(6-4) We've seen that if there are no forces acting, so the potential energy is everywhere zero, the the Schrödinger equation simplifies to:
① So the second derivative of ψ is the same as ψ itself, except that it is multiplied by a constant. What kind of functions behave like that?
② Consider a solution of the form ψ(x) = a sin(kx) + b cos(kx)
Insert this wave function back into the free particle Schrödinger equation above and hence confirm that it is, indeed, a solution.
③ Deduce an expression for the total energy, E, of the particle.
④ Recalling (from chapter 5) that k is the angular wavenumber, given by 2π/λ, use the de Broglie equation to confirm that your expression is consistent with the kinetic energy of the particle.