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The Quantum Theory of the
Ground State Energies of
Helium-Like Atoms

Niels Bohr's model of the atom provided a wonderfully accurate explanation of the spectrum of hydrogen, but when it was applied to the spectrum of helium it failed. Werner Heisenberg developed a modification of Bohr's analysis but it involved half-integral values for the quantum numbers. Bohr considered non-integral values of quantum numbers nonsensical in terms of the logic of quantum theory. There was a crisis in quantum theory that ultimately led to the creation in the 1920's by Heisenberg of a new quantum theory which became known as Matrix Mechanics.

The issue of the whether quantum numbers have to be integers was resolved in 1927 by the work of Erwin Schrödinger in which he established that the discreteness of quantum numbers arises because they are eigenvalues of certain partial differential equations. Heisenberg disparaged Schrödinger's Wave Mechanics, but Schrödinger showed it gave the same predictions of quantum phenomena as did Heisenberg's Matrix Mechanics.

The Model of Electrons in Atoms

It is easy to formulate the physics of a two-electron atom, and that will be done shortly, but it is wise to first formulate the physics of a one electron atom. One electron atoms include not only the hydrogen atom but also the singly ionized helium atom, the doubly ionized lithium atom, the triply ionized beryllium atom and so forth.

For purposes of simplicity it is assumed that the nuclei of atoms are so massive compared with their electron(s) that the motion of the nuclei can be ignored.

One Electron Atoms

The Hamiltonian for a hydrogen-like atom is

H = p²/(2m) − KZe²/r

where p is the momentum of the electron, m is its mass, K is the constant for the electrostatic force, Z is the number protons in the nucleus, e is the electrostatic charge unit and r is the distance from the center of the nucleus to the center of the electron.

The time independent Schrödinger equation derived from the above Hamiltonian has eigenvalues of

En = −mZ2e4/(2hn²)

where h is Planck's constant divided by 2π and n is an integer, called the principal quantum number. The value of En is the energy of the state corresponding to n.

The lowest energy is E1 and the other energies of the other states can be expressed in terms of it; i.e.,

En = E1/n²

There is a more cogent simplification of the formula for En. Let me4/(2hn²) be denoted as γ. Then

En = −γZ²/n²

The quantity γ is the absolute value of the ground state energy of the electron in a hydrogen atom (Z=1).

The spectrum is derived from the change in energy resulting from the electron falling from a state n1 to a state n2. Thus the energy of the emitted quantum is proportional to [1/n2² − 1/n1²].

The wave function for the state n can be found as a solution to its Schrödinger equation.

Two Electron Atoms

For a two-electron atom the Hamiltonian is

H = p1²/(2m) − KZe²/r1 + p2²/(2m) − KZe²/r2 + Ke²/r12

where r12 is the distance between the two electrons.

The above Hamiltonian may be thought of as

H = H1 + H2 + Ke²/r12

with the last term considered as a perturbation to the solution for two independent single electron atoms. Solutions for the ground (minimum) state of an electron in a hydrogen-like atom can be combined with a general state for an electron. The general state of an electron is given by three quantum numbers, (n, l, ml, where n≥1, l<n and |ml|≤l. The ground state of an electron is where n=1, l=0 and ml=0.

Thus the ground state of a helium-like atom is the state in which both electrons are in their ground states; i.e., E1,1.

The expected value of the energy involved in the interaction can be approximated by using the wave function for the ground state of the corresponding hydrogen-like atom. This expected value is found to be (5/4)Zγ. Thus

E1,1 = −2Z²γ + (5/4)Zγ

Several physicists have computed the ground state energies of helium-like atoms. Here are two such comparisons.

Ground State Energies of Helium-like Atoms
Atom or
L. Pauling and E.B. Wilson, Introduction to Quantum Mechanics, p. 165

Ground State Energies of Helium-like Atoms
Atom or
J.C. Slater, Quantum Theory of Atomic Structure,, p. 339-342

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