Quantum Mechanics II Lecture 5

Transcription

1 Quantum Mechanics II Lecture 5

2 Nobelpriset i fysik 2015 Kungl. Vetenskapsakademien har beslutat utdela Nobelpriset i fysik 2015 till Takaaki Kajita, Super-Kamiokande Collaboration, University of Tokyo, Kashiwa, Japan och Arthur B. McDonald, Sudbury Neutrino Observatory Collaboration, Queen s University, Kingston, Kanada för upptäckten av neutrinooscillationer, som visar att neutriner har massa.

3 Super-Kamiokande NEUTRINOS FROM COSMIC RADIATION COSMIC RADIATION KAMIOKA, JAPAN ATMOSPHERE m PROTECTING ROCK SUPER- KAMIOKANDE Muon-neutrinos give signals in the water tank. Muon-neutrinos arriving directly from the atmosphere 40 m SUPER- KAMIOKANDE MUON- NEUTRINO Light detectors measuring Cherenkov radiation Muon-neutrinos that have travelled through the Earth Figure from nobelprize.org CHERENKOV RADIATION

4 In the atmosphere, cosmic rays create electron and muon neutrinos!! π ± µ ± +ν µ (ν µ ) µ ± e ± +ν e (ν e )+ν µ (ν µ ) Super-Kamiokande has measured electron and muon neutrinos and see a deficit of 600 muon neutrinos sub-gev e-like sub-gev µ-like multi-gev e-like multi-gev µ-like + PC cos! cos! Figure from Super-Kamiokande

5 Consider nu_mu to nu_tau oscillations only. The Schrödinger Equation and its Hamiltonian can be written (hbar=1) i d dt e µ = H V e µ H V = m 2 4E cos2 sin2 sin2 cos2 The mass difference squared and the mixing angle θ are constants determined from data

6 We can solve this equation for the eigenstates nu_1 and nu_2 and determine what the probability is that a muon neutrino created in the atmosphere is detected as a tau neutrino The neutrinos travel with the speed of light, so replace time with distance travelled L P(ν µ ν τ )= sin 2 2θ sin 2 (1.27 Δm2 L E ) with masses in ev, distance in kilometer and energy in GeV

7 With best-fit parameters for mass differences and mixing angles we get (at 1 GeV) For small distances (neutrinos from above, the probability is about zero, for larger distances we lose some of the muon neutrinos First maximium in P (dip in nu_mu flux at around 500 km) as seen by Super-Kamiokande

8 Problem 1 In the basis of eigenstates to Sz { >, >}, the Sx operator looks like 0 1 Ŝ x = ~ How does it look like in the basis consisting of the eigenstates to the Sx operator,!i = 1 p ; i = 1 p Hint 1: you can do this in a simple way by reasoning, or by actually calculating the matrix elements. Hint 2: A ij = he i  e ji

9 Problem 2 Consider a particle with spin 1/2. The eigenstates to S 2 and Sz are given by = ( 1 0 ) ; = ( 0 1 Assume that we have a Hamilton operator, Ĥ = Ŝx ; = real constant If the system at time t=0 starts in the state >, how does it look like at a later time? )