For a particular multi-electron atom, application of the variational approach using an effective nuclear charge parameter Z 0 results in the following expression for E(Z 0 ), where a is a constant and energy is in atomic units (Hartree): E(Z 0 ) = −Z 0 e −aZ0 Determine the variational parameter Z 0 and the (minimum) variational energy.

Aparticle of mass m is placed in a one-dimensional box of length l. the box is so small that the particle’s motion is relativistic, so that e p2/2m is not valid. (a) derive an expression for the energy levels of the particle using the relativistic energy–momentum relation and the quantization of momentum that derives from connement. (b) if the particle is an electron in a box of length l 1.00 1012 m, nd its lowest possible kinetic energy. by what percent is the nonrelativistic formula for the energy in error?