3 - The Leap from Classical to Quantum

Summary

Everyone has lost their marbles!

Anonymous

Before we formally present quantum mechanics in all its wonders, we shall spend time providing some basic intuitions behind its core methods and ideas. Realizing that computer scientists feel comfortable with graphs and matrices, we shall cast quantum mechanical ideas in graph-theoretic and matrix-theoretic terms. Everyone who has taken a class in discrete structures knows how to represent a (weighted) graph as an adjacency matrix. We shall take this basic idea and generalize it in several straightforward ways. While doing this, we shall present a few concepts that are at the very core of quantum mechanics. In Section 3.1, the graphs are without weights. This will model classical deterministic systems. In Section 3.2, the graphs are weighted with real numbers. This will model classical probabilistic systems. In Section 3.3, the graphs are weighted with complex numbers and will model quantum systems. We conclude Section 3.3 with a computer science/graph-theoretic version of the double-slit experiment. This is perhaps the most important experiment in quantum mechanics. Section 3.4 discusses ways of combining systems to yield larger systems.

Throughout this chapter, we first present an idea in terms of a toy model, then generalize it to an abstract point, and finally discuss its connection with quantum mechanics, before moving on to the next idea.