Question-

a) The time-independent Schrodinger equation for a free particle moving in one dimension (x) is written as (d^2 Psi(x) / dx^2) + k^2 Psi(x) = 0, where k is a constant.

(i) Show that the following function is a solution to this equation: Psi(x) = A sin(k x)

(ii) Briefly outline why constraining the particle within a square well with infinitely high walls at x = 0 and x = L gives rise to discrete stationary states and associated quantized energy levels of the particle. State clearly any important properties of the wavefunction that you employ.

b) Draw two diagrams, each showing both the potential energy function Epot and the lowest energy time-independent wavefunction for a particle confined in a one-dimensional square well of width D, and where in the first diagram the potential energy outside the well is infinite, and in the second diagram it is finite. (Take the potential energy inside the well to be zero in each case.)

Explain the interpretation of the wavefunctions shown in your sketches, in terms of the location of the particle, highlighting any differences that would be expected from treating the particle classically.

c) (i) Sketch the third lowest eigenfunction, Psi 3, for a particle in an infinite square well between x = 0 and x = L.

(ii) Sketch the corresponding probability distribution for finding the particle along the x-axis.

(iii) What is the probability of finding the particle between x = 0 and x = 2L/3?

How would this probability change if the height of the walls of the potential well was reduced to a finite value?

(iv) The mass of the particle in the infinite well is 2.00 Ã- 10^-30 kg, and the width of the well is 1.00 Ã- 10^-9 m. If the particle makes a transition from the third eigenstate to the second eigenstate, what will be the wavelength of the emitted light?

Psi = Wavefunction symbol (looks like a candle holder)