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Q.M3 Home work 1 Due date 8.11.15 1

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Q.M3 Home work 1 Due date 8.11.15 1
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
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