Consider a particle trapped in a one-dimensional finite potential well. Assuming that the well admits exactly three energy eigenstates, do the following: a)  Sketch the form of the potential, and the approximate positions of the energy eigenvalues. b)  Add to your sketch the values of the wavefunction, ψ(x), and the probability density, ρ, for each of the three states. c)  Explain the features of your sketch that do not occur in classical physics.

# quantum mechanical particde in a harmonic osci lator potential has the initial wave function y,)+4,(x), where Y. and Y, are the real wavefunctions in the ground and fist exci ted state of the harmonic osciclator Hamiltonian- for Convenience we take mzhzw= 1 for the oscillator- What ở the probabilpty den sity of finding the par ticke at x at time tza?

We will consider the Schrödinger equation in this problem as well as the analogies between the wavefunction and how boundary conditions are an essential part of developing this equation for various problems (situations). a) Write the form of the time-independent Schrödinger equation if the potential is that of a spring with spring constant k. Write the form of the time-dependent Schrödinger equation with the same potential. Briefly describe all the terms and variables in these equations. b) One solution to the time independent Schrödinger equation has the form Asin(kx). Why might it be called the wavefunction? If this form represents a wave of light, what is the energy for one photon? (Notek here stands for the wavevector and not the spring constant.) c) Why must all wavefunctions go to zero at infinite distance from the center of the coordinate system in all systems where the potential energy is always finite?

Please don't provide handwritten solution .....    Determine the normalization constant for the wavefunction for a 3-dimensional box (3 separate infinite 1-dimensional wells) of lengths a (x direction), b (y direction), and c (z direction).