The Structure of the Atom and Its Interaction with an Electromagnetic Field

Abstract

The Hamiltonian of an atom composed of a nucleus (of charge Z) and N electrons consists of three summations: these are sums over the kinetic and potential energy operators for each separate electron moving in the field of the nucleus and also over the interelectronic Coulomb interaction terms:

$$ \widehat H = \sum\limits_{n = 1}^N {\left( {\frac{{\widehat p_n^2}}{2} - \frac{Z}{{{r_n}}}} \right)} + \frac{1}{2}\sum\limits_{n > q = 1}^N {\frac{1}{{\left| {{r_n} - {r_q}} \right|}}} ,{\widehat p_n} = - i{\nabla _n} = - i\frac{\partial }{{\partial {r_n}}} $$

((2.1))

where r _n is the radius vector of the nth electron. For a neutral atom N = Z. In (2.1) all relativistic effects have been neglected so all interactions represented are of Coulombic origin. The state of an atom is determined by its wavefunction Ψ _E (x ₁ ⋯ x _N ), which is a solution of the stationary Schrödinger equation:

$$\hat{H}\Psi_E(x_1\cdot\cdot\cdot x_N)=E\Psi_E(x_1\cdot\cdot\cdot x_N)$$

((2.2))

where Ψ _E depends on the spatial coordinates and spin projection of each electron, i.e., x _n ≡ r _n , σ _n .