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# p3-342 - E = mc 2 for a particle at rest(i Find the...

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Problem Set 3 Physics 342 Due January 28 Some abbreviations: S - Shankar. 1 . S15.2.2 2 . S15.2.3 3 . S15.2.7 4 . We can construct an oscillator model of angular momentum as follows: consider a system of two creation and two annihilation operators, a α ,a α with α = ± and h a α ,a β i = δ αβ . (i) Show that the operators J - = a - a + , J + = a + a - , J z = 1 2 ± a + a + - a - a - ² 1 2 ( N + - N - ) (1) satisfy the su (2) algebra. (ii) Show that the states | j,m i = ( a + ) j + m ( a - ) j - m p ( j + m )!( j - m )! | 0 i (2) with m = - j, - j + 1 ,...,j form a representation with total angular momentum j . In particular, compute J ± | j,m i , J 2 | j,m i and compare the result to the general expectation. 5 . Let’s consider a quantum mechanical two state system consisting of the electron neutrino ν e and the muon neutrino ν μ . In some theories, these particles have very small masses m e and m μ and there is a small amplitude for transitions between these particles. Namely, h ν μ | H | ν e i = A is non-zero where H is the Hamiltonian and A is a real number. In this problem, we will use units in which c = 1 so that energy and mass have the same units using

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Unformatted text preview: E = mc 2 for a particle at rest. (i) Find the masses (i.e. energies) m 1 and m 2 of the states which are eigenvectors of H . Let the diagonal elements of H be h ν μ | H | ν μ i = m μ , h ν e | H | ν e i = m e . 1 You may assume that | A | << | m e-m μ | . (ii) The eigenstates ν 1 and ν 2 of H can be written as linear combinations of the states | ν μ i and | ν e i | ν 1 i = cos θ | ν e i -sin θ | ν μ i , | ν 2 i = sin θ | ν e i + cos θ | ν μ i . Find the mixing angle θ in terms of A,m e ,m μ with again | A | << | m e-m μ | . (iii) Suppose at time t = 0, a ν μ is produced at rest in the laboratory. What is the probability as a function of time that this state will actually be observed as a ν e ? Express the answer in terms of m 1 ,m 2 and θ . 2...
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p3-342 - E = mc 2 for a particle at rest(i Find the...

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