"Consider the time-dependent Schrodiner equation ..." 1. The problem statement, all variables and given/known data Consider the time-dependent Schrodinger equation ih2ψt = [-h2/(2m)]ψxx + V(x)ψ which is the underlying equation of quantum mechanics. Here V(x) is a given potential, h is the Planck's constant, and m is the mass of the particle. ψ(x,t) is the amplitude of the wave that the particle traces out. i=√(-1) is the imaginary unit. (a) Use the separation of variable ψ(x,t)=u(x)exp(-iEt/h) to derive the time-independent Schrodiner equation which governs u(x). Show that the resulting equation is a Sturm-Louiville eigenvalue problem with p(x) = h2/(2m), q(x) = V(x), r(x) = 1, λ=E. The eigenvalue E represents the energy of the particle. (b) Solve the Sturm-Louiville eigenvalue problem in the domain -1 < x < 1 with zero potential and homogenous Dirichlet boundary conditions. Sketch the ground state (the lowest non-zero energy state). What is the energy? (c) Without further calculation, explain what would happen to the eigenfunctions and eigenvalues if the domain is cut in half, i.e. 0 < x < 1, with new boundary conditions u(0)=u(1)=0. 2. Relevant equations A Sturm-Louville eigenvalue problem has the form -p(x)u''(x)-p'(x)u'(x)+q(x)=λr(x)u(x) 3. The attempt at a solution Part (a) is trivial. For part (b), doesn't "zero potential" mean V(x)=0, in which case I seem to get a trivial solution?