and S is in the state ( )

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1 Physics 517 Homework Set #8 Autumn 2016 Due in class 12/9/16 1. Consider a spin 1/2 particle S that interacts with a measuring apparatus that is another spin 1/2 particle, A. The state vector of the system is the product of the state vectors of the two systems A, S At time t = 0, A is in the state ( ) 1 0 A and S is in the state ( ) a b S, where a, b are complex numbers with a 2 + b 2 = 1. During the time interval 0 t T, S and A are coupled via the interaction Hamiltonian H = 1/σz S σy A g(t) where g(t) is a continuous real function that is non-zero only for 0 t T, and satisfies T 0 g(t)dt = π. Also σs z, σy A are Pauli matrices that refer to the systems S and A and thus commute with each other. Determine the state vector for the combined system S, A at time T. 2. Suppose we have a beam of photons whose polarization state is unknown to us. (a) What measurements would determine the polarization state, and what is the minimum number necessary? (b) What is the density matrix ρ for unpolarized light? (c) Find ρ for a mixed state consisting of 50 % linearly polarized light in the x direction and 50 % positive helicity circular polarized light. 3. Consider a non-relativistic electron that is trapped in a state above a semi-infinite flat insulator. Due to the attractive interaction between the electron and insulator, the electron experiences a potential V (z) = A/z, z > 0, V (x) = +, z < 0, where A > 0 is a constant, and z is the distance between the electron and insulator (the insulator is the plane at z = 0). The form of the potential suggests that the system is a one-dimensional analog of the hydrogen atom. Assume that the electron can not penetrate into the insulator. In parts (a), (b), consider only the z dependence of the wave function. (a) Determine the spatial part of the wave function for the ground state up to a normalization constant. Hint: what are the boundary conditions? (b) Compute the ground state binding energy in terms of A,, and m. (c) State all of the quantum numbers for all of the states bound in the z direction and determine the energies in terms of these quantum numbers. Take into account electron spin, and that the electron can move in all three directions. (d) Is the total energy necessarily smaller than 0 for a state bound in the z-direction? Explain.. Which of the following proposed density matrices are legitimate density matrices? If any represent pure states, ] [ 9 ] find the state vector. ρ 1 =, ρ 2 = 25 25, ρ 3 = 1 0, ρ 5 = 1 3 u u v v [ ( u v + v u ) with u, v each representing orthonormal states. 5 Time Dependence and Neutron-Antineutron nn oscillations The discovery of neutron-antineutron nn oscillations, in which a neutron spontaneously becomes an anti-neutron, could answer crucial questions of particle physics and cosmology. For example, why do we observe more matter than antimatter in the Universe? In the presence of n n osciullations the Schroedinger equation is given by i Ψ t = ( ) En α Ψ, (1) α E n where Ψ is tne neutron-antineutron state vector (a two-component spinor), E n is the neutron energy, E n is the anti-neutron energy, α is the mixing parameter predicted by totally new physics, yet to be discovered. In free space E n = E n = Mc 2 in the center of mass. In nuclear matter or under an external magnetic field these two energies differ. It is useful to define a quantity V 1 2 (E n E n ). (a) Suppose the system initially (t = 0) consists only of a neutron in the presence of magnetic or nuclear fields (V 0). Consider the transition probability P n (t) to generate anti-neutrons at t 0. Without doing a calculation, figure out what conditions P n (t) should respect. Two correct conditions is sufficient for an answer. (b) Suppose the system initially (t = 0) consists only of a neutron in the presence of magnetic or nuclear fields (V 0). Show the transition probability to generate anti-neutrons at t > 0 is given by P n n n = α2 α 2 +V sin 2 ( α 2 + V 2 t/ ). 2 (c) Suppose the following conditions are valid: αt/ 1 and also the system is in vacuum (there are no external fields). Determine P n (t). (d) In nuclei, the condition V t/ >> 1 holds. Determine P n (t). Comment on whether it is better to search for n n oscillations in free space or using nuclei. 1

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