QUANTUM MECHANICS IPHYS 516Problem Set # 4Distributed: Mar. 3, 2010Due: Mar. 15, 20101. Coherent States and Minimum Uncertainty: A particle in a har-monic oscillator potential is in a coherent state |αi, for which |αi = e−α∗α/2eαa†|0iso that a|αi = α|αi (c.f., Ballentine pp. 541-548).a. Compute x = hxi = hα|x|αi.b. Compute ∆x2= h(x − x)2i.c. Compute ∆x.d. Compute p, ∆p2, ∆p.e. Compute ∆x∆p.2. Matrix Mechanics - Discretization: Discretize the Schr¨odinger equa-tion for a particle in a box of length L = 1. Set m = ¯h = 1. State explicitlywhat your step size is and the size of the matrix you 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).e. Compare these eigenvectors with the analytically available eigenvectors(c.f., Dicke & Wittke, Chap. 3). What are the similarities and differences?What can you say about signs and normalization?3. Matrix Mechanics - More discretization: Discretize the Schr¨odingerequation for a particle in a harmonic oscillator potential. Set m = k = ¯h =1. State explicitly what your step size is and the size of the matrix you arediagonalizing.a. Sort and plot all energy eigenvalues from small to large.b. Compare to the eigenvalues that can be obtained analytically.c. Which eigenfunctions might you believe and which would you definitelynot believe?1d. Plot the “lowest” five eigenfunctions (this means the eigenfunctions be-longing to the five smallest eigenvalues) (c.f., Ballentine, Chap. 6).e. Compute the matrix elements hn + 1|x|ni and hn + 1|p|ni for n = 0 usingthe numerical eigenvectors.4. Wave Mechanics - Matrix Mechanics Comparison:a. For the harmonic oscillator plot the five lowest eigenfunctions ψn(x) =Hn(x)e−x2/2/p2nn!√π, n = 0, 1, 2, 3, 4.b. Plot the numerical eigenfunctions from Problem 3d. on the same graph.c. Say something clever about the sign and normalization problems thatyou encounter.d. Compute the matrix elements hn + 1|x|ni and hn + 1|p|ni for n = 0 usingthe analytic eigenvectors. Compare with your answer to Problem 3e..5. Matrix Mechanics - More discretization: Discretize the Schr¨odingerequation for a particle in a bimodal potential V (x) =14x4− αx2. Set m = k =¯h = 1. Set α = 2. State explicitly what your step size is and the size of thematrix you are diagonalizing.a. Sort and plot all energy eigenvalues from small to large.b. Which eigenfunctions might you believe and which would you definitelynot believe?c. Plot the “lowest” six eigenfunctions (this means the eigenfunctions be-longing to the six smallest eigenvalues) (c.f., Ballentine, Chap. 6).d. Which of these eigenfunctions do you believe? Why?6. Matrix Mechanics All Over Again: The one-dimensional potentialV (x) is symmetric and it is desired to compute the eigenvalues and eigenfunc-tions of a particle confined by this potential. Set m = k = ¯h = 1. Choose aninterval −L ≤ x ≤ +L and choose basis functions coskπxLfor k = 0, 1, 2, ···, Nand sinkπxLfor n = 1, 2, ···, N .A. Choose V (x) =12x2.a. What is your reasonable choice of L? What value of N are you using?b. Compute the matrix elements hk|H|k0i in the cosine basis.c. Compute the eigenvalues of this matrix.d. Plot the four lowest eigenvectors.e. Do you believe these eigenfunctions? Why/not?f. If you don’t believe them, what must you do to compute a more believableset? Do it.g. Repeat these calculations in the sine basis.h. Why don’t you need to use the mixed sine/cosine basis?B. Choose V (x) =14x4− αx2.i. Repeat steps a. through g. for this bimodal