problem 1: Show that wavefunction ψ1(x) = (2/L)^1/2 sin(πx/L) of a particle in a box is an eigenfunction of the operator d²/dx². And also find out the corresponding eigenvalue.

problem 2: A particle is in the two dimensional potential well (V(x,y) =  ∞ for the outside of the well, V(x,y) = 0 for the inside of the well).

a) Starting with the shrodinger equation, find out the state function using L₁ and L₂ as the length of each side of the box.
b)  find out the lowest energy that this particle would have if the dimension of the box is 3 nm by 5 nm.
 
problem 3: The π electrons of a conjugated molecule can be modeled as a particle in a box, where the box length is a little more than the length of the conjugated chain. The Pauli exclusion principle allows no more than two electrons to occupy each energy state. For a conjugated hydrocarbon chain molecule with three double bonds (CH₂=CH-CH=CH-CH=CH₂), find out the wavelength of radiation absorbed when the π electron is excited from the highest-occupied state to the lowest vacant state.

problem 4: For a particle confined to a one-dimensional box of length L, find out the probability that the particle is in the central third of the box (the middle section when the box is divided into three sections with equal length).

problem 5: An electron was accelerated by 1.0 volt, and traveled to the right in one dimension and entered the potential well with V(x) = V with finite width, 50 pm (pm = 10^-12 m). Assume that there is zero potential outside of the region.

a) find out the initial kinetic energy of the electron.

b) prepare down the general form of the state function for this electron for the inside of the potential well and the outside of the potential well.

c) Assume that the potential energy of the well is 1.5 times of the initial kinetic energy of the electron, find out the probability of the electron penetrating the potential barrier.