QUANTUM MECHANICS IPHYS 516Problem Set # 4Distributed: Feb. 7, 2011Due: Feb. 14, 20111. Elementary Wave Mechanics: Place a particle of mass m in a one-dimensional box (infinitely deep potential) of length L. Set m = ¯h = 1 andL = 10.a. Compute the lowest 5 eigenstates and their energy eigenvalues.b. Normalize the eigenfunctions correctly. (What does that mean?)c. Plot the five lowest eigenvectors.2. Matrix Mechanics - Discretization and Analytic Solution: Dis-cretize the Schr¨odinger equation for a particle in a box of length L = 10. Setm = ¯h = 1. State explicitly what your step size is and the size of the matrixyou are diagonalizing.a. Show that the discrete equation consists of a tridiagonal matrix. Whatis this matrix?b. Plot the eigenvalue spectrum for this discrete problem with the analyticsolutions you computed in Problem #1. You can use the eigenfunctions weconstructed in Class. Point out where the numerical and analytic solutionsbegin to diverge.c. Plot the five lowest eigenfunctions.d. How do you normalize these eigenfunctions?3. Matrix Mechanics - Discretization and Numerical Solution:Discretize the Schr¨odinger equation for a particle in a box of length L = 10. Setm = ¯h = 1. State explicitly what your step size is and the size of the matrixyou are diagonalizing.a. Sort and plot all energy eigenvalues.b. Compare to the eigenvalues that can be obtained analytically.c. Which eigenfunctions might you believe and which would you definitelynot believe?d. Plot the “lowest” five eigenfunctions (this means the eigenfunctions be-longing to the five smallest eigenvalues).1e. Compare these eigenvectors with the analytically available eigenvectors(c.f., your solutions to Problem #1). What are the similarities and differences?What can you say about signs and normalization? How do you normalize thenumerical eigenfunctions so you can compare them with the analytic eigenfunc-tions?4. Wave Mechanics - Harmonic Oscillator and Analytic Solution:Place a particle of mass m in a one-dimensional harmonic oscillator potential.Set m = k = ¯h = 1.a. Write down the energy eigenvalues for the five lowest states.b. Write down the eigenfunctions with the five lowest energy engenvalues.Be sure the get the normalization correct.c. Plot the five lowest eigenfunctions ψn(x).5. Matrix Mechanics - More discretization and Numerical Solu-tion: Discretize the Schr¨odinger equation for a particle in a one-dimensionalharmonic oscillator potential. Set m = k = ¯h = 1. State explicitly what yourstep size is and the size of the matrix you are diagonalizing.a. Sort and plot all energy eigenvalues from small to large.b. Compare to the eigenvalues that can be obtained analytically (c.f., Prob-lem # 4).c. Which eigenfunctions might you believe and which would you definitelynot believe?d. Plot the five “lowest” eigenfunctions (this means the eigenfunctions be-longing to the five smallest eigenvalues).e. Compare these eigenvectors with the analytically available eigenvectors(c.f., your solutions to Problem #4). What are the similarities and differences?What can you say about signs and normalization? How do you normalize thenumerical eigenfunctions so you can compare them with the analytic