(6-6)  We have our wave functions for the particle in a box:

Now, the Schrödinger equation is going to give us their energies:

We have agreed that ψ(x) is zero in the region where the potential energy would be infinite, in the walls. In the allowed region, between the walls, the potential is zero and so the V(x)-dependent term vanishes and the equation again simplifies to the one we had for a free particle. Consequently, the energy expression we derived in Question 6-4 still applies: 

However the boundary conditions have now restricted the possible values of k.

①  Use the wave function expression you derived in Question 6-5, with its implication for the possible values of k, and combine it with the energy expression you derived in Question 6-4, to write down an expression for the energies of the possible wavefunctions of our particle in a box.

②  Explain how the expression you have found correlates with the idea of localisation energy which we derived from the uncertainty principle in Chapter five.

③  Consider the case where n=0. Why does this not give a possible wavefunction?  What are the implications for the possible energies of the particle?

④  Looking at the wavefunction shapes and thinking about our discussion of the kinetic energy term in the Schrödinger equation, explain why the energy increases as the value of n increases.

⑤  What is the relationship between the number of nodes (points where the wavefunction goes to zero – not counting the ends) and the value of n in the wavefunction?