Consider a charged scalar particle of mass m with charge q and describe a suitable modification of the derivative operator àμ →μ+qªμ that will yield a Lorentz invariant, real Lagrangian.
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- For a one dimensional system, x is the position operator and p the momentum operator in the x direction.Show that the commutator [x, p] = ihProblem 3: A mass m is thrown from the origin att = 0 with initial three-momentum po in the y direction. If it is subject to a constant force F, in the x direction, find its velocity v as a function of t, and by integrating v find its trajectory. Check that in the non-relativistic limit the trajectory is the expected parabola. Hint: The relationship F = P is still true in relativistic mechanics, but now p = ymv instead of p = mv. To find the non-relativistic limit, treat c as a very large quantity and use the Taylor approximation (1+ x)" = 1 + nx when a is small.Consider a spacetime diagram in which for simplicity, we will omit the space coordinate z. That is, only take into account three axes, two axes that define the xy plane characterized by the spatial coordinates x and y, and a third axis perpendicular to the xy plane that corresponds to the temporal coordinate t (ct so that the three axes have the same dimensions ). This coordinate system S is in which an observer O describes the events that occur in the Universe from his point of viewc) Now drop the spatial coordinate y. On the plane (x, ct) that describes the events seen by the observer O in his system S and considering the Lorentz transformations, draw the axes for a system S' in which an observer O' would describe the events that occur in the universe. We must create a space-time diagram for the scenario presented in the problem. Only the x, y, and ct-axis must be taken into consideration. Thus, we have our three dimensions (ct, x, y). We'll remain with relativistic units for the…
- Suppose that we want to solve Laplace’s equation inside a hollow rectangular box, with sides of length a, b and c in the x, y and z directions, respectively. Let us set up the axes so that the origin is at one corner of the box, so that the faces are located at x = 0 and x = a; at y = 0 and y = b; and at z = 0 and z = c. Suppose that the faces are all held at zero potential, except for the face atz=c,onwhichthepotentialisspecifiedtobeV(x,y,c)=V0 =const. a) Find the electrostatic potential V at a generic point inside the box.b) Find the expression for the electrostatic potential evaluated at the center of the box, i.e. deter- mine V (a/2, b/2, c/2). Simplify your answer as much as you can! c) Suppose now that a = b = c, i.e. the box is a cube. Give a simple argument which gives theexact (and simple) expression for the potential at the center of the cube. (No calculations are asked here. Use physics, wave your hands, etc. and say “the answer is such and such because ...”)A stationary observer O is standing on a platform of length 65 meters on earth. A rocket passes at a velocity of – 0.80c, parallel to the edge of the platform. The observer O notes that at a particular instant the front and back of the rocket simultaneously line up with the ends of the platform. (a) According to O, what is the time necessary for the whole rocket to pass a particular point on the platform? c = 3 × 10% m/s. (b) What is the rest length of the rocket according to an observer O' on the rocket? (c) According to O', what is the length of the platform? 65 m 0.8c O' Figure 2: Problem 4.[General Theory of Relativity]