Physics 220 Homework #3 Spring 2015 Due Wednesday 4/26/15 1. A particle of mass m is moving in one dimension in a potential V ( x,t ) . The wave function for the particle is Ψ ( x,t ) = Axe and k are constants. ⎛ km ⎞ 2 ⎛3 k ⎞ −⎜ x −i⎜ t ⎝ 2! ⎠⎟ ⎝ 2 m ⎠⎟ e for −∞ < x < ∞ , where A a. Show that V is independent of t , and determine V ( x ) . b. Normalize the wave function and determine the constant A . c. Using the normalized wave function, calculate x , x 2 , p , and p 2 . 2. Determine which of the following one-dimensional wave functions represent state of definite momentum. For each wave function that does correspond to a state of definite momentum, determine the momentum. a. ψ ( x ) = eikx b. ψ ( x ) = xeikx c. ψ ( x ) = sin ( kx ) + i cos ( kx ) d. ψ ( x ) = eikx + e−ikx 3. Griffith’s problem 1.4. 4. Griffith’s problem 1.15. 5. ! Suppose that a wave function Ψ ( r,t ) is normalized. Show that the wave function ! eiθ Ψ ( r,t ) , where θ is an arbitrary real number, is also normalized. 6. Griffith’s problem 1.17 parts a – e. 7. Griffith’s problem 2.3 8. Suppose that ψ 1 and ψ 2 are two different solutions of the time-independent Schrodinger wave equation with the same energy E . a. Show that ψ 1 + ψ 2 is also a solution with energy E . b. Show that cψ 1 is also a solution of the Schrodinger equation with energy E . 9. A particle of mass m is moving in one dimension near the speed of light so that the p2 relation for the kinetic energy E = is no longer valid. Instead, the total energy is 2m given by E 2 = p 2 c 2 + m 2 c 4 . So, we can no longer use the Schrodinger equation. Suppose that the wave function for the particle Ψ ( x,t ) is an eigenfunction of the energy operator and an eigenfunction of the momentum operator, and also assume there is no potential energy V . Derive a linear differential equation for Ψ ( x,t ) . 10. Determine the odd solutions to the finite square well. Determine the energy of the single bound state with E < V0 . Normalize your solutions in each region to determine the unknown coefficient A in each region. Plot your solution for ψ 2 (x) . 11. Determine the normalization coefficients for the second energy state of the even solutions to the finite square well. That is, renormalize the solutions and determine B in each region for E3 . Plot your solution for ψ 3 (x) , along with the solutions for ψ 2 (x) from above and ψ 1 (x) from class. 12. Griffith’s problem 2.40 13. Griffith’s problem 2.47