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Unformatted text preview: TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK F ¨ ORDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHANICS Kvantmekanik f¨ ordjupningskurs SI2380 f¨or F4 Thursday December 18, 2008, 8.00 – 13.00 Write on each page: Name, study program and year, problem number Motivate in detail! Insufficient motivation leads to reduction of points Allowed material: Formula collection, BETA (or equivalent), pocket calculator Grading system: Max 3 points per problem Examiner: Jack Lidmar, tel 5537 8715 1 Scattering Calculate in the Born approximation the differential cross section dσ ( θ,φ ) d Ω , for particles scattering against a Gaussian potential, V ( r ) = V exp (- r 2 / 2 a 2 ) . 2 Time-dependence of harmonic oscillator states A harmonic oscillator is at t = 0 in the state | ψ (0) i = 1 √ 3 ( | 1 i + | 2 i + | 3 i ) . Find the expectation values of position h ˆ x ( t ) i and energy D ˆ H ( t ) E at time t ! 3 Hydrogen molecule H + 2 The simplest molecule consists of only two protons and one electron. Assume that the positions of the protons are fixed and separated a certain distance d apart. In the limit d → ∞ the electron could be localized on either one of the protons, hence the system would be doubly degenerate (let us ignore the spin degree of freedom). At finite but large separation d one would still expect the Hydrogen ground state wave functions to be a good approximation. Use the ground state hydrogen wave functions to suggest variational wave functions for the ground state and the first excited state of the molecule. Based on the symmetries of the problem, what can you conclude about the form of the wave function? Do not bother calculating the energies of the states! 1 4 Two coupled spin- 1 2 particles Two identical spin- 1 2 particles are placed in a harmonic potential. The particles interact only via their spin, ˆ H = ˆ H 1 + ˆ H 2- k ˆ S 1 · ˆ S 2 , where ˆ H 1 and ˆ H 2 are the ordinary harmonic oscillator Hamiltonians for particle 1 and 2, respectively. What are the “good” quantum numbers characterizing the energy eigenstates? Determine the possible ground states, and express them in the basis | n 1 ,m 1 i⊗| n 2 ,m 2 i , where n i label the harmonic oscillator states and m i = ± label the spin. How does your result depend on k ?...
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This note was uploaded on 12/12/2009 for the course FIZIK 201 taught by Professor Belmaşimsek during the Spring '09 term at Çukurova University.

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