Question

Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2 0.4 0.6 0.8 1.0 Figure 3: Particle in a box of size L in 1D. The first three energies E(O) eigenfunctions |n(0)) are illustrated (n 1,2,3). The unit for z is L. and (a)[9 pts] Caleulate Ef for this perturbed problem. (b)[9 pts] Calculate Ex for this perturbed problem.

Answer 1

state wavefunction = squareroot 2/L sin pi x/L E'_1 = integral^L_0 squareroot 2/L sin pi x /L V delta {(x - L/2)^2 - a^2} squareroot 2/L sin pi x/L E'_1 = 2v/L [integral^L_0 sin pi x/L delta {x - L/2 - a)} + integral^L_0 sin^2 pi x/L delta {x - (L/2 -a)} = 2v/L [sin^2 pi/L (L/2 + a) + sin^2 pi/L (1/2 - a)] = 2b/ L [sin^2 {pi/2 + a pi/L} + sin^2 (pi/2 - a pi/L)] = 2v/L [{sin (pi/2 + a pi/L)}^2 {sin^2 (pi/2 - a pi/L)}^2] = 2v/L [cos^2 a pi/L + cos^2 a pi/L] = 2v/L 2 cos^2 a pi/L = 4v/L cos^2 a pi/L because L = 1 E'_1 = 4 cos^2 a pi b) E'_2 = 2/L integral^L_0 sin^2 2 pix/L [delta (x - L/2 - a) + delta (x - L/2 + a)] = 2v/L [{sin 2 pi/L (L/2 + a)}^2 {sin 2pi/L (L/2 - a)}^2]