Download Quantum Mechanics II Final Exam: Solutions and Problems and more Exams Quantum Mechanics in PDF only on Docsity! CTH/GU Fysik FKA081/FYN190 Quantum mechanics II Final exam Time: Tisdag 19 oktober, 14:00-18:00 Place: V-huset Examiner: Stellan Östlund (0708-723201) (present 14:00 and 16:30) Ling Bao (031-7723184) Additional material: Your course text. You may not have extensive handwritten notes in the book. If you do, you will be asked to exchange the book for one without notes. You may also borrow one of the two copies BETA , which will be available at the exam. If you do, return it promptly. Instructions: I strongly suggest you go directly to the problems you know how to do and get them out of the way. Think about substituting a symmetry or physical argument instead of messy algebra; I always prefer such derivations and the algebra can be drastically reduced in many of these problems with a bit of insight. Problem 1 (10pt) - The two Stern-Gehrlach device This problem is from the textbook (1.13) and from RP2. A beam of spin 12 atoms goes through a series of Stern Gerlach experiments as follows: The first measurement accepts sz = h̄ 2 and rejects sz = − h̄2 . The second apparatums is identical, but oriented at an angle β relative to the z axis. The third apparatus is identical to the first but oriented at 1800 relative to the first. • (4pt) Without calculating, only from physical arguments, deduce for what values of β the outgoing beam must vanish • (6pt) Compute the relative intensity of the outgoing beam as a function of β and compute the β where the outgoing beam is strongest. Problem 2 (12pt) - Projectors and symmetries Consider the 4 × 4 periodic discrete translation matrix T = 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 In the following, argue as compactly as possible what the right answer is to each question. Each question can be answered with one line if you use symmetries or other physical argu- ment; i.e. use whatever trick you know to avoid doing messy algebra. A brute force solution will not only cost a lot of time, but a few points as well. • (2) What is characteristic equation of T • (2) What are the the eigenvalues? • (2) Explictly write down the four eigenvectors. • (2) Write down explictly the 4 × 4 matrices | −1 〉〈−1 | and | −1 〉〈 1 |. • (2) Write down the matrix | −1 〉〈−1 | show it is Hermitian. Show it is a projection operator and indicate why you don’t really need to do the matrix multiplication. • (2) Argue as concisely as possible how can you know that | −1 〉〈 1 | is x4 but that the operator cannot be diagonalized. 3 (10 pt) - Unitary Galilean transformations Let the unitary operator G be given by G = eiv(mx−pt)/h̄ where v is a constant and x and p are operators. • (5pt) Show that G generates the galilean transformation x → x − vt • (5pt) Compute the result of transforming p with G. 4 (10 pt ) - Angular momentum addition Assume you have two spin 1 objects (A) and (B) and the Hamiltonian H = αSA · SB + h · (SA + SB). • (6) Compute the the eigenvalues of the H as a function of α and h. • (2) Draw an energy level diagam of the energy levels as a function of h and label the levels with | jm 〉. • (2) What is the lowest nonzero value of |h/α| for which a degeneracy occurs. 5 (10pt) - Harmonic Oscillator, (Sakurai 2.13, RP3-4) Consider a one-dimensional simple harmonic oscillator, taking values of ω, m and ω so that a = √ 1 2 (x + ip) and similarly for a†. Recall a |n 〉 = √n |n − 1 〉 and a† |n 〉 = √ n + 1 |n + 1 〉. Consider 〈m |x |n 〉, 〈m |p |n 〉, 〈m |x2 |n 〉 and 〈m |p2 |n 〉. 2
