The primary motivation in studying quantum mechanics is that the key
constituents of biochemical reactions, namely electrons, have properties
that are described by quantum mechanics rather than classical mechanics.
Quantum mechanics will serve as the foundation for understanding the
properties of electrons and how different spectroscopies can be used to
probe proteins and other biological samples. In this chapter, Schrödinger’s
equation is solved for the hydrogen atom. The concepts of atomic orbitals
are interpreted in terms of the resulting wavefunctions of the equation.
As was done for the particle in a box and simple harmonic oscillator,
the wavefunction allows the calculation of all of the properties of the
hydrogen atom, such as the average distance or energy of any given orbital.
These results for the hydrogen atom with one electron are extended to
understanding the properties of multi-electron atoms in terms of both empir-
ical constants and as detailed calculations. Finally, the solutions are used
to understand the organization of elements in the periodic table. The bio-
logical question addressed in this chapter is how the use of hydrogen by
biological organisms can contribute to energy policies.
Schrödinger’s equation for the hydrogen atom
The hydrogen atom is solved using Schrödinger’s equation, as has been
used for the other problems – the particle in a box and the harmonic
oscillator. First, we must determine the potential to be used in the prob-
lem. For the hydrogen atom, the potential is assumed to be due to the
electrostatic interaction between the negatively charged electron and the
positively charged nucleus, giving a potential, V(r), of:
Vr (12.1)
e
r()=
−^2
4 πε 0