...Kindly help me

Problem 1.2 Consider the one-dimensional normalized wave functions wo(x), VI(x) with the properties vo(-x) = vo(x) = vo(x). VI(x ) = N- dwo dx Consider also the linear combination V (x) = cIvo(x) + czyI(x) with |cil2 + |c2|2 = 1. The constants N, CI, C2 are considered as known. (a) Show that wo and v are orthogonal and that w(x) is normalized. (b) Compute the expectation values of x and p in the states wo, , and v. (c) Compute the expectation value of the kinetic energy 7 in the state wo and demonstrate that and that ( *,IT|VI) 2 (WITly) > (volTwo) (d) Show that (wolx' Ivo)(wilp'141) 2- (e) Calculate the matrix element of the commutator [x2, p] in the state .Problem 1.2 Consider the one-dimensional normalized wave functions vo(x), VI (x) with the properties vo(-x) = vo(x) = vi(x), VI(X)=NYO dx Consider also the linear combination y (x ) = cIvo(x) + cavi(x) with |cil2 + |c2|2 = 1. The constants N, CI, c2 are considered as known. (a) Show that wo and v are orthogonal and that v(x) is normalized. (b) Compute the expectation values of x and p in the states vo, , and v. (c) Compute the expectation value of the kinetic energy 7 in the state vo and demonstrate that and that (d) Show that ( wolx ? Ivo ) ( wilp * 141 ) 2 7 (e) Calculate the matrix element of the commutator [x2, p ] in the state y.Problem 1.1 Consider a particle and two normalized energy eigenfunctions y (x) and 12(x) corresponding to the eigenvalues E1 # E2. Assume that the eigenfunc- tions vanish outside the two non-overlapping regions $2, and $2 respectively. (a) Show that, if the particle is initially in region $2, then it will stay there forever. (b) If, initially, the particle is in the state with wave function V (x, 0) = # [WI( x) + 42(x)] show that the probability density | w(x, /)|2 is independent of time. (c) Now assume that the two regions $2, and $22 overlap partially. Starting with the initial wave function of case (b), show that the probability density is a periodic function of time. (d) Starting with the same initial wave function and assuming that the two eigenfunctions are real and isotropic, take the two partially overlapping regions $, and $2 to be two concentric spheres of radii R1 > R2. Compute the probability current that flows through $1