Question

4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or annihilation) operator â and the raising (or creation operator at, which are defined as = (ip + mwe), and at = - 2ħmw =(-ip + mwe). 2ħmw Show that can be written as glâ +ât)?hw/4. (c) Using the time-independent non-degenerate perturbation theory, find the first order and second order correction to the eigen-energies of the perturbed Hamiltonian û = Ho + Ñ'. (d) In fact, the problem can be solved exactly. i. By introducing a new angular frequency w', show that the perturbed Hamiltonian can be expressed as po h = h + ' = +mw2.2 (10) 2 m 2 What is w' in terms of w? ii. Write down the eigen-energies of the perturbed Hamiltonian in Eq. (10). Compare the result with the result obtained from the perturbation theory in part (c).

Answer 1

Hamiltonian of harmonic oscillator in ID: h = 1 + 1 - 2 meter (a) Eigen - energies of uperturbed Hammiltonian mwan is given by E = (+4)70), :0,1,2, where n = o for ground State nål for first existed State and so on. (6) The given by perturbation is H = 1/2 g mo na 2 h Given, a = t (if & m02) o f me (-18 + mwá) in atât. 2 mwê $2ħ mw mw

Cât êt) = 4 an war (Qu" 2.mwa 2 t mw 2 mw 2 mo in * = 1 gm wrx ħ (atâtr 2 mo The perturbation is h = f g more 19 So, first order Correction to the energy O = (m) À I m) 3.thom (m] (a + alv) (m) - Thu [kalatât) (@+at] [mm] . ? ( ) ( ) (4% |-) + (n+1/4)

OPT (+) + n] EU = 9 to (2m +1) Formula uses: âlny & in 1m-1) at my = nt nt) and {nl my = 1 for m=n = 0 for min that is calm-y=0 So, first order correction to the energy NU Second-order Correction energy, to the FREE Ixon = (10'z Kul (á mah) | (4) mpra E

Noo, (mCât atyaya) - Koml Cârât) (In 1m-1) + WMH Inti) (1-2) {m} m-2) + Jeon +75 (mm) + m {mm) + (mt) (n+2) [ro] n +2) 2+2 NOO, EO-ES = [nt£ - "+2-]tre = 2 hw and to EU = [1 + 1/2-1-2-12 ] hw n +2 - 2 hw

x2 TREME 69:23 - 0 +7 (920) 140 [-V-20 - -*-30 – 2] This is second-order energy of th Correction of State. So, energy after Corsection, = (0 £) 40 + 24 (90 a) The Cave) (d) NOW, 0 À 8,4W LED + Lurrar o initial, it is the most + 2 quant + (1+9)**

Compairing equation 0 and ②, on = (1+9) cov (1) So, energy-eigen values of perturbed Hamiltonian, 5(haya ng puting (2017) - Airport (50 +2) 324
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