Questionnaire for Advanced Quantum Mechanics 1 Uni Graz WS 19/20, Übungen von Quantenmechanik

Nur auf Docsity: Lade Questionnaire for Advanced Quantum Mechanics 1 Uni Graz WS 19/20 und mehr Übungen als PDF für Quantenmechanik herunter! Questionnaire for Advanced Quantum Mechanics 1 1. Compute for the orbital angular momentum operator L̂ the commu- tation relation [L̂i, L̂j ] and discuss the implications using Heisenberg’s uncertainty relation. 2. Consider the eigenstates |m, j〉 of the angular momentum operator Ĵ . What is the physical interpretation of m, j? Compute the action of Ĵ · Ĵ , Ĵz, and Ĵ± on |m, j〉. 3. Consider two angular momentum operators Ĵ1, Ĵ2. Write down the operators and states characterizing the uncoupled and coupled bases. Show how these two bases are connected via the Clebsch-Gordan co- efficients. 4. Consider two angular momentum operators for two spins 12 . Write down the uncoupled and coupled bases using the singlet and triplet states. Determine the corresponding Clebsch-Gordan coefficients. 5. The dimensions of a Hilbert space for j1, j2 can be written in the form (2j1 + 1)(2j2 + 1) = j1+j2∑ j=|j1−j2| (2j + 1) . Discuss the meaning of the left- and right-hand side of the above equa- tion, as well as the range of values for j. 6. The spin-orbit interaction of an electron moving in a potential V (r) can be evaluated to Ĥso = Λ [ 1 r dV (r) dr ] Ŝ · L̂ , where Λ is a constant. Derive within lowest order perturbation theory an expression for the energy renormalization of the 2s and 2p states. Consider the different orientations between Ŝ and L̂. 7. What is the physical origin of the hyperfine interaction? Discuss the energy splittings originating from this coupling. 8. What is the physical meaning of the Wigner-Eckart theorem? Provide a brief example. 9. Discuss for a generic quantum-mechanical operator the von-Neumann measurement postulate. 10. Discuss the EPR-paradox. Why did Einstein, Podolsky and Rosen think that quantum mechanics was incomplete? How did Bohr react to this attack on quantum mechanics? 11. Write down a maximally entangled state between two spin 12 particles. What happens when the two particles are measured in a x-basis? What happens in a z-basis? 12. Discuss Bell’s inequalities. What are the assumptions underlying a description in terms of local hidden variables? Discuss the setup of an experiment that can discriminate between the local hidden variable theory and quantum mechanics? Be as detailed as possible. 13. What are the assumptions underlying the Lagrange formalism in clas- sical mechanics? Discuss how to derive the Hamilton function from the Lagrange function. Write down the equations of motion within Hamilton’s formalism using Poisson brackets. 14. What is the idea behind the concept of canonocal quantization? Pro- vide the necessary equations and discuss the concept in words. 15. Write down the Lagrange function for a point particle interacting with classical electromagnetic fields. Use the concept of canonocal quanti- zation to derive the minimal coupling Hamiltonian. 16. Show how to modify the electromagnetic potentials and the electron wavefunction in case of gauge transformations. Show explicitly that observables do not depend on the chosen gauge. 17. Discuss the Aharonov-Bohm effect. Sketch the stup and compute the phase difference between electron trajectories passing on either side of a solenoid, and express the result in terms of the magnetic flux ΦB = ∫ B · dS. 18. Consider a hydrogen atom inside a constant and static magnetic field B. What types of energy shifts occur? 19. Consider a vector potential of the form A = Bxŷ (Landau gauge). Write down the Hamilton operator and compute the eigenstates, the so-called Landau levels. Express the final result in terms of the cy- clotron frequency ωc and magnetic length `0 defined through ωc = qB m , `0 = √ ~ qB . 20. What is the physical motivation behind the electric dipole approxima- tion. Show how to obtain a light-matter coupling of the form −qr ·E.