- However the position of the central peak does not depend on the velocity, and it is therefore not shifted if the neutron beam has some velocity dispersion.. - On the contrary, in the method of question 4.1.5, the position of the cen- tral peak depends directly on the velocity. - A dispersion in v will lead to a corresponding dispersion of the position of the peak we want to measure.. - Actually, one can improve the accuracy considerably by analysing the shape of the peak. - 5.1 Preparation of the Neutron Beam. - A beryllium crystal acts as a filter to eliminate harmonics, and the vertical ex- tension of the beam is controlled by two gadolinium blocks, which are opaque to neutrons, separated by a thin sheet of (transparent) aluminum of thickness a, which constitutes the collimating slit, as shown in Fig. - Preparation of the neutron beam. - One observes the impacts of the neutrons on a detector at a distance L = 1 m from the slit. - The vertical extension of the beam at the detector is determined by two factors, first the width a of the slit, and second the diffraction of the neutron beam by the slit. - We recall that the angular width θ of the diffraction peak from a slit of width a is related to the wavelength λ by sin θ = λ/a. - For simplicity, we assume that the neutron beam is well collimated before the slit, and that the vertical extension δ of the beam on the detector is the sum of the width a of the slit and the width of the diffraction peak.. - What is the corresponding width of the beam on the detector?. - What is the observed width of the beam at the detector?. - Comment on the respective effects of the slit width a and of diffraction, on the vertical shape of the observed beam on the detector?. - The extension of the beam corresponds to the distribution of neutron im- pacts along the z axis. - Since the purpose of the experiment is not only to observe the beam, but also to measure its “position” as defined by the maxi- mum of the distribution, what justification can you find for choosing a = 5 µm?. - Figure 5.2 is an example of the neutron counting rate as a function of z.. - The horizontal error bars, or bins, come from the resolution of the measur- ing apparatus, the vertical error bars from the statistical fluctuations of the number of neutrons in each bin. - Measurement of the beam profile on the detector. - 5.2 Spin State of the Neutrons. - both its spin state and its spatial state, we consider the eigenbasis of the spin projection along the z axis, ˆ S z , and we represent the neutron state as. - where the respective probabilities of finding the neutron in the vicinity of point r with its spin component S z. - h/2 when measuring S z irrespective of the position r. - the expectation value of the x com- ponent of the neutron spin S x in the state | ψ(t). - What are the expectation values of the neutron’s position r and momentum p in the state | ψ(t). - We assume that the state of the neutron can be written:. - We assume that the components of the magnetic field are B x = B y = 0 B z = B 0 + b z . - b y and B y B z over the region of space crossed by the neutron beam), one can settle this matter, and arrive at the same conclusions.. - Magnetic field setup in the Stern–Gerlach experiment. - The magnetic moment of the neutron ˆ µ in the matrix representation that we have chosen for | ψ is. - Hereafter, we denote the neutron mass by m.. - What is the form of the Hamiltonian for a neutron moving in this magnetic field?. - Show that the Schr¨ odinger equation decouples into two equations of the Schr¨ odinger type, for ψ + and ψ − respectively.. - We assume that, at t = 0, at the entrance of the field zone, one has. - and that r = 0, p y = p z = 0 and p x = p 0 = h/λ, where the value of the wavelength λ has been given above.. - The above conditions correspond to the experimental preparation of the neutron beam discussed in Sect. - Give the physical interpretation of the result, and explain why one observes a splitting of the initial beam into two beams of relative intensities | α. - Express the result in terms of the kinetic energy of the incident neutrons (we recall that L = 1 m and b = 100 T/m).. - Given the experimental error δz in the measurement of the position of the maximum intensity of a beam, i.e. - δz m as discussed in question 5.1.3, what is the accuracy on the measurement of the neutron magnetic mo- ment in such an experiment, assuming that the determination of the magnetic field and the neutron energy is not a limitation? Compare with the result of magnetic resonance experiments:. - In the same experimental setup, what would be the splitting of a beam of silver atoms (in the original experiment of Stern and Gerlach, the atomic beam came from an oven at 1000 K) of energy E J? The magnetic moment of a silver atom is the same as that of the valence electron. - Show that, quite generally, in order to be able to separate the two outgoing beams, the condition to be satisfied is of the form. - Section 5.1: Preparation of the Neutron Beam. - The spreading of the beam on the detector is then equal to the Heisenberg minimum δ = 2. - In other words, the spreading of the wave packet, which increases as a decreases com- petes with the spatial definition of the incoming beam.. - The reason for making this choice is that the shape of the diffraction peak is known and can be fitted quite nicely. - Therefore, this is an advantage in determining the position of the maximum. - However, one cannot choose a to be too small, otherwise the neutron flux becomes too small, and the number of events is insufficient.. - Section 5.2: Spin State of the Neutrons 5.2.1. - 2 is the probability density of finding the neutron at point r. - Section 5.3: The Stern–Gerlach Experiment 5.3.1. - The matrix form of the Hamiltonian is. - If we write it in terms of the coordinates ψ ± we obtain the uncoupled set i¯ h. - Since both ψ + and ψ − satisfy Schr¨ odinger equations, and since ˆ H ± are both hermitian, we have the usual properties of Hamiltonian evolution for ψ + and ψ − separately, in particular the conservation of the norm. - The quantities A + and A − are the expectation values of the physical quantity A, for neutrons which have, respectively, S z. - Consequently, the expectation values of the vertical positions of the neutrons which have µ z = +µ 0 and µ z. - µ 0 diverge as time progresses: there is a separation in space of the support of the two wave functions ψ + and ψ. - The intensities of the two outgoing beams are proportional to | α. - µ 0 b | L 2 /2E where E is the energy of the incident neutrons.. - The error on the position of each beam is δz = 5 µ m, that is to say a relative error on the splitting of the beams, or equivalently, on the measurement of µ 0. - Hence, in the same configuration, one would obtain, for the same value of the field gradient and the same length L = 1 m, a separation ∆z = 3.4 cm, much larger than for neutrons. - by an appro- priate inspection of the line shape, one may lower this limit). - This is nothing but one of the many forms of the time–energy uncertainty relation. - The right-hand side is not the standard ¯ h/2 because we have considered a rectangular shape of the incident beam (and not a Gaussian). - (a) First, it shows that the effort that counts in making the experiment feasible is not to improve individually the magnitude of the field gradient, or the length of the apparatus, etc., but the particular combination of the product of the energy transferred to the system and the interaction time of the system with the measuring apparatus.. - (b) Secondly, this is a particular example of the fundamental fact stressed by many authors 2 that a measurement is never point-like. - The Stern–Gerlach experiment is actually a very good example of a measuring apparatus in quantum mechanics since it transfers quantum information – here the spin state of the neutron – into space–time accessible quantities – here the splitting of the outgoing beams.. - Here it emerges as a consequence of the spreading of the wave packet. - It is a simple and fruitful exercise to demonstrate rigorously the above property by calculating directly the time evolution of the following expectation values:
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