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Unformatted text preview: TEORETISK FYSIK KTH TENTAMEN I KVANTMEKANIK F ¨ ORDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM ME- CHANICS Kvantmekanik f¨ordjupningskurs 5A1385/5A1329 f¨or F4 Monday December 18, kl. 14.00-19.00 Write on each page: Name, study program and year, problem number Motivate in detail! Insufficient motivation leads reduction of points Allowed material: Summary of lectures, BETA, pocket calculator Grading system: Max 3 points per problem Examiner: Mats Wallin tel 5537 8475 1. Particle in a parabolic perturbation Consider a particle with mass m in a square well potential V ( x ) = braceleftbigg for − a/ 2 < x < a/ 2 ∞ otherwise Calculate the shift in the ground state energy due to a weak perturbation H 1 = cx 2 2. Orbital angular momentum A particle is in an eigenstate of L 2 and L y with eigenvalues 2 planckover2pi1 2 and 0 respectively. What are the possible results and corresponding probabilities in a measurement of L z ? 3. Born approximation Calculate the differential scattering cross section in the Born approximation for scattering by a Yukawa potential V ( r ) = V e − ar ar SEE NEXT PAGE! 1 4. Rigid rotator At low temperatures, a diatomic molecule can be regarded as a rigid rotator with Hamil- tonian H = L 2 2 I where the moment of inertia is I = μr 2 , μ is the reduced mass, and r the distance between the nuclei. In a weak electric field along the z axis the Hamiltonian is perturbed by H 1 = − dE cos θ where d is the electric dipole moment of the rotator. Suppose that the perturbation is turned on at time t = 0, when the rotator is in its groundstate. To what states can the perturbation induce transitions? 5. Two dimensional harmonic oscillator A two dimensional harmonic oscillator has a degenerate energy eigenvalue E = 3 planckover2pi1 ω What happens to this energy level due to the perturbation H 1 = Cxy where C is a constant?...
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- Spring '09