A particle of mass m moves in a one-dimensional potential box for |z|>3|u|, for a 3Jal) of length 6a. Use the first order perturbation method to calculate the energy of the ground state. Vo -3a За
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- The difference of the scalar potential squared and the modulus of the vector potential squared, Φ2 - |A|2, is Lorentz invariant (a Lorentz scalar). why the statement true?The Brachistochrone Problem: Show that if the particle is projected withan initial kinetic energy 1/2 m v02 that the brachistochrone is still a cycloidpassing through the two points with a cusp at a height z above the initialpoint given by v02 = 2gz.Please obtain the same result as in the book.
- A particle experiences a potential energy given by U(x) = (x² - 3)e-x² (in SI units). (a) Make a sketch of U(x), including numerical values at the minima and maxima. (b) What is the maximum energy the particle could have and yet be bound? (c) What is the maximum energy the particle could have and yet be bound for a considerable length of time? (d) Is it possible for a particle to have an energy greater than that in part (c) and still be "bound" for some period of time? Explain. ResponsesHELP!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 ...”)