4. Single-Particle Systems

AbstractIn this chapter, the machinery to deal with single particles is worked out, culminating in the vital solutions for the hydrogen atom and hydrogen molecular ion.

The first section covers the harmonic oscillator. This vibrating system is a simple model for such systems as an atom in a trap, crystal vibrations, and electromagnetic waves.

Next, before the hydrogen atom can be discussed, first the quantum mechanics of angular momentum needs to be covered. Just like you need angular momentum to solve the motion of a planet around the sun in classical physics, so do you need angular momentum for the motion of an electron around a nucleus in quantum mechanics. The eigenvalues of angular momentum and their quantum numbers are critically important for many other reasons besides the hydrogen atom.

After angular momentum, the hydrogen atom can be discussed. The solution is messy, but fundamentally not much different from that of the particle in the pipe or the harmonic oscillator of the previous chapter.

The hydrogen atom is the major step towards explaining heavier atoms and then chemical bonds. One rather unusual chemical bond can already be discussed in this chapter: that of a ionized hydrogen molecule. A hydrogen molecular ion has only one electron.

But the hydrogen molecular ion cannot readily be solved exactly, even if the motion of the nuclei is ignored. So an approximate method will be used. Before this can be done, however, a problem must be addressed. The hydrogen molecular ion ground state is defined to be the state of lowest energy. But an approximate ground state is not an exact energy eigenfunction and has uncertain energy. So how should the term

lowest energybe defined for the approximation?To answer that, before tackling the molecular ion, first systems with uncertainty in a variable of interest are discussed. The

expectation valueof a variable will be defined to be the average of the eigenvalues, weighted by their probability. Thestandard deviationwill be defined as a measure of how much uncertainty there is to that expectation value.With a precise mathematical definition of uncertainty, the obvious next question is whether two different variables can be certain at the same time. The

commutatorof the two operators will be introduced to answer it. That then allows the Heisenberg uncertainty relationship to be formulated. Not only can position and linear momentum not be certain at the same time; a specific equation can be written down for how big the uncertainty must be, at the very least.With the mathematical machinery of uncertainty defined, the hydrogen molecular ion is solved last.

- 4.1 The Harmonic Oscillator
- 4.1.1 The Hamiltonian
- 4.1.2 Solution using separation of variables
- 4.1.3 Discussion of the eigenvalues
- 4.1.4 Discussion of the eigenfunctions
- 4.1.5 Degeneracy
- 4.1.6 Noneigenstates

- 4.2 Angular Momentum
- 4.2.1 Definition of angular momentum
- 4.2.2 Angular momentum in an arbitrary direction
- 4.2.3 Square angular momentum
- 4.2.4 Angular momentum uncertainty

- 4.3 The Hydrogen Atom
- 4.3.1 The Hamiltonian
- 4.3.2 Solution using separation of variables
- 4.3.3 Discussion of the eigenvalues
- 4.3.4 Discussion of the eigenfunctions

- 4.4 Expectation Value and Standard Deviation
- 4.4.1 Statistics of a die
- 4.4.2 Statistics of quantum operators
- 4.4.3 Simplified expressions
- 4.4.4 Some examples

- 4.5 The Commutator
- 4.5.1 Commuting operators
- 4.5.2 Noncommuting operators and their commutator
- 4.5.3 The Heisenberg uncertainty relationship
- 4.5.4 Commutator reference

- 4.6 The Hydrogen Molecular Ion