Question

Write the solution of the 1D PIB problem in terms of the eti functions. Check to make sure the solution is normalized (don't forget to use the complex conjugate of the function in Ψ*)

Answer 1

The Schrodinger equation -h^2/2m d^2 psi/dx^2 + U(x) psi = E psi so, U(x) = 0, 0 < x < L = infinity > L and < 0, so, in the region 0 to L Schrodinger equation, U(x) = 0 -h^2/2m middot d^2 psi/dx^2 = E psi Here, psi = Ae^ikx + Be^-ikx cab be a solution [why only Ae^ikx or e^-ikx is not only the solution ? Ans Because if psi(x) = Ae^ikx then boundary condition that psi(x) = 0 at x = 0 & x = L not satisfied \r\n The avobe term can be written as in terms of sines and cosines psi(x) = C sin kx + D cos kx psi(0) = psi(L) = 0 [Boundary condition] putting x = 0 psi(0) = D = 0 putting L = 0 psi(L) = 0 = c sin kL kL = n pi k = n pi/L Therefore psi_n (x) = L sin kx psi(x)

Note: Particle in a box model considered as a particle confined in a box of length L . Inside the box the potential energy of that particle is zero whereas outside the particle potential energy is infibite. There are two boundary condition for the wavefunction of the particle , the wavefunction is zero at the each edge of the box .i.e . at x= 0 and x= L , wave functiin Y=0.