Answer Happy

uestion 11 This question concerns a particle of mass m in a one-dimensional infinite square well, described by the potential energy function so for -L/2 < x <L/2 V(x) = elsewhere. = In the region -L/2 < x <L/2, the normalized energy eigenfunctions take the form I V E cos ("E") for n = 1,3,5, ... Vn(x) = n. sin for n = 2,4,6,... (a) Write down the time-independent Schrödinger equation for this system in the region -L/2 x <L/2. Verify that Vz(x) and U2 (2) (as defined above) are solutions of this equation, and find the corresponding energy eigenvalues. (6 marks) (6) State the boundary condition that the energy eigenfunctions Vn(x) must satisfy at x = = -L/2 and x = L/2 and show that V1(2) and U2(x) satisfy this condition. (3 marks) (c) Show that the expectation value of position r is equal to zero in the state described by V2(x), and calculate the expectation value of x2 in this state. (5 marks) (d) Calculate the probability that the particle is found in the region -L/4Sx <L/4 in the state 02(r). (3 marks) You may use the standard integrals (T/2 7T sinºu du = " *sinº u du = u 2' 이녀 3