1. Funny Quantum energy bank: you can get a no interest loan ΔE for a certain time Δt as long as ΔE Δt < h x
where ħ = h/2π, and σx, σp are the standard deviations of position (r) and momentum 

2.
no potential energy in this scenario, there is a discrete set of wave numbers since an integral number of wavelength need to fit into the box
for any kind of calculation normalize
Some “trajectories” of a particle in a box (infinite square well/ infinitely deep well) according to Newton's laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics. In (B-F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.