On the Mean Value of the Potential and Kinetic Energy of an Electron in a Hydrogenlike Atom

Abstract

The mean values of the potential \( \left\langle \hat{\varPi}\right\rangle \) and kinetic \( \left\langle \hat{\mathrm{T}}\right\rangle \) energy of an electron in a hydrogenlike atom are found. It is found by direct calculation that \( \left\langle \hat{\varPi}\right\rangle =2E \) and \( \left\langle \hat{\mathrm{T}}\right\rangle =\mid E\mid \) for arbitrary states with set of quantum numbers {n,l,m} . Such relations for the ground state {n = 1, l = m = 0} are well known and are a particular case of this general result. Thus, this work can have methodological value as a helpful supplement to the traditional university course in quantum mechanics. Moreover, on the scientific plane, it is possible to apply these results to a calculation of the energy of a two-electron atom by the variational method in spaces with number of dimensions D = 3,2, and 1.