- Relative accuracy ∆ω/ω of a fountain atomic clock as a function of the number of atoms N sent in each pulse. - With its own (quartz) clock, it compares the times at which different “clicks”. - What is the minimum number of satellites that one must see at a given time in order to be able to position oneself in latitude, in longitude, and in altitude on the surface of the Earth?. - What is the order of magnitude of the accuracy of the positioning just before the clocks undergo a new synchronization?. - Some cosmological models predict a (small) variation in time of the fine struc- ture constant α = e 2. - By comparing a rubidium and a cesium clock at a one year interval, no sig- nificant variation of the ratio R = ω 0 (Cs) /ω (Rb) 0 was observed. - Section 2.1: Hyperfine Splitting of the Ground State. - The Hilbert space of the ground state is the tensor product of the electron spin space and the nucleus spin space. - Energy levels of the hyperfine Hamiltonian.. - The determination of the eigenvalues of ˆ H therefore consists in diagonalizing the series of 2 × 2 matrices corresponding to its re- striction to these subspaces. - The matrix corresponding to the restriction of ˆ H to the subspace E m n is the same as given in the text.. - The eigenvalues given in the text are actually independent of m n . - In the case s n = 1/2 (hydrogen atom), these two eigenvalues are A/4 and − 3A/4.. - We do recover the dimension 2(2s n + 1) of the total spin space of the ground state.. - The square of the total spin is:. - Section 2.2: The Atomic Fountain. - In the limit → 0, the final state vector of the atom is simply the matrix product:. - which corresponds to crossing the cavity, at time t = 0, then to a free evolution between t = 0 and t = T , then a second crossing of the cavity at time t = T.. - We therefore obtain the state vector of the text.. - Since the atoms are assumed to be independent, the distributions of the random variables N 1 and N 2 are binomial laws. - 2.2 that the accuracy of the clock improves like N − 1/2 , as N increases. - For N = 10 6 and T = 0.9 s, the above formula gives s. - Section 2.3: The GPS System. - With two of them, the difference between the two reception times t 1 and t 2 of the signals localize the observer on a surface (for instance on a plane at equal distances of the two satellites if t 1 = t 2. - three satellites localize the observer on a line, and the fourth one determines the position of the observer unambiguously (provided of course that one assumes the observer is not deep inside the Earth or on a far lying orbit).. - If the clock of the satellite has drifted, the signal is not sent at time t 0 , but at a slightly different time t 0 . - an error on the position of 2.5 meters.. - Section 2.4: The Drift of Fundamental Constants. - Using the expression given in the text for the dependence on α of the frequen- cies ω Cs and ω Rb , we find that a variation of the ratio R would be related to the variation of α by:. - If we extrapolate this variation time to a time of the order of the age of the universe, this corresponds to a variation of 10 − 4 . - Remark: a more precise determination of the α dependence of ω Cs , for which the approximation Zα 1 is not very good, gives for the quantity inside the bracket a value of 0.45.. - In the late 1970s, Overhauser and his collaborators performed several neutron interference experiments which are of fundamental importance in quantum mechanics, and which settled debates which had started in the 1930s. - We study in this chapter two of these experiments, aiming to measure the influence on the interference pattern (i) of the gravitational field and (ii) of a 2π rotation of the neutron wave function.. - For a particular value of the angle of incidence θ, called the Bragg angle, a plane wave ψ inc = e i( p·r− Et. - h , where E is the energy of the neutrons and p their momentum, is split by the crystal into two outgoing waves which are symmetric with respect to the perpendicular direction to the crystal, as shown in Fig. - |p | since the neutrons scatter elastically on the nuclei of the crystal. - In the interferometer shown in Fig. - The neutron beam actually corresponds to wave functions which are quasi- monochromatic and which have a finite extension in the transverse directions.. - In order to simplify the writing of the equations, we only deal with pure monochromatic plane waves, as in (3.1).. - The measured neutron fluxes are proportional to the intensities of the waves that reach the counters. - Defining the intensity of the incoming beam to be 1 (the units are arbitrary), write the amplitudes A 2 and A 3 of the wave functions which reach the counters C 2 and C 3 , in terms of α and β (it is not necessary to write the propagation terms e i( p·r− Et. - Calculate the measured intensities I 2 and I 3 in terms of the coefficients T and R.. - Suppose that we create a phase shift δ of the wave propagating along AC, i.e. - in C the wave function is multiplied by e iδ. - The phase difference δ between the beams ACD and ABD is created by rotating the interferometer by an angle φ around the direction of incidence.. - This creates a difference in the altitudes of BD and AC, which both remain horizontal, as shown in Fig. - The difference in the gravitational potential energies induces a gravitational phase difference.. - Show that the side L of the lozenge ABCD and its height H, shown in Fig. - (a) Calculate the difference ∆p of the neutron momenta in the beams AC and BD (use the approximation ∆p p). - Express the result in terms of the momentum p along AC, the height H , sin φ, M , and the acceleration of gravity g.. - Evaluate the phase difference δ between the paths ABD and ACD.. - (a) Compare the path difference between the segments AB and CD.. - (b) Compare the path difference between the segments BD and AC.. - The variation with φ of the experimentally measured intensity I 2 in the counter C 2 is represented in Fig. - Deduce from these data the value of the acceleration due to gravity g.. - The plane of the setup is now horizontal. - The phase difference arises by placing along AC a magnet of length l which produces a constant uniform magnetic field B 0 directed along the z axis, as shown in Fig. - 3.5: the beam is along the y axis, the z axis is in the ABCD plane, and the x axis is perpendicular to this plane.. - We neglect any transient effect due to the entrance and the exit of the field zone.. - The incident neutrons are prepared in the spin state. - which is the eigenstate of ˆ µ x with eigenvalue +µ 0 . - (a) Write the magnetic interaction Hamiltonian of the spin with the magnetic field.. - (b) What is the time evolution of the spin state of a neutron in the magnet?. - h, calculate the three components of the expecta- tion value µ ˆ in this state, and describe the time evolution of µ ˆ in the magnet.. - When the neutron leaves the magnet, what is the probability P x (+µ 0 ) of finding µ x = +µ 0 when measuring the x component of the neutron magnetic moment? For simplicity, one can set T = M lλ/(2π¯ h) and express the result in terms of the angle δ = ωT /2.. - For which values b n = nb 1 (n integer) of the field B 0 is this probability equal to 1? To what motion of the average magnetic moment do these values b n correspond?. - Write the state of the neutrons when they arrive on C 2 and C 3 (note p 2 and p 3 the respective momenta).. - Express the difference of intensities I 2 − I 3 in terms of δ and of the coefficients T and R.. - The experimental measurement of I 2 − I 3 as a function of the applied field B 0 is given in Fig. - A numerical fit of the curve shows that the distance between two maxima is ∆B = (64 ± 2. - Difference of counting rates (I 2 − I 3 ) as a function of the applied field
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