Question

Consider an infinite 1D square well with a length L with a particl of mass M trapped inside. (a) Calculate the expectation values(x), (2,2), (p), and (pay for the ground state where n 1. Note that the first three quantities may be calculate by using symmetry argu- ments, or by appealing to the fact that the kinetic energy is known exactly. On the other hand, to find (2) one will need to evaluate an integral. (b) Calculate ΔΧ-V(22)-(zy2 and ΔΙΕ V(p2)-(p)? (c) Find Δ2Ap. Commient on whether your result is cornpatible with the Heisenberg Uncertainty relation:

Answer 1

1-D Square will: Let a potential well with infinite potential A particle of mass M, trapped inside the box The potential inside The box is Zero Let energy of particle is E. v(x) = 0 for 0 lessthanorequalto x lessthanorequalto L infinity Other wise Since schrodinger equation d^2 phi/dx^2 + 2m(E - V)/h^2 phi = 0 Here v = 0 for 0 lessthanorequalto x lessthanorequalto L d^2 phi/dx^2 + 2mE/h^2 phi = 0 Let k^2 = 2mE/h^2 d^2 phi/dx^2 + k^2 phi = 0 phi = A sin kx + B cos kx \r\n let us now use boundary condition in phi = 0 at x = 0 & x = L Hence o = A sin theta + B cos 0 B = 0 Here phi = A sin cx Also phi = 0 at x = L 0 = A sin kL sin kL = sin n pi kL = n pi k = n pi/L