Quantum numbers are sets of parameters that produce physically acceptable solutions to the Schrodinger equation. They often take integer or half-integer values. They are used to label the different eigenstates of a quantum system. The number of quantum numbers in a system depends on the system, but there will be equal to or greater than the number of dimensions in the problem.
Infinite Square well
The problem of an infinite square well demonstrates quantum numbers. For a one dimensional box, the possible eigenstates (states in which the we might find the system in when we make a measurement of it) are:
Here, is our quantum number and can take integer values. Notice how in this 1 dimensional problem there is only 1 quantum number. The infinite square well problem can be extended to consider a particle trapped inside a 3 dimensional box. This produces a solution of the form:
where , and are the lengths of each side of the box. Now that we are working in three dimensions, the number of quantum numbers has increased to 3. They are: , , and again take integer values.
The Hydrogen Atom
Four quantum numbers are required to describe the state of an electron in a hydrogen atom. As the electron is a fermion, the Pauli exclusion principle applies and states that no electron can have the same four quantum numbers. The four quantum numbers are:
Principle quantum number - corresponds to the main electron shell in which the electron resides. Can have the value n=1, 2, 3..., corresponding to shells with increasing amounts of energy. However, in stable atoms, this tends to be less than or equal to 7
Azimuthal quantum number - corresponds to the electron subshell of the electron. Can have the number l=0, 1, 2, 3 up to n-1. It is also sometimes called the "angular momentum quantum number", due to its relationship with angular momentum
Magnetic quantum number - corresponds to the orbital of the electron, the orbital is the different orientations of the electron subshell around the atom. It is normally denoted by m and varies from -l to l, taking integer values, i.e. -l, -l + 1,..., -1, 0, 1, ..., l-1, l
Spin quantum number - corresponds to the spin of the electron, can have the value 1/2 or -1/2, indicating that each orbital of an atom can only hold 2 electrons. This is an example of a quantum number that takes half-integer values.
Although this is an example with three dimensions, there are 4 quantum numbers due to spin. Considering the electron as a planet orbiting the proton can be a useful analogy to understand spin. In this analogy, the angular momentum of a planet about the sun corresponds the azimuthal quantum number and the rotation of the planet about its axis to spin. However, the analogy is not perfect; an electron is a point particle and therefore it makes no sense for it to rotate about its own axis. Instead, the electron is said to have an intrinsic angular momentum.