5 - 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 Friday June 2 2006, kl. 08.00-13.00 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 square well A particle of mass m is in the ground state of an infinite square well of width a , given by V ( x ) = 0 for 0 < x < a , and V = ∞ otherwise. Suddenly the right wall moves to a point b > a . (a) Determine the probability for the particle to be in the new ground state. (b) What is the probability to be in the new ground state if the wall is instead moved adiabatically (infinitely slowly)? 2. Anharmonic oscillators Calculate the shift in the ground state energy of a one dimensional harmonic oscillator from the small perturbations (a) H 1 = Ax 4 (b) H 1 = Bp 4 where A, B are constants. 3. Angular momentum expectation values Let L be an angular momentum operator and let L z | m ) = m planckover2pi1 | m ) . Calculate the expec- tation values: (a) ( m | L x | m ) and ( m | L y | m ) (b) ( m | L 2 x − L 2 y | m ) (c) ( m | L x L y + L y L x | m ) SEE NEXT PAGE! 1 4. Particle in a spherical delta function potential Consider a particle of mass m in a three-dimensional potential V ( r ) = − Aδ ( r − a ), where A, a are positive constants. Find wave functions (you do not need to normalize) and energies of bound states with zero angular momentum ( s-states). How many s-states are there? 5. Fermi golden rule At time t = 0 a particle of mass m in one dimension is in the ground state of an attractive δ-function potential V ( x ) = − αδ ( x ) where α > 0 is a constant. For t > 0 the particle is exposed to a periodic perturbation V ( x, t ) = − xF cos ωt . Use the Fermi golden rule to calculate the rate of transitions out of the ground state. Assume that the final states are plane waves....
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5 - TEORETISK FYSIK KTH TENTAMEN I KVANTMEKANIK EXAMINATION...

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