Chapter 2: Postulates of Quantum Mechanics
Figure 2.1 – Titus Lucretius Carus gazing at the Milky Way galaxy [authors] built from an image of Titus Lucretius in the public domain and an image of the Milky Way galaxy [NASA]
In the first two books of his six-book poem De Rerum Natura (On the Nature of Things), Titus Lucretius Carus, a Roman poet and philosopher, discusses life and love and explains the basic principles of Epicurean physics, a Greek way of understanding the world before Christ [Lucr_1]. He put forward the idea that matter is both active and indeterminate [Lucr_2], a very "quantum" way of thinking to say the least.
Using an analogy of dust particles in a sunbeam, Lucretius described what is now known as Brownian motion [Lucr_3]. He talked about matter and used concepts such as mostly empty space to describe it. It would take more than 2 millennia for these ideas to become widely adopted and put into the postulates of quantum mechanics. We reviewed the milestones of the late 1800s and early 1900s that lead to the postulates of quantum mechanics in Chapter 1, Introducing Quantum Concepts.
The five postulates of quantum mechanics are not considered the law of nature and cannot be shown to be true, neither mathematically nor experimentally. Rather, the postulates are simply guidelines for the behavior of particles and matter. Even though it took a few decades for the postulates to be formulated and a century to be utilized by the broader scientific community, the postulates remain a powerful tool for predicting the properties of matter and particles and are the foundation of quantum chemistry and computing.
In this chapter, we will cover the following topics:
- Section 2.1, Postulate 1 – Wave functions
- Section 2.2, Postulate 2 – Probability amplitudes
- Section 2.3, Postulate 3 – Measurable quantities and operators
- Section 2.4, Postulate 4 – Time independent stationary states
- Section 2.5, Postulate 5 – Time evolution dynamics, Schrödinger's equation
In this chapter, we primarily focus on the significance of Postulate 1, Wave functions, because we think that this postulate has powerful repercussions for useful innovations. Traditionally, Postulate 1 is hard to grasp conceptually and has been a scientific challenge to represent mathematically and artistically. We have taken active steps to overcome this artistically, as shown in Figure 1.4 and in Figure 2.2. The other four postulates support Postulate 1. We do not go into as much detail with these postulates as we do with Postulate 1 in this chapter; however, will be utilizing them in subsequent chapters. Readers who are not familiar with linear algebra or with Dirac notation are invited to refer to Appendix A – Readying Mathematical Concepts.
Figure 2.2 – Artistic image of a hydrogen atom wave function [authors]