hw2solutions - HOMEWORK ASSIGNMENT 2 Solutions PHYS852...

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HOMEWORK ASSIGNMENT 2 Solutions PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: 1st order perturbation theory 1. The two-level Rabi model is ubiquitous in quantum mechanics. In order to compare the results of perturbation theory with the exact results, find the exact eigenvalues and eigenstates of the Hamil- tonian H = δS z + Ω S x . Use as your basis states: | ↑) for spin up and | ↓) for spin down, relative to the z-axis. Compare your answer to the results from first-order perturbation theory derived in class. Answer : In matrix form we have H = planckover2pi1 2 parenleftbigg δ Ω Ω δ parenrightbigg With E = planckover2pi1 ω , the eigenvalue equation is thus det vextendsingle vextendsingle vextendsingle vextendsingle δ 2 ω Ω 2 Ω 2 δ ω 2 vextendsingle vextendsingle vextendsingle vextendsingle = 0 which gives ω 2 δ 2 - Ω 2 2 = 0, which has the solutions ω 0 = 1 2 radicalbig δ 2 + Ω 2 ω 1 = 1 2 radicalbig δ 2 + Ω 2 The normalized eigenvectors are found to be | ω 0 ) = Ω | ↑) + ( δ + δ 2 + Ω 2 ) | ↓) radicalBig 2( δ 2 + Ω 2 + δ δ 2 + Ω 2 ) and | ω 1 ) = ( δ + δ 2 + Ω 2 ) | ↑) + Ω | ↓) radicalBig 2( δ 2 + Ω 2 + δ δ 2 + Ω 2 ) Note that in the limit Ω 0 we have ω 0 → − δ/ 2, | ω 0 ) → | ↓) , and ω 1 δ/ 2, | ω 1 ) → | ↑) . Equating the unperturbed eigenstates with the limiting value of the perturbed eigenstates fixes the global phase of the perturbed eigenstates. Thus only the global phase choice given above will match exactly with the results of perturbation theory. 1
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2. The relative motion of two identical particles in a harmonic potential is described by the Hamiltonian H 0 = P 2 2 + 1 2 X 2 , where X and P are in dimensionless units. If the particles are spin-polarized (i.e. same spin ori- entation) bosons, then symmetry requires that only the even energy levels be allowed, while if the particles are spin-polarized fermions then only the odd levels are allowed.
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