# Quantum numbers

Put simply, a quantum object could be a subatomic particle, atom, or molecule. Every quantum object can be described with four quantum numbers (QNs): principal, orbital angular momentum, magnetic, and spin magnetic QN. While these numbers apply to all quantum objects, I will be using an electron to give an example.

Name | Numbers | Quantizes... |
---|---|---|

n | 1, 2, 3,... | Electron energy |

l | 0, 1, 2,...(n-1) | Magnitude of orbital angular momentum |

ml | 0, ±1, ±2,...±l | Magnitude of orbital angular momentum component along an applied magnetic field |

ms | ±1/2 | Spin angular momentum component along an applied magnetic field |

### Principal quantum number: n

This number tells how much energy the electron has. n can be any integer value from 1 on. As n increases, so does the energy of the electron. For a single electron, the electron’s energy is completely defined by n.

### Orbital angular momentum quantum number: l

When an object orbits something, it has an angular momentum **L** associated with it. This can be seen as a quantity of angular motion that is going out of the plane of rotation (more accurately, orthogonal to the angular motion as defined by the right hand rule). It’s like linear momentum, but with something in circular motion. An electron orbiting something, say a nucleus, has an orbital angular momentum **L**. Interestingly, an electron’s **L** can only have specific values. In a classical system, an orbiting object can have any value of **L**.

### Magnetic quantum number: m_{l}

An electron has an electric charge and can thus react to a magnetic field. When a magnetic field is applied, its angular momentum shifts and has a component along the magnetic field’s direction. This number quantizes the value of this component.

### Spin magnetic quantum number: m_{s}

Quantum particles have a property called spin. The term “spin” shouldn’t be taken as a literal spin, since particles don’t work like that (see details section below). Think of it as an intrinsic angular momentum that is quantifiable. This QN quantizes the spin orbital angular momentum component along the magnetic field. That means that there are certain numbers defining how the spin reacts to a magnetic field.

# Details

I left out some important details in the explanations above to make the concepts simple. In this section, we move on to the fun stuff!

First off, particles act like waves. Electrons are *not *little balls moving around everywhere. They act like particles *and *waves. The same goes for all subatomic particles (and all bulk mass, technically, but that’s for another post). They are described with wave functions, or mathematical functions that represent particles as waves. So a **quantum state** is a wave function defined by the four quantum numbers. The QNs n, l, and m_{l }describe the spatial extent of the particles while m_{s }determines the general direction of the particle’s motion.

The principal QN can’t be 0, since the probability amplitude for the particle would be 0; i.e., it wouldn’t exist. However, it can be shown through mathematics that even if n could be 0, the energy would *not *be 0. This is called **zero-point energy**. This tells us that at absolute zero, atomic motions actually would *not *stop. At 0K, particles would still be moving! We could say that 0K is impossible because particles cannot stop moving, or say that at 0K, particles would keep moving. Either way, 0K is impossible.

Since electrons are charged particles, their motion alone generates magnetic fields. Remember that a moving electric charge (corresponding to a changing electric field) generates a (changing) magnetic field. A magnetic field could be the collective magnetic dipoles of electrons in bulk matter. The effect on electrons is much stronger with an *applied* magnetic field.

Not every state has a unique energy. As n goes up, the number of states that could exist with the same energy increases. This is called **degeneracy**. For example, a threefold degenerate electron could be in three possible states at a certain energy. When a particle changes state by emitting or absorbing a photon, it has to do so in a specific way. The change is defined by **selection rules**. It can change its l number by ±1, or m_{l }by 0, ±1. It may seem redundant to say m_{l }can change by 0, but the details lie in understanding shells and orbital theory.

Shell | Subshell | Orbitals |
---|---|---|

K | 1 | s |

L | 2 | s, p |

M | 3 | s, p, d |

N | 4 | s, p, d, f |

O | 5 | s, p, d, f, g* |

The subshell is defined by n (subshell 1 for n=1), and the orbitals are defined by l (s for l=0, p for l=1, and so on).

*There is currently no element that exhibits a g orbital. Theoretically, element 121 would be the first element with a g orbital.