where is the ground-state energy of a conventional hydrogen atom.
Here, is a non-negative integer, and .
where is the hydrogen ground-state energy. Note that this energy lies well above the energy of the ground-state of singly-ionized helium, which is . This means that a helium atom excited to the state has the option of decaying into a free electron and a singly-ionized helium ion, with the energy of the ejected electron determined by energy conservation. This process is known as autoionization.
Note that this estimate is much closer to the experimental value ( ) than our previous estimate ( ).
where is a non-negative integer, and . Demonstrate that
Suppose that is a trial solution to the previous equation that is not properly normalized. Prove that
where is the lowest energy eigenvalue.
where . Let
where is the hydrogen ground-state energy. For the case of a negative hydrogen ion (i.e., ), this formula gives . The experimental value of this energy is . For the case of a singly-ionized lithium ion (i.e., ), the previous formula gives . The experimental value of this energy is .
Obviously, we need to perform a more accurate calculation for the case of a negative hydrogen ion. Following Chandrasekhar , let us adopt the following trial wavefunction:
show that the expectation value of (i.e., ) is
The previous table shows the numerically determined values of and that minimize for various choices of and . The table also shows the estimate for the ground-state energy ( ), as well as the corresponding experimentally measured ground-state energy ( ) [63,66]. It can be seen that our new estimate for the ground-state energy of the negative hydrogen ion is now less than the ground-state energy of a neutral hydrogen atom, which demonstrates that the negative hydrogen ion has a positive (albeit, small) binding energy. Incidentally, the case , yields a good estimate for the energy of the lowest-energy spin-triplet state of a helium atom (i.e., the spin-triplet state).
where , , is the Bohr radius, and an adjustable parameter . Show that the energy of the hydrogen molecule ion, assuming a molecular wavefunction that is even under exchange of proton positions, can be written
where is the proton separation, the hydrogen ground-state energy, and
It can be shown, numerically, that the previous function attains its minimum value, , when and . This leads to predictions for the equilibrium separation between the two protons, and the binding energy of the molecule, of and , respectively. (See Figure 9.1.) These values are far closer to the experimentally determined values, and , than those derived in Section 9.8.