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AUGUSTANA PH 401 - Homework in Physics

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Homework assignments PHYS 401, 2010-11, part IIIDue Wednesday, Feb 2:1) You may leave your answers to the following questions in terms of .a) A photon is a spin-1 particle, in other words it has s=1. Find the magnitude of its spin angular momentum. b) The Delta-baryon has spin s=3/2. Find all possible values of the z-component of its spin.2) Given the commutation relations of the different components of spin on p 159,a) Show that S2 commutes with Sx.b) Prove that an electron can be in a simultaneous eigenstate of S2 and Sx.c) Prove that [Sx, SySz]= i(Sz2 - Sy2).3) Evaluate each of the following. Each answer should be a single term (not a sum or difference or commutator)a) [[Sx, Sy], Sz]b) [[Sx, Sy], Sx]Consider the following questions, to be discussed in class on Wednesday:A) Suppose that Schrödinger’s cat really is a quantum system. The eigenstates of its liveliness could be written as |alive> and |dead>. If we observe the cat, we always collapse its wavefunction into one of these eigenstates. Once we have collapsed the wavefunction, is there any way to un-collapse it? If we find the cat is dead, is there any way to get a second chance?B) Suppose two electrons are produced from a total-spin zero state. This means the sum of their spin vectors is zero, so in particular the sum of the z-components is zero. ms1+ms2=0. If one is spin-up, the other is spin-down. Further suppose that both spins areuncertain, each electron is in a superposition state of spin-up and spin-down. Now we measure the z-component of one spin, and find that it is spin-up, thus collapsing its wavefunction. Does the wavefunction of the other, unmeasured electron collapse? Why or why not?Due Wednesday, Feb 9:Problems 7.5, 8.11, also….1) On p 326, the answer to question 8.5 gives the spin-component operator matrices for a particle that has spin s=1 (rather than s=1/2 like an electron).a) Prove that 001, 010, and 100 are eigenstates of Sz and find the eigenvalues.b) Find a matrix representation of the spin state which is an eigenstate of Sx, with eigenvalue 0. Be sure to normalize it. You may assume that the first matrix element is real and positive (a common convention).c) If this particle is in the spin state which is an eigenstate of Sx, with eigenvalue 0, what is the probability of each of the possible values of Sz? (Do they add to 1? I hope so.)b) Find a matrix representation of the spin state which is an eigenstate of Sy, with eigenvalue . Be sure to normalize it. You may assume that the first matrix element is real and positive (a common convention).c) If this particle is in the spin state which is an eigenstate of Sx, with eigenvalue 0, what is the probability that an Sy measurement will yield ?Due Friday, Feb 11, full credit by noon on Monday, Feb 14 <3:Problems 11.2*, also….1) An oscillating electric field in the z-direction cannot change the value of m; can an oscillating electric field in the x-direction? In this case, Vpert=Ccos(t)x.2) A system is perturbed by an oscillating perturbing potential with frequency, . We are interested in the transition from an initial state with E=2eV to a final state with E=2.5 eV in a time of 2 fs. Consider the dependence of the transition probability in eqn 11.14.a) Evaluate this for =o. (This is the max/resonance)b) By what % is this smaller than the max if  is 10% larger thano?c) By what factor is this smaller than the max if  is 10 times larger thano?*use eqn

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