(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.