Answer Happy

4. A particle of mass 7 moves in a one-dimensional infinite-well potential of width 2a such that its potential energy is cf. leftmost panel of the figure at the bottom of the page] X for za region I) U(z) - 0 for - a - x - a (region II) X for za region III). (a) Show that the position-picture wavefunctions of the energy eigenstates of the problem have the form On(x) = A + Be-ianz, where A, B e C and a,, = √2mEn/h with En the 7th energy eigenvalue. (b) Imposing that , (r) - 0 at the boundaries and outside of region II and normalisation of the wavefunction, show that En ²²,² Sma2 and On (1) 1 2√a EPEN € 2a + +(-1)"+¹e) (n = 1,2,...). (7) (c) At time - 0), the infinite-well potential is subjected to an instantaneous expansion that doubles the width of the well. As a consequence, region II now extends from r = -2a to □ = 2a [cf. figure at the bottom of the page]. Show that the energy eigenvalues E and wavefunctions of the energy eigenstates o(r) of the modified system are related to E, and On (1) as En 1 E₁ July On (x/2). 4 (7) (d) Assume that, before the instantaneous expansion, the particle is prepared in 01 (2) and that the state of the particle immediately after the expansion remains unchanged. Assume that. immediately after the expansion takes place, the energy of the particle is measured. Show that the probability that such energy measurement results in the eigenvalue modified system is of the 64 P2n+1 π² [(2n + 1)² − 4]² while P₂ = 0), Vn = 0, 1, 2, ... (7) region II region I [ x U=0 a 0 a instantaneous expansion 2a region II U-0 20