Postulates of Quantum Mechanics

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1 EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 1: Stern-Gerlach experiment Postulates of Quantum Mechanics AStern-Gerlach(SG)deviceisabletoseparateparticlesaccordingtotheirspinalonga given axis. Consider a beam of spin 1/ particles. We call positive filter along the z axis a SG that selects spin +,whileameasurer SGẑ measures the value of the spin along the same axis. We perform the following experiments. i) We apply a positive filter along the z axis followed by a measure with SGẑ. ii) We apply a filter along z followed by a measurer SGˆx along the x axis. iii) We apply a positive filter along z followed by another positive filter along x, and we finally measure with SGẑ. Discuss the result in each case. Exercise : Commutation and diagonalization of operators Show that if two operators commute, they can be simultaneously diagonalized. Can they be simultaneously diagonalized if they anticommute, i.e., {A, B} AB + BA =0? Exercise 3: Uncertainty relations i) Given two observables A and B, deducetheuncertaintyrelation ψ A ψ B 1 ψ [A, B] ψ, where the expectation value is A ψ ψ A ψ and the uncertainty (average quadratic dispersion) is ψ A [ ψ (A A ψ ) ψ ] 1/ =[ A ψ A ψ ]1/. ii) Use the commutation relation [X, P] =i I to prove Heisenberg s relation ψ X ψ P. iii) Derive the energy time uncertainty relation τ ψ ψ E, where τ ψ ψ A/ d A ψ dt is the time scale for a change in the observable A and ψ E is the uncertainty in the energy of ψ.

2 EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 4: Eigenvalues, eigenvectors, expectation value and density matrix The Hamiltonian H and the physical observables A and B are given by the matrices A = H = ω λ 0 λ λ, B = where λ and µ are real numbers different from zero. µ µ 0 µ 0 i) Find the eigenvalues and eigenvectors of H, A and B. ii) Determine all the subsets of operators that define a CSCO and find in each case the common basis of eigenstates., iii) Consider a system in the state ψ = c 0 i. (a) Find the normalization constant c. (b) What are the possible values of the energy and their probability when we measure it to ψ?whatisthestateafterthemeasureineachcase? (c) Find the expectation values of H, A and B for the state ψ. Whatistheuncertainty in the energy? iv) Consider a pure ensemble with all its elements in the state ψ. (a) Find the density matrix that describes this ensemble. (b) Find the density matrix that describes the collectivity after the measurement of the energy to all its elements.

3 EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 5: Test exercise, September 014 [3P] Consider a system in the state ψ and the observables A y B given in a certain basis by 1 ψ = c 0 ; A = ; B = 0 0 0, where c in the normalization constant. i) If we measure A and right after that we measure B, findtheprobabilitytoobtain the values 1 and 1, respetively. ii) Find the probability to obtain the same values 1 and 1 if the sequence is taken in the opposite order, first we measure B and then A. iii) In a pure statistical ensamble with all its elements in the state ψ we measure A and B in a random order, and we select those states where we find 1 and 1 disregarding any other possibility. Find the density matrix thatdescribesthe ensamble that we have selected. Exercise 6: Baker-Campbell-Hausdorff formula and quatization rules Consider two operators A and B that satisfy [A, [A, B]] = [B,[A, B]] = 0 and a function f that admits a Taylor expansion. i) Show that [A, f(b)] = [A, B] df(b) db. ii) If G(t) =e ta e tb being t arealvariable,applythepreviousresultto dg(t) and dt prove the relation e A e B = e A+B+[A,B]/. iii) Consider the quantization rules in the Weyl form: U α V β = e i δ ijαβ V β U α, where U α e i αx i,v β e i βp j, with α and β real, X i the position cartesian coordinates and P j the conjugate momenta. Using the previous result, show that the infinitesimal formoftheserules correspond to the canonical quantization rules [X i,p j ]=i δ ij I.

4 EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 7: Neutrino oscillations Consider two families of neutrinos with a non-zero mass. Let us assume that the neutrinos that are produced through weak interactions, ν e and ν µ,donotcoincidewiththe mass eigenstates, ν 1 and ν : ( ν e ν µ ) = ( cos θ sin θ sin θ cos θ Suppose that at t =0asourceproducesaneutrinoν e of momentum p, ψ(0) = ν e, i) Solve Schrödinger equation for ψ(t) if the energy of a state of well defined mass and momentum is E i = p c + m i c4 pc )( ν 1 ν ) (1+ m i c ii) What is the probability to observe an electron neutrino ν e at a distance L from the source? And a muon neutrino ν µ? iii) What is the optimal distance from the source to observe these neutrino oscillations? p. ). Exercise 8: The neutral kaon system Neglecting CP violation, the interaction eigenstates K 0 and K 0 of the neutral kaons can be expressed in terms of the mass eigenstates K S y K L as K 0 = 1 ( K S + K L ), K 0 = 1 ( K S + K L ). These two unstable states can be described by an effective hamiltonian that is not hermitian and has then complex eigenvalues: ES 0 = m S c i Γ S, EL 0 = m L c i Γ L, where ES,L 0 are the energy eigenvalues at rest, m S,L the masses, Γ S,L = /τ S,L the decay widths and τ S,L the lifetimes of K S,L.Ifatt =0we produce a kaon at rest in the state ψ(0) = K 0,

5 EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 i) What is the probability to find the kaon in the same state K 0 after a time t?andin the state K 0? ii) Is the addition of both probabilities constant? iii) Discuss the similarities and differences between these kaons and the neutrino system studied in the previous exercise. Exercise 9: Test exercise, February 011 [.5P] Consider a quantum system associated to a 3-dimensional Hilbert space with basis { u 1, u, u 3 }.ThematricesoftheHamiltonianH and of the observable A in this basis are H = ω ; A = a The initial state of the system is ψ(0) = c ( u u + 1 u 3 ),wherec is a normalization constant. i) Find the expectation value of A in that instant. What sthe probability to obtain a if we measure A? ii) Suppose that we measure A and obtain a. Whatvaluesmaywefindifwemeasure the energy to the resulting state? With what probability? iii) Determine this state (with measured value a) ataanarbitrarytimet. Doesthe expectation value of the energy change with time? And the expectation value of A? Exercises 4-9 must be turned in by the end of the semester. Exercises will be discussed in class. Solution to exercises -6-8:

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