The corrected eigenvalue 73 Brzatie des) . ) ada ( 10 ) Hat ) - 19 (-24 +24+2) = x2 -21-2/ The corrected eigenvale de a 2artia