Let i = √(V-I), L gt; 0 and 0 ∈ C([0,L]) such that 0(0) = 0(L) = 0. The mathematical model known as the particle in a box is described by the initial value problem for the (free) Schrödinger equation with Dirichlet boundary conditions: UT = c^2 ∂^2u/∂t^2 + ∂^2u/∂x^2 = 0, t ∈ [0,L], x ∈ [0,L] (Sch) u(0,t) = u(L,t) = 0, t ∈ [0,L] (BC) By using the method of separation of variables, show that the formal solution of (Sch) is given by: u(t,x) = Σ bn sin(nπx/L) exp(-i(cπn/L)^2t), t ∈ [0,L], x ∈ [0,L]; where bn = ∫[0,L] u(0,x) sin(nπx/L) dx, n ∈ N. Assume that |bn| ≤ C/n^2 for some constant C gt; 0. (17) Use the Weierstrass criterion to show that the function given in (16) belongs to C([0,L]) and that it is actually a solution of (Sch). Assume (17). Prove that ∀t ∈ [0,L]: ∫[0,L] |u(t,x)|^2 dx = 2 ∑ |bn|^2, and use this fact to show that the problem (Sch) is well-posed, i.e., that (Sch) has a unique solution and depends continuously on the initial data.
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