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a proton that has a positive charge, that is, a charge of +1, and an electron that has a negative charge, a charge of -1. When we say a charge of 1, it actually is shorthand for the charge that is on one proton. The charge in real units is 1.6 × 10-19 C, where C is the Coulomb, the unit of charge. Ernest Rutherford (1871-1937) did experiments in 1911 that showed that atoms were composed of a small, heavy positively charged nucleus and one or more electrons outside of the nucleus. Rutherford won the 1908 Nobel Prize in Chemistry “for his investigations into the disintegration of the elements, and the chemistry of radioactive substances.” Rutherford’s findings applied to the hydrogen atom mean that the proton is the nucleus and a single electron is found outside the nucleus. Even for hydrogen where the nucleus is composed of a single proton, the nucleus is much heavier than an electron. The mass of a proton is mp = 1.67 × 10-27 kg, while the mass of the electron is only me = 9.1 × 10-31 kg. A proton weighs about 1836 times as much as an electron.

In the Bohr model, the electron orbited the proton like a planet orbiting the sun. The lowest energy state of the hydrogen atom, n = 1, has the electron going around the proton in a circle. Higher energy states of the electron, with n greater than 1, could have different shapes. Some were still circles, but some were ellipses. This picture of the electron orbiting the proton should immediately set off danger warnings based on the material covered in earlier chapters. In Chapter 6, the Heisenberg Uncertainty Principle was discussed. We know that an absolutely small particle cannot have a classical trajectory. To have a trajectory, it is necessary to know the position and the momentum of a particle simultaneously over all time. But the Heisenberg Uncertainty Principle says that it is not possible to know both the position and the momentum precisely simultaneously. The uncertainty relation states that ΔxΔp ≥ h/4π, where h is Planck’s constant. Absolutely small particles are described in terms of probability amplitude waves, not trajectories. Of course in 1913, when Bohr came out with his mathematical treatment of the hydrogen atom, the nature of absolutely small particles was not known.

The failure of Bohr’s approach became apparent when it was applied to systems other than the hydrogen atom. While Bohr’s method could predict very accurately the energy levels, and therefore the spectrum of the hydrogen atom, it could not do so for the next simplest atom, the helium atom. Nor could it properly predict the properties of the simplest molecule, the hydrogen molecule, which is composed of two hydrogen atoms. The Bohr method could not account for the strength of the chemical bond that held the two hydrogen atoms together to form the hydrogen molecule. Although Bohr made giant steps in the right direction, the failures of his approach ultimately led to the development of true quantum theory in 1925.

10

The Hydrogen Atom: Quantum Theory

IN 1925 SCHRÖDINGER AND HEISENBERG separately developed quan tum theory. Their two formulations are mathematically different, but their theories are rigorous and form the underpinning of modern quantum theory. At about the same time, Dirac made major contributions as well. First, he presented a unified view of quantum theory that showed that the Schrödinger and Heisenberg theories, while mathematically different, were equivalent representations of quantum mechanics. In addition, he developed a quantum theory for the hydrogen atom that is also consistent with Einstein’s Theory of Relativity. The formulation by Schrödinger is the most often used to describe atoms and molecules. Therefore, most of our discussions, starting with the hydrogen atom and then going on to larger atoms and molecules, will be based on the concepts and language that is inherent in the Schrödinger approach.

THE SCHRÖDINGER EQUATION

We used a very simple but correct mathematical method for obtaining the energy levels of the particle in the box and the wavefunctions, but