Question

Quantum Mechanics.

Find the energies, degenerations and wave functions for the first three energy levels (ground state
and first two excited states) of a system of two identical particles with spin , which move in a one-
dimensional infinite well of size .

Find corrections of energies to first order in $V_{0}$ if an attracting potential of contact $H_{I}=-V_{0}\delta (x_{1}-x_{2})$ is added.

Show that in the case of "spinless" fermions, the previous perturbation has no effect.

Step by step process with good handwriting, please. It can be hard to read the answer sometimes.

Thank you a lot.

Answer 1

The particle in one dimensional box is equivalent to one particle in two dimensional box. psi = psi(x_1) psi(x_2) E = E_x_1 + E_x_2 V(x) = 0 0 < x < a = infinity otherwise using schrodinger wave equation for 0 < x < L d^2 psi/dx^2 + 2m E psi/h^2 = 0 (d^2 psi_1/dx_1^2 + 2m E_x_1 psi_1/h^2) + (d^2 psi_2/dx_2^2 + 2m E_x_2 psi_2/h^2) = 0 equation(1) d^2 psi_1/dx_1^2 + 2m E_x_1 psi_1/h^2 = 0 d^2 psi_1/dx_1^2 + k_1^2 psi_1 = 0 where k_1 = squareroot 2m E_x_1/h^2 psi(x_1) = A sin k_1 x_1 + B cos k_1 x_1 using boundary condition psi_1 (0) = psi_1 (L) = 0 implies B = 0 psi(x_1) = A sin k_1 x_1 using Normalization condition integral^L_0 Vertical-bar psi(x_1) Vertical-bar ^2 dx = 1 A = squareroot 2/L psi(L) = 0 implies A sin kL = 0 kL = n_1 pi k = n_1 pi/L \r\n psi(x_1) = squareroot 2/L sin k_1 x_1