1. Consider a one-dimensional rectangular potential barrier V(x): V(2) – {v. So z(0, z)a 0(z(a (1) where Vo is positive. A particle with energy E < V, approaches the barrier from the left (see figure Fig. 1). E 3 x=a Figure 1 i Find in each region the wave function by solving the Schrodinger equation. ii By definition, the transmission coefficient T is defined as the transmitted current divided by the incident current: Jranamatted T Jmesdent where j 2mi evaluate T in terms of the amplitudes of transmitted and incident wave functions. iii Using the boundary conditions prove that: T = where ka 1+ sinh kya is the wave vector in region 2.

There is an electron, in a 1-d, infinitely deep square potential well with a width of d. If it is in ground state,  1. Draw the electron's wavefunction. Show the position of the walls of the potential well. 2. Explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.

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 .