# 6 - TEORETISK FYSIK KTH TENTAMEN I KVANTMEKANIK EXAMINATION...

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Unformatted text preview: TEORETISK FYSIK KTH TENTAMEN I KVANTMEKANIK EXAMINATION IN QUANTUM MECHANICS Kvantmekanik f¨ ordjupningskurs 5A1329 f¨ or F4 Thursday December 15 2005, 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. Piecewise constant potential Obtain an expression for the energy levels for a particle in the potential V ( x ) =    ∞ , x < , x > a + b , < x < a V , a < x < a + b where V > , a > , b > 0 are constants. Assume E > V . Simplify as far as possible, but do not attempt to solve the final equation for the energy levels (it must be solved numerically). 2. Perturbation theory Formulate an assumption that allows Problem 1 to be analyzed by perturbation theory. Estimate the ground state in Problem 1 using first order perturbation theory. 3. Hyperfine interaction Calculate the splitting of the ground state energy due to a hyperfine coupling between the electron spin and the proton spin in a hydrogen atom. The interaction is given by H 1 = A S 1 · S 2 where A > 0 is a constant. SEE NEXT PAGE! 1 4. Sequential Stern-Gerlach experiment A beam of spin- 1 2 particle is sent through a sequence of Stern-Gerlach devices. The first device has its field in the z-direction and transmits only the | S z = +1 / 2 i state. The second device has its field along a direction n 1 which is rotated an angle θ 1 away from the z-axis in the xz-plane, and transmits only the | S n 1 = +1 / 2 i state. The third device has its field along a direction n 2 which is rotated an angle θ 2 with respect to the z-axis in the xz-plane, and transmits only the | S n 2 = +1 / 2 i state. What is the transmission probability through the last two devices?...
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6 - TEORETISK FYSIK KTH TENTAMEN I KVANTMEKANIK EXAMINATION...

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