1. a) What happens to a wavefunction at a node?

b) What are the general geometric shapes of nodes for wavefunctions in 1, 2 and 3 dimensions?

c) The slope of a wavefunction is discontinuous at a certain position. What causes this?

d) A wavefunction has cusps at 3 different positions in space. Name a system for which this is true?

2) a) Where are the nodes for a particle in the first excited state of a 1-d box positioned from 0 to L?

b) Why does it make physical sense that eigenstates of energy for a particle in a box are superpositions of momentum states?

c) Write the eigenstates for 2 non-interacting particles in a 1D box using the particle in a box wavefunctions (you do not need to derive them).

d) What are the quantum numbers for a particle in a 3D box?