Question

8. Compare the expression derived for translational energy for a particle confined to a 2D plane, Em..., to one confined to a 1D line, E, or E. Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related.

Answer 1

For a particle of mass m in a 1-D box of length a, the expression for energy eigenvalue of a state with quantum number n, is given by n'h2 En = mo2, n = 1,2,3,... (h=Planks constant) For a particle of mass m in a 2-D box of length and breadth a,b (x and y directions), the expression for energy eigenvalue of a state with quantum number nx and ny , is given by nzah2 n2h2 Enxny = Sma2 + 3mb2 nx = 1,2,3, ... and ny = 1,2,3,... From the expression of En and En,n, it is evident that the energy of the particle in a 2-D box of dimensions a and b are same as sum energies of two independent but identical particles (same mass) confined in 1-D boxes of lengths a and b respectively. Physically, it means that the motion of a particle (confined in a 2-D BOX) along the mutually perpendicular directions are not correlated and are hence independent.