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Unformatted text preview: TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK F ¨ ORDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik f¨ ordjupningskurs SI2380 f¨or F4 Thursday December 20, 2007, 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 (handed out with the exam), BETA, pocket calculator Grading system: Max 3 points per problem Examiner: Jack Lidmar, tel 5537 8715 1 Variational calculation of a 1D bound state A particle moves in a potential V ( x ) = ( ∞ if x < ax if x ≥ where a > . Suggest a variational ansatz for the ground state wave function and calculate an estimate of its energy. Calculate also an estimate of the expectation value of the position. 2 Spin in a magnetic field Consider a spin S in a uniform magnetic field B , with a Hamiltonian ˆ H =- γ B · ˆ S . Derive the equation d dt D ˆ S E =- γ B × D ˆ S E . Specialize now to a spin- 1 2 particle and assume that the magnetic field is aligned in the z-direction, B = B e z . Assume further that the state at t = 0 is | ψ (0) i = 1 √ 2 ( | + i + |-i ) , where |±i are the eigenstates of ˆ S z with eigenvalues ± ¯ h/ 2, respectively. What is the state vector at time t = 2 π/γ | B | ? What is the expectation value D ˆ S x ( t ) E at the same time? You may need the commutation relations for the components of the spin: [ ˆ S x , ˆ S y ] = i ¯ h ˆ S z , and cyclic permutations of x,y,z. SEE NEXT PAGE! 1 3 Identical particles Two identical non-interacting spin- 1 2 particles are placed in a 1D potential V ( x ) = ( if 0 ≤ x ≤ L ∞ otherwise A strong magnetic field is applied so that the projection of the total spin on the z-axis is maximized. Write down the ground state wave function! 4 Perturbed harmonic oscillator A two dimensional harmonic oscillator has a degenerate eigenvalue E = 3¯ hω . What happens to this energy level due to the perturbation ˆ H 1 = C ˆ x ˆ y where C is a constant. 5 Time-dependent perturbation Assume that a system, described by a time-independent Hamiltonian ˆ H , is perturbed by ˆ H 1 ( t ) = ( ˆ H e t/τ for t < for t > , where ˆ H is small. At time t =-∞ the system is in an energy eigenstate | n i of ˆ H . Calculate, using first order time-dependent perturbation theory, the state of the....
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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.
- Spring '09