Question

4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is represented by the wavefunction v(x) 0, otherwise a. Sketch the wavefunction. Write down the (time independent) Schrodinger equation. Show whether or not the wavefunction is a solution to the equation. b. What does it mean physically if the wavefunction of the particle is NOT a solution to the Schrodinger equation? Explain. c. Determine the normalization constant A. 5. Same system. Find the average or expectation value of the energy E. a. b. Find the average or expectation value of the position x. c. Find the average or expectation value of the momentum p. 6. Same system. a. Find the average or expectation value of x2 b. Find the average or expectation value of p2 c. Find the uncertainty or standard deviation in x, ??. 7. Same system. a. b. c. Find the uncertainty or standard deviation in p, ??. Show whether your results are consistent with the Uncertainty Principle. What would it mean if your results were NOT consistent with the Uncertainty Principle? Explain.

Answer 1

"4(x) = {4(a^2-x^2);-a lessthanorequalto x lessthanorequalto a 0;otherewise a) H psi = E psi H = -h^2/2m d^2/dx^2 hence(-h^2/2m d^2/dx^2) .[4(a^2-x^2)] = E[A(a^2-x^2)] 1 complete Schrodinger equation (Time independent) solving 1, we get + h^2/2m (2A) = E psi implies h^2A/m = E psi implies psi (x) is not the solution of Schrodinger equation b) As wave function is not the solution of Schrodinger Equation, it means that the solution is not the normal mode. However, there can be more solutions to full Schrodinger equation, which can't i.e described by a single normal mode solution can be described by linear combination of normal modes. C) integral^+ a_-a psi (x) psi (x) dx = integral^+ a_-a(psi (x))^2 dx = 1 implies