Normal helium atom

A normal helium atom, also known as helium-4, is the most common and stable isotope of helium. In quantum mechanics, the description of a helium atom involves the application of the Schrödinger equation to model the behaviour of electrons within the atom. The helium atom consists of two electrons orbiting around a nucleus containing two protons and two neutrons. Here are some key aspects of the quantum mechanical description of a helium atom:

1. Electron Configuration:

   • Each electron in a helium atom is described by a set of quantum numbers that specify its energy, angular momentum, magnetic moment, and spin.

   • The electron configuration of helium is 1s², indicating that both electrons occupy the 1s² orbital, the lowest energy level.

2. Schrödinger Equation:

   • The time-independent Schrödinger equation for the helium atom can be written as: Hψ=Eψ where H is the Hamiltonian operator, ψ is the wave function, and E is the total energy of the system.

3. Many-Body Problem:

   • Solving the Schrödinger equation for a helium atom involves addressing the many-body problem, as it requires considering the interactions between both electrons and their interaction with the nucleus.

   • The exact solution for the helium atom is not analytically tractable due to the electron-electron repulsion term, leading to the need for approximation methods.

4. Variational Method:

   • One common approach to solving the many-body Schrödinger equation is the variational method. This method involves using trial wave functions and adjusting parameters to minimize the energy, providing an approximate solution.

5. Hartree-Fock Approximation:

   • The Hartree-Fock approximation is another method used to approximate the wave function of a many-electron system, treating electrons as if they move in an average field created by other electrons.

6. Correlation Effects:

   Electron-electron correlation effects play a significant role in helium due to the repulsion between electrons. Methods beyond the Hartree-Fock approximation, such as configuration interaction or correlated wave function methods, are employed to better account for these effects.

7. Helium Ion (He+):

   The helium ion (He+) is often used as a simplified system for theoretical calculations, as it involves only one electron.

   Compared to the hydrogen atom, helium atom has a nucleus with z = 2 and two electrons. Because it is a three-body problem, it is more complicated. There is a strong repulsive electrostatic force between the two electrons e2/ r1 - r2 when the electron coordinates are r1 and r2 with the nucleus at the beginning, as illustrated in Figure 1

The unperturbed wavefunction is

where 𝜓 are hydrogen atom wavefunction

For ground state

where E is the ionisation energy of normal hydrogen atom (n=1, z=1). The experimental value -78.98 eV

The first order perturbation theory gives

where 𝜉_12 = l r_1 - r_2 l and 𝜎_1 = z(ri/ao). For spherically symmetric l = 0 wavefunction 8 represents the mutual electrostatic energy of this overlapping infinite spherical charge distributions.

Quantum mechanics provides a powerful framework for understanding the behaviour of electrons in atoms, but the complexities arising from the many-body nature of the helium atom necessitate the use of approximation methods to obtain practical solutions. Advanced computational techniques are often employed to model and analyze the quantum mechanical properties of helium and other atoms.

This note is a part of the Physics Repository.