For (i) the infinite square well, (ii) the finite square well and (iii) the quantum harmonic oscillator, carefully sketch each of the following: a) The potential and the energy levels of the two lowest energy eigenstates. b) The wavefunctions for the two lowest energy eigenstates. c) The probability densities for the two lowest energy eigenstates. In the case of the finite square well, assume that the well admits exactly two energy eigenstates.

Consider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)

A particle is trappend in a one-dimensional well. Two of its wavefunctions are shown below. (a) Identify wether the well is finite or infinite. (b) Identify the quantum number n associated with each wavefunction; (c) Overlay a sketch of the probability density for each wavefunction. n = n =

An Infinite Square Well of width L that is centred around x = 0 is shown in the figure. At t = 0, a particle exists in this system with the wavefunction provided, where Ψ0 is √(12/L), and Ψ = 0 for all other values of x.  Calculate the probability density for this particle at t = 0, and state the position at which it takes its maximum value. then, calculate the expectation value for the position of this particle at t = 0, i.e. ⟨ x⟩. Compare the results of the positions found and explain why they are different.

If you double the width of a one-dimensional infinite potential well, (a) is the energy of the ground state of the trapped electron multiplied by 4, 2, , , or some other number? (b) Are the energies of the higher energy states multiplied by this factor or by some other factor, depending on their quantum number?

The curve in the attached figure is alleged to be the wave function for the fifth energy level for a particle confied in the one-dimensional potential energy well shown. Quantitately prove which aspects of the wave function are correct, as well as which are not correct.

1.The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ ? ≤ L/2, are given by :  (see figure)  and have  Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc  a) Sketch the potential of this system , including in your sketch the positions of the lowest three energy levels . Indicate  in your sketch the form of the wavefunction for a particle in each of these energy levels , and state which of the wavefunctions you have drawn could be decirbed by the Ψn  written above (see figure) .  b) Calculate the expectation value of momentum , ⟨p⟩ for a particle with n=2 c) Calculate the expectation value of momentum squared ⟨p 2⟩ , for a particle with n = 2 .  Hint : you may use the mathematical identiy  sin2 x = 1/2 (1 − cos 2x) without proof .