Quantum Numbers Primer - Oregon State University

Quantum Numbers Primer
The quantum numbers (n, l, ml, and ms) in the wave model each has physical significance and
rules for use.
n is the principal quantum number (n = 1, 2, 3..., ∞)
n designates the principal energy level—it tells us something about the
energy (En=-RH/n2 in hydrogen) and the radii (rn=n2a0 in hydrogen).
Orbitals with the same value of n are said to be in the same shell.
Consider the hydrogen atom. When the electron occupies the n = 1 orbital (the lowest
value allowed), the atom is at the lowest energy and the radius is the smallest. We call
this lowest energy state the ground state.
We can promote an electron in the n = 1 energy level to a higher level; such as n = 2.
This excitement was demonstrated in lecture by use of the hydrogen lamp. When the
electron returns to the ground state (n = 1) energy is emitted in the form of
electromagnetic radiation (radio, microwave, IR, visible, UV, X-ray).
l is the sublevel (sometimes called the orbital or angular momentum) quantum number (l = 0, 1, 2..., n-1)
l designates the shape of the electron cloud (orbital) (the region of space that represents
the volume that the electron occupies 90% of the time).
When l = 0 (called an s-orbital) the shape is a sphere. When 1 = 1 (called a p-orbital) the
shape is dumb bell. When l = 2 (called the d-orbital) the shape is, well butterfly and
ml is the magnetic quantum number (ml = -ℓ, …, –2, -1, 0, +1, +2, …, +ℓ)
(note: ℓ is lowercase L... it was used here so it is not confused with the number one).
ml determines the number and orientation of the orbital.
When n = 1, l must be 0. When l = 0, ml = 0. Because ml has only one value (the value
0), there is only one 1s orbital. The label 1s comes from n = 1 and l = 0.
When n = 2, l can be 0 or 1. When l = 0, ml = 0. Because ml has only one value (the
value 0), there is only one 2s orbital. The label 2s comes from n = 2 and l = 0. When l =
1, ml = -1, 0, +1. Because ml has three values (the values –1, 0, and +1), there are three
2p orbitals. These three 2p orbitals are orthogonal. The label 2p comes from n = 2 and l
= 1.
ms is the spin quantum number (ms = +1/2 or –1/2)
Spin is an intrinsic quantum number, which causes electrons to behave as if they are
spinning on their axis, generating a magnetic field. One result of spin is that electrons
orient with an external electric or magnetic field. The ms quantum number takes the
values +1/2 (spin up) and -1/2 (spin down). Electrons really do not spin on their axis
(why would they all orient themselves in space to spin in only 2 directions?).