4.1: Particles in Quantum Mechanics

When we talk about a “particle” in quantum mechanics, we mean something that behaves as if it were just a single body. However, we are often also talking about a particle as it is understood in the Standard Model of Particle Physics. In the Standard Model, a fundamental particle is something that is effectively a mathematical point. As far as we can tell, the fundamental particles have no spatial extent. The most common everyday example of a particle from the Standard Model is the electron. You may be familiar with electrons if you have taken any chemistry classes in the past. Atoms are made of of electrons orbiting nuclei. Nuclei themselves are made up of protons and neutrons. Protons and Neutrons may be treated as particles in quantum mechanics, but in fact they are not fundamental particles. Rather, they are themselves made up of quarks, which are (at least as far as we understand) fundamental particles.

If a fundamental particle doesn’t have a size, what can it have? Well, it can have a position, and it can have a momentum. Later, we will find out that there must be some uncertainty associated with one or both of these quantities for any given particle, but these are quantities that you can figure out for the particle. However, they aren’t really fundamental to the particle; they just say where the particle is, or, effectively, how fast it’s moving relative to something you’ve chosen to measure speeds relative to. Similarly, if the particle is an electron in an orbital in an atom, it can have angular momentum as a result of that orbit. Again, this isn’t a fundamental property of the particle, but there result of its interaction with the atomic nucleus.

The mass of the particle is a fundamental property of the particle. Likewise, the electric charge of the particle. The electric charge on the electron, in SI units, is \(\ -1.602 \times 10^{-19} \mathrm{C}\). In fact, often when we are dealing with atomic and subatomic particles, we’ll measure charge in terms of the elementary charge \(\ e\), which is defined as the absolute value of the charge on the electron: \(\ e=+1.602 \times 10^{-19} \mathrm{C}\). (It is unfortunate that the notation for the elementary charge is the same letter as \(\ e\), the natural exponential that shows up, for instance, in the mathematical model for radioactive decay. You need to be careful about the context whenever you see an \(\ e\), so that you can figure out whether we’re talking about the natural exponential, the charge on the electron, or something else..)

Another property of fundamental particles is their angular momentum. Because this is fundamental to the particle itself, we refer to it as the spin of the particle. As an analogy, consider the Earth orbiting the Sun. The Earth has orbital angular momentum as a result of the circle it makes yearly about the Sun. It also has spin angular momentum as a result of its daily rotation about its own axis. Where the analogy breaks down, however, is that the Earth is indeed an extended ball; the electron, on the other hand, is a point particle, and has no spatial extent. As such, there really isn’t anything spinning around anything else to create this angular momentum. This is conceptually difficult; how, then, can the electron have angular momentum? Alas, the best answer we can give is that it just does. Experiments have shown that indeed electrons behave as if they have angular momentum, and that they can transfer angular momentum to other particles and systems when they interact with them.

Just as every electron has exactly the same mass and exactly the same electric charge, every electron has exactly the same total angular momentum. (We will see later what the value of that angular momentum is.) You can’t cut off a piece of an electron to leave behind a particle that is a part of an electron, with a lower mass and possibly a lower electric charge. Similarly, you can’t speed up or slow down the spin of an electron, the way you can get a top spinning faster or slower. All electrons are effectively spinning at the same rate— only, remember, they’re not really little balls spinning at all, but rather angular momentum is just one of the properties associated with those quantum particles we call electrons.