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Q.M3 Home work 1 Due date 8.11.15 1 Let us define a state using a hardness basis |hi, |si, where: Ôhardness |hi = |hi , Ôhardness |si = −|si and the hardness operator Ôhardness is represented (in this basis) by 1 0 Ôhardness = 0 −1 (1) (2) Suppose that we prepare a state |Ai = cos(θ)|hi + sin(θ)eiϕ |si (3) where θ and ϕ are parameters 1) Is this state normalized? If yes,prove it. If not, normalize it. 2)Find a state |Bi that is orthogonal to |Ai. Make sure |Bi is normalized. 3) Express |hi and |si in the {|Ai, |Bi} basis. 4) What are possible outcomes of a hardness measurement on the state |Ai, and with what probability will each occur? 5) Express the hardness operator in the {|Ai, |Bi} basis. 2 A particle of mass m moves in one dimension subject to a harmonic oscillator potential 1 2 2 2 ω mx . The particle is perturbed by an additional weak anharmonic force described by the potential ∆V = λ sin(kx) where λ 1. Find the first order correction for the ground state. (Hint: You can use Zassenhaus 1 formula eÂ+B̂ = e eB̂ e− 2 [Â,B̂] · · · You need to prove the formula if you are using it!) 1 3 A system of spin 1/2 is place in constant magnetic field in the z direction (Bz ) - namely, the unperturbed Hamiltonian H0 is: 1 0 (4) H0 = Bz 0 −1 At time t = 0 the spin was at the | ↑i state Then at time t = 0 weak (Bx Bz ) varying magnetic field applied at the x direction, namely the perturbation V (t) is: 0 1 V (t) = Bx cos(ωt) (5) 1 0 Find the probability to find the system at | ↓i at time t. 4 Read the subject of WKB approximation. Using the WKB approximation find the energy spectrum for the harmonic potential, Namely the potential is: 1 V (x) = mω 2 x2 2 Compare your answer to the known energy spectrum of harmonic potential. 2