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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 August 25 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 Examinator: Mats Wallin tel 5537 8475 1. Spin-1 particle A spin-1 particle is in the state | Ψ i = 1 √ 14 3 2 i in the S z basis. (a) What is h S x i ? (b) What is the probability that a measurement of S x will give the value ¯ h for this state? 2. Commuting operators Show that if two Hermitean operators ˆ A and ˆ B have a complete set of eigenstates in common, then they commute. 3. Time evolution Assume that a system has a time dependent Hamiltonian ˆ H ( t ) that commutes at different times: ˆ H ( t ) ˆ H ( t ) = ˆ H ( t ) ˆ H ( t ). Use the time dependent Schr¨ odinger equation to show that the time evolution operator is given by ˆ U ( t ) = exp- i ¯ h Z t dt ˆ H ( t ) SEE NEXT PAGE! 1 4. Born approximation Use the Born approximation to calculate the differential cross section for the potential energy V ( r ) = V e- r/a where V and a are constants. 5. Time dependent force on the harmonic oscillator A harmonic oscillator is subject to a time dependent spatially uniform external force F ( t ) = Ce- λt where C, λ are constants. If the oscillator is in the ground state at t = 0, calculate the probability to find the oscillator at time...
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