Math 115 Spring 2015 Problem Set 7 Prof. Bernoff Separation of Variables Due date: Wednesday, March 11th in class. For these problems, you may use MAPLE when appropriate. G1: Suppose that the temperature, u(x, t), of a bar of unit length satisfies the heat equation, ut = Duxx , 0 < x < 1, 0<t where D is the thermal diffusivity. Also, suppose that initially the temperature is zero throughout the bar, that the temperature at the left end is fixed and at t = 0 the temperature at the right end is raised to unity u(x, 0) = 0, u(0, t) = 0, 0<x<1 u(1, t) = 1, 0 < t. (a) Solve for a steady-state solution us (x) that satisfies the heat equation and the boundary conditions. (b) Show that v(x, t) = u(x, t) − us (x) satisfies a homogeneous heat equation with homogeneous boundary conditions and non-zero initial conditions. (c) Solve for v(x, t) using separation of variables. Use this and (b) to write the complete solution for u(x, t). (d) Graph the solution at some sample times. Describe the behavior as t → ∞. G2: In this problem we will first consider an eigenvalue problem and then a separation of variables problem from which it might arise. (a) Consider the eigenvalue problem −y 00 (x) = λy(x), 0<x<1 y(0) = 0; y 0 (1) = 0, with mixed Dirichlet and Neumann boundary conditions. Show that the eigenvalues for the problem are real and positive. Show that if two eigenfunctions are associated with different eigenvalues, that they are orthogonal (using the usual L2 inner-product). Finally, find the eigenvalues and corresponding eigenfunctions explicitly. (b) Consider a bar of unit length with one insulated end whose temperature u(x, t) satisfies the heat equation: ut = uxx 0<x<1 u(0, t) = ux (1, t) = 0 Initially the temperature is given by u(x, 0) = x. Use separation of variables to solve for u(x, t) as an infinite series in terms of the eigenfunctions you found above with timedependent coefficients. Make sure you find explicit expressions for all the coefficients. (please turn over) Math 115 Spring 2015 Problem Set 7 Prof. Bernoff G3: Consider a bar of length π with insulated ends. At t = 0 a spatial heating that varies like cos(x) is turned on. The temperature u(x, t) satisfies the inhomogeneous heat equation, ut = uxx + cos(x) 0 < x < π, t > 0. with boundary conditions ux (0, t) = ux (π, t) = 0 t > 0. The temperature is initially given by u(x, 0) = 0 0 < x < π. Solve for u(x, t) and graph your solution at various times. Make sure you identify the limiting behavior as t → ∞. G4: The time-dependent version of Schr¨odinger’s equation for the wavefunction Ψ(x, t) is h ¯2 Ψxx + V (x)Ψ = i¯hΨt − 2m where V (x) is an external potential. If we look at an infinite square well potential, ( 0 when 0 < x < L, V (x) = ∞ when x ≤ 0 or x ≥ L , we get the equivalent problem for Ψ(x, t) i¯h Ψxx 0 < x < L, 2m Ψ(0, t) = 0, Ψ(L, t) = 0 Ψt = (1) (2) because the probability density must drop to zero outside the potential well. In addition, because |Ψ(x, t)|2 is a probability density, it satisfies Z L |Ψ(x, t)|2 dx = 1. (3) x=0 (a) The stationary states of the infinite square well satisfy (1, 2) take the form Ψ(x, t) = ψn (x)e−iωn t where the function ψn (x) and the angular frequency ωn are real. In addition, the stationary states should satisfy (3). Find the stationary states of the infinite square well. (please turn over) Math 115 Spring 2015 Problem Set 7 Prof. Bernoff (b) The ground state of the system is the stationary state, Ψg , with the lowest energy value, E=h ¯ ω. Show that r πx 2 h ¯ π2 sin e−iωg t ωg = Ψg (x, t) = L L 2mL2 (c) Suppose the system is initially in the ground state, Ψg and at time t = 0 the well is suddenly expanded to be twice as wide. The wave function now satisfies the initial value problem i¯h Ψxx 0 < x < 2L, t > 0 2m Ψ(0, t) = 0, Ψ(2L, t) = 0 t>0 (q 2 sin πx when 0 < x < L, L L Ψ(x, 0) = 0 when L < x < 2L , Ψt = Solve for Ψ(x, t) using the ideas of separation of variables and superposition. Your answer should take the form Ψ(x, t) = ∞ X cp ψp (x)e−iωp t p=1 where ψp (x)e−iωp t are stationary states of the interval of length 2L. Hint: Be careful with the first few integrals for cp . (d) The total probability density in the interval 0 < x < 2L is conserved, that is Z 2L |Ψ(x, t)|2 dx = 1. x=0 what does this tell you about the cm ’s ? Verify this identity numerically. (e) Show that the wave function is periodic in time. It may help to let T = ω1 t where ω1 = h ¯ π2 8mL2 Why does this mean that the particle is localized in the left side of the well (where 0 < x < L) infinitely often? (f) (Harder) Show that the particle is also localized in the right side of the well (where L < x < 2L) infinitely often also. Explain why this suggests the probability density is sloshing from side to side of the box.