Entanglement, Its Challenges and Its Applications

When we dealt with the H atom in Chap. 17, we started from the classical formulation in terms of an effective particle having a reduced mass (see Sect. 2.5.1). The same argument works in Quantum Mechanics, with the momenta replaced with the quantum operators, starting from the Hamiltonian $$ H=-\frac{\hbar ^{2}}{2 m_{e}}\nabla _{e}^{2}-\frac{\hbar ^{2}}{2 m_{p}}\nabla _{p}^{2}+V(\rho ); $$ here, \(\nabla _{e}\) acts on \(r_{e}\), \(\nabla _{p}\) acts on \(r_{p}\), and \(\overrightarrow{\rho }=\overrightarrow{r_{e}}-\overrightarrow{r_{p}}\). The wave function \(\psi =\psi (\overrightarrow{r_{e}},\overrightarrow{r_{p}})\) depends on the coordinates and spins of both particles, however we omit spins, since H does not depend on them in the non-relativistic limit.