In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we exhibit various approximations of the electron distribution which facilitate the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could therefore also serve as an interesting pedagogical introduction to Feynman's formulation of quantum mechanics for university students dealing with the foundations of quantum mechanics.
The quantum mechanical problem of diffraction and interference of massive particles is discussed, though without detailed formulas, by Feynman in his famous lecture notes. 1 A more exact treatment, though still lacking in detail, is in his book with Hibbs. 2 It was first observed experimentally by Jönsson in 1961. 3 Moreover, there are also experiments for neutrons diffraction for single and double slit (see 4 and the references therein) and in quantum optics about interference between photons, (e.g., 5 ). The crucial point of this paper is to deal with different optical regimes: the usual Fraunhofer regime, and (less commonly taught to students) the Fresnel regime and intermediate regimes. Recall that these regimes depend on the distance between the slits and the screen, where the Fraunhofer regime corresponds to the case when the distance between the slits and the screen is infinite and the other regimes appear when this distance is finite, the intermediate and Fresnel regimes being distinguished by the value of the Fresnel number N F ≡ 2a 2 /λL, where 2a is the width of the slit, L is the distance between the screen and the slit and λ is the wave lenght of the electron. Thus, the purpose of this article is firstly to present the theory of the slit experiment using the Feynman path integral formulation of quantum mechanics, which may be of pedagogical interest as compared with the optical Young experiment (Section II, III, IV), secondly to give an analytical derivation of the final formulas for the intensity of the electron on the screen, and to analyse these formulas using some approximations based on the asymptotic behavior of the Fresnel functions 6 occurring in these expressions (Section V).
In particular we show how the physical parameters, especially the Fresnel number, affect the form of the diffraction and interference images. There exists some pedagogical papers about the multiple slit experiments (see e.g. 7 for experimental and numerical discussions and, 8 for theoretical discussions using path integral formulation), but surprisingly the approximations obtained in the Section V has never been published.
Students are often surprised to learn that under certain physical conditions the experimental behavior of matter is wave-like. In fact, this can present a didactical obstacle in teaching the first course of quantum mechanics where the principle of wave-particle duality can appear mysterious, especially with electron diffraction experiments for one-and two slits. Despite the interesting historical ramifications, students often have many metaphysical questions which are not answered satisfactorily in introductory quantum mechanics courses.
A formal and complete solution of the electron diffraction problem based on Feynman’s path integral approach may be helpful in this regard. It could demystify the diffraction experiment and clarify the principle of duality between wave and particle by analogy with wave optics. In addition, it could to be an interesting introduction to quantum mechanics via the Feynman approach based on the notion of path integral rather than the Schrödinger approach, highlighting the analogy between quantum mechanics and optics. Moreover, there is an interesting article that has recently been published in the European Journal of Physics Education that discusses using the Feynman path integral approach in secondary schools to teach the double slit experiment. 9 We begin by outlining Feynman’s formulation of quantum mechanics. We recall some of the fundamental equations before studying the electron diffraction problem. For more detail and for historical remarks about this theory, see Ref. 2 , 10 , 11 . We will give an equivalent formulation to that based on Schrödinger’s equation, but which is related to the Lagrangian formulation of classical mechanics rather than the Hamiltonian formulation. Consider a particle of mass m under the influence of an external potential V [x(t), t] where x(t) = (x (1) (t), x (2) (t), .., x (d) (t)) is the (d-dimensional, d ≥ 1) coordinate of the particle at time t.
The Lagrangian of the particle has the simple form
where ẋ(t) = dx(t)/dt denotes the velocity of the particle at time t. If the particle is at position x i at time t i and at x f at time t f > t i then the action is given by
In classical mechanics, the total variation of the action δS for small variations of paths at each point of a trajectory δx(t) is zero, which leads to the Euler-Lagrange equations well known to students of physics:
However, in quantum mechanics the Least Action Principle as described above is generally not true. Thus, the concept of classical trajectory is also no longer valid: the position as a function of time is no longer determined in a precise way. Instead, in quantum mechanics the dynamics only determines the probability for a particle to arrive at a position x f at time t f knowing it was at a given position x i at time t i < t f . In other words, knowing
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