- Comparing the values b n of question 3.3 with this experimental result, and recalling the result of a measurement of µ x for these values, explain why this proves that the state vector of a spin-1/2 particle changes sign under a rotation by an odd multiple of 2π.. - The intensities at the two counters are. - The fact that I 2 + I 3 does not depend on the phase shift δ is a consequence of the conservation of the total number of particles arriving at D.. - Section 3.2: The Gravitational Effect 3.2.1. - (a) Since there is no recoil energy of the silicon atoms to be taken care of, the neutron total energy (kinetic+potential) is a constant of the motion in all the process. - 2gH is of the order of 0.5 m/s, and the neutron velocity is v = h/M λ 2700 m/s. - (a) The gravitational potential varies in exactly the same way along AB and CD. - The neutron state in both cases is a plane wave with momen- tum p = h/λ just before A or C. - The same Schr¨ odinger equation is used to determine the wave function at the end of the segments. - This implies that the phases accumulated along the two segments AB and CD are equal.. - (b) When comparing the segments AC and BD, the previous reasoning does not apply, since the initial state of the neutron is not the same for the two segments. - The relative precision of the experiment was actually of the order of 10 − 3. - For δ = nπ the magnetic moment has rotated by 2nπ around the z axis by Larmor precession.. - The phase of the upper component of the spinor written in the. - z } basis, is shifted by +δ, that of the lower component by − δ:. - Altogether, we obtain the following intensities of the total neutron flux in the two counters:. - If δ = nπ, whatever the integer n, one is sure to find the neutrons in the same spin state as in the initial beam. - 10 − 4 T confirms that if the spin has rotated by 4nπ, one recovers a constructive interference in channel 3 as in the absence of rotation, while if it has rotated by (4n + 2)π, the interference in C 3 is destructive. - We shall analyse it in the specific case of a neutron beam, where it can be used to determine the neutron magnetic moment with high accuracy, by measuring the Larmor precession frequency in a magnetic field B 0. - for the eigenstates of the z projection ˆ S z of the neutron spin, and γ for the gyromagnetic ratio of the neutron: ˆ µ = γ S ˆ , ˆ µ being the neutron magnetic moment operator, and ˆ S its spin.. - The neutrons are initially in the state. - In all parts of the chapter, the neutron motion in space is treated classically as a linear uniform motion. - We are only interested in the quantum evolution of the spin state.. - What is the Hamiltonian ˆ H (t) describing the coupling of the neutron magnetic moment with the fields B 0 and B 1. - γB 1 , write the matrix representation of H ˆ (t) in the basis. - Treating B 1 as a perturbation, calculate, in first order time-dependent perturbation theory, the probability of finding the neutron in the state. - (far from the interaction zone) if it was in the state. - One measures the flux of neutrons which have flipped their spins, and are in the state. - Show that this probability has a resonant behavior as a function of the applied angular frequency ω. - as a function of the distance from the resonance ω − ω 0 . - How does the width of the resonance curve vary with v and a?. - The existence of this width puts a limit on the accuracy of the measurement of ω 0 , and therefore of γ. - On the path of the beam, one adds a second zone with an oscillating field B 1 . - Show that the transition probability P. - across the two zones can be expressed in a simple way in terms of the transition probability calculated in the previous question.. - Why is it preferable to use a setup with two zones separated by a distance b rather than a single zone, as in question 4.1.2, if one desires a good accuracy in the measurement of the angular frequency ω 0 ? What is the order of magnitude of the improvement in the accuracy?. - Suppose now that the neutrons, still in the initial spin state. - propagate along the z axis instead of the x axis. - Suppose that the length of the interaction zone is b, i.e. - that the oscillating field is given by (4.1) for. - In practice, the neutron beam has some velocity dispersion around the value v. - Which of the two methods described in questions 4.1.3 and 4.1.5 is preferable?. - Numerical Application: The neutrons of the beam have a de Broglie wavelength λ n = 31 ˚ A. - In order to measure the neutron gyromagnetic ratio γ n , one proceeds as in question 4.1.3. - One can assume that the accuracy is given by. - The most accurate value of the neutron gyromagnetic ratio is currently γ n. - where q is the unit charge and M p the proton mass. - be the neutron state at time t. - 1 in the integral. - The width of the resonance curve is of the order of v/a. - This quantity is the inverse of the time τ = a/v a neutron spends in the oscillating field. - h, when an interaction lasts a finite time τ the accuracy of the energy measurement δE is bounded by δE. - Therefore, from first principles, one expects that the resonance curve will have a width of the order of ¯ h/τ in energy, or 1/τ in angular frequency.. - In the two-zone case, the transition amplitude (in first order pertur- bation theory) becomes. - If we make the change of variables t = t − b/v in the second integral, we obtain. - which is the same formula as previously but multiplied by 1 + e i(ω 0 − ω)b/v . - However, owing to the extra oscillating factor, the half-width at half- maximum of the central peak is now of order πv/(2b). - The parameter which now governs the accuracy is the total time b/v that the neutron spends in the apparatus, going from one zone to the other.. - In spectroscopic measurements, it is important to locate the exact position of the maximum of the peak. - Multiplying the width of the peak by a factor a/b ( 1 since a b) results in a major improvement of the measurement accuracy. - The neutron (more generally, the particle or the atom) has some transition amplitude t for undergoing a spin flip in a given interaction zone. - The total amplitude T is the sum. - where e iφ is the phase shift between two zones.. - We now set z = vt, and x = y = 0 for the neutron trajectory. - This will modify the phase of the field (Doppler effect). - 2 , which has a width of the order of b/v but is centered at. - Comparing with question 4.1.2, we find that the resonance frequency is dis- placed: The neutron moves in the propagation direction of the field, and there is a first order Doppler shift of the resonance frequency.. - If the neutron beam has some velocity dispersion, the experimental result will be the same as calculated above, but smeared over the velocity distribution.. - In the method of question 4.1.3, the position of lateral fringes, and the width of the central peak, vary with v
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