Question

In the lecture notes, we only solved the TISE for the quantum harmonic oscillator 1 Now, write down the actual solution of the wavefunction of the quantum harmonic oscillator, i.e. the solution that solves TDSE not TISE. 2. We consider the Quantum Harmonic Oscillator In Heisenberg Picture: (a) Hamiltonian to use is the quantum harmonic oscillator Hamiltonian Solve the Heisenberg equations of motion for the operators X (t) and P(t) where the Calculate the commutator [X(t), X (0)] and show that you get the expected result (b) in the limit t -> 0 Calculate the commutator [X(t), P(0)] and show that you get the expected result in (c) the limit t> 0 (d) picture and the Heisenberg picture with respect to the ground state |0) Calculate the time-dependent expectation value (X (t) via both the Schrodinger

Answer 1

I have solve the first four part of your problem since it was going long. If you have any doubt in that, please comment below.

Salution : 1:- Time dependent schrödinger wave equations is given by, in d4 = h 0²4 +v4 - an at am. For Quantum hasmonic oscillator, ube the time independent quantity So, the right hand term is constant and it ich equivalent to energy of system. Now , 4 = 4 GC, it) = 4! (SC) T (t) By seperation al variables. Yoo in 12) = – t2 TH. 8240 + V400 It) 2 m . dividing whale equation. 4 (0) TH) * dI = - D2400 + v 0. T at 2m.400 So, we sees that in egn RHS and RHS are equations of different variables so they must be constant. Let constant be "E"

dI = E + I dI = dt o at ik toge T = - i E t + C. integrating F = A T = ec eitt A e-i Estla T(t) = lec= O =some constant 4 be, t) = 400 A e 12th ( 4 (s, t) = 4a e-etth Absorving in 400 Time dependent solution whore 40 is the ti TISE salution, 4 n = (mwly 1 Halg) eh where Hng) is Hermite palynomial. and = mw x V Synor, t) = ( mw jry. I Halg) e de Ent/A) (Tk) anni

23 () Heisen berg equation of motion Operator As so in Heisenberg bicture herritorion needs to be time dependent therefore, HALt) - Prit) & tm w d (t) 2m 2 . subscript it means, Heisenberg picture Heisenberg equation of motion is given by it d  (t) = [Arkt), HH W] + ih ja Asht)). at 14 As in schrödinger representation, and. È are time independent. Honce it a HGH) = { ÂM H, Halt)] dt For nylt :- PH + Y) LÀ H. it 5,5) = (B), ), but + 4 musi; ] * * * = 4), Lucu, Pull 2m DH = I 2 let) xit = PHA) Uh 2 m )

dolyg tion the am dt 2. Ben * Equation For momentum operator: - - [Fico, et mw ] - 1 (fra f mu? a) = mu? zānu EFH, Am W] = + mw2 â nit, ait ů k la PNG) = -m w²8CH (t)) = 2nd equation at [6] Let the general case of it wo aperators in Heisenberg picture À 4) and (d) At = 7 t it) ÅR Î (4) BA) = tit) Bo î let) [A4), i (H)] = À 4) Bld) – B (t) Act) = pt (t) Å. TH) 9 td) Bo. d) - † +3) 6. 10 . T)

in limit ett tit) 7 t = T4 T * (t) = 1 - G 4) . 6, 7 ) . (1) 8, P. TA) - 1 ) TP., 30 T4) Th 4, Sa)] = T* (1) Tư, 6 7T4) Saln ch aur post equations and equation (1) = Cos + + + 3in1 Đ. ht) - Î was wt - mwõ sinut - ①. Tów, w w ] - jr tout mw = a las ut t how Å sinust, 2010)] [xx), the CA sin wit, 5 to ] ( [a ces ut,7) lim to. [ã,70] = [0, 0] lin (a 4), 10] =0 mw mw [X4), Plo)] = a coswt & mon & Sinust, & ] mw

4P(0] = [cê cos wt, } Ef simot, F] lim. x H), Pro) = txo ( Los o = 1] Ilim [X (H), Plo)] = it ou st the expected result for Hence we got bath the cases.