Exam in Quantum Mechanics (phys201), 2010, Allowed: Calculator, standard formula book and up to 5 pages of own handwritten notes.

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Peder Tollefsen
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1 Exam in Quantum Mechanics (phys01), 010, There are 3 problems, 1 3. Each problem has several sub problems. The number of points for each subproblem is marked. Allowed: Calculator, standard formula book and up to 5 pages of own handwritten notes. Eksamen i Kvantemekanik, phys01, 010 Det finns 3 oppgave, merk vekt! Tillatte hjelpemidler: Kalkulator, formelsamling, og intill 5 sider med egne handskrevne notater

2 Problem 1) Spin ½ particle is at rest in magnetic field along the z axis (Bz) so that the Hamiltonian is H o = k B S= kbz Sz a) What are the eigenvectors of this hamiltonian and to what energies they correspond (p)? b) Particle is in m=1/ state for t=0. There arrives a short lived perturbation of the form: V t = kb x Sx sin t 0 t Use first order time dependent perturbation theory: Calculate the amplitude a_s(t) of m= 1/ state for small t>0 (p). Calculate the whole state vector for small t>0 (3p). a s t = 1 t i ħ t 0 [e i E s E b ħ V sb ]d S x = ħ [ ] S y = ħ [ 0 i i 0 ] S z = ħ [ ] 1

3 Oppgave 1) Spin ½ partikkel sitter i magnetisk felt lang z akse B= (0,0,Bz). Partikkel hamiltonen er: H o = k B S= kbz Sz a) Hva er hamiltonens egenvektorene og hva er tillsvarende energi egenverdier (p)? b) Partikkel er i m=1/ tillstand for t=0. Det kommer et kortlivende perturbasjon: V t = kb x Sx sin t 0 t Bruk tidsavhengig perturbasjon teori : beregn amplitude a_s(t) for m= 1/ tillstand ved kort tid t>0. (p). beregn tilstand vektor ved kort tid t>0 (3p). a s t = 1 t i ħ t 0 [e i E s E b ħ V sb ]d S x = ħ [ ] S y = ħ [ 0 i i 0 ] S z = ħ [ ] 1

4 Problem ) (6 points) a) Solve one dimensional time-independent Schroedinger equation with delta-function potential and find the energy of the bound states. Specify continuity properties of the wave function and its first derivative (4p) [ ħ m d x ] x =E x d x b) Explain why parity is conserved in this case. What is the conclusion for the symmetry of wave function of the bound state? (p)

5 Problem ) (6 points) a) Finn losningen av Schroedingers ligning i en dimensjon med delta-funksjon potensjal og beregne energien av bundne tilstander. Hvilke kontinuitet krav ma bolgefunksjon og forste derivative tilfredsstille? (4p) [ ħ m d x ] x =E x d x b) Forklar hvorfor paritet er en bevart storrelse og finn bundne tillstand sin paritet. (p)

6 Problem 3 N non interacting identical particles each with mass M and unknown spin s are sitting in a infinite one dimensional square well with length L. Energy levels (orbitals) for single particle in such a well fulfill the equation: E n =n E 1 where E 1 = ħ L M ; n 1 a) Lets assume that spin of each particle is zero. What is the lowest energy state of the whole system? What is the excitation energy? ( The energy difference between lowest energy state and the next to lowest energy state) (1p) b) Now there is N=5 (five) particles in the well and the excitation energy was measured to be 3*E_1. With what spins of the particles this measurement is compatible (p)? a) s=0 b) s=1/ c) s=1 d) s=3/ Note that there is more than one possibility. c) Now there is N=10 (ten) the same type of particles in the well and the excitation energy was measured to be 5*E_1. Determine the spin of particles. (p) d) We consider now two identical non interacting particles with spin=0. The state function was written in the form: r1, r = r where r = r1 r Is r an eigenfunction of parity operator? What is the eigenvalue? Explain why. (p) 3

7 Oppgave 3 N ikkevekselvirkende identiske partikler med masse M og ukjent spin s sitter i et uendlig endimesjonelt bronn med lenge L. Tiltallte energieverdien (orbitalen) for et enkel partikkel er: E n =n E 1 where E 1 = ħ L M ; n 1 a) Anta det s=0. Hva er det laveste energitilstand for det hele systemet? Hva er eksitasjon energi? ( Energi forskjel mellom laveste energi og nesten laveste energi) (1p) b) Anta N= 5 (fem) partikler i bronn. Eksitasjon energi var malt og er 3*E_1. Med hvilken partikkelspinn er det kompatibel (p)? a) s=0 b) s=1/ c) s=1 d) s=3/ Merk det finnes mer enn en mulighet. c) Det finnes N= 10 det same partikler (b) i bronn og eksitasjon energi var malt og er na 5*E_1. Hva er spin s av disse partikler? (p) d) Vi har na to identiske ikke vekselvirkende partikkler med spin=0. Topartikkel bolgefunksjon er skrivet: Er r r1, r = r where r = r1 r en egenfunsjon til paritetoperator? Hva er egenverdi? Forklar. (p) 3

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