Themes > Science > Chemistry > General Chemistry > Atomic Structure > Electronic Structures of Atoms > Atomic Structure Index > Quantum numbers


The solution to the Schrodinger equation for the hydrogen atom yields a set orbitals described by three quantum numbers, n, l and ml. An additional one, ms arises from an extension of the theory. Every electron in an atom resides in an orbital and according to the Pauli Exclusion Principle must have a unique set of quantum numbers.

  • n is the principal quantum number, and describes the principle energy level of the electron. This is the same number that Bohr used in his model of the hydrogen atom. n is an integer that can range from 1 to infinity, with larger n corresponding to higher energy orbitals.
  • l is the angular momentum quantum number. l is an integer that ranges from 0 to n-1. These describe the one or more sublevels to the principle energy level. The sublevels are often given letter designations as follows
    • l = 0 is an s subshell
    • l = 1 is an p subshell
    • l = 2 is an d subshell
    • l = 3 is an f subshell
    (The letters come from old spectroscopic notation.)
  • ml is the magnetic quantum number. Each l sublevel is made up of one or more orbitals. ml is an integer than can range from -l to +l
  • ms is the spin quantum number. Each electron acts like a small spinning top, although it is not actually spinning. The spin can be aligned in one of two directions, up or down. These are denoted +1/2 and -1/2.
Orbitals are denoted as n(l letter)ml description . In other words, an orbital denoted 2pz has n=2, l=1 and ml =0. Don't worry about knowing the designations of the ml number, but understand what a 4f orbital means, for example.

Models of two orbitals are shown below along with their quantum numbers. (The models are not to scale: the 1s orbital is much smaller than the 3dxy) Areas with lots of dots have a high probability of an electron being there, areas with few dots have a low probability.
1s orbital:
  • n = 1
  • l = 0
  • ml = 0
 3dxy orbital
  • n = 3
  • l = 2
  • ml = -2
A set of quantum numbers tell us everything we can know about an electron for an atom. For example, an electron with n=2, l=0, ml = 0 and ms = +1/2 is in a 2s orbital with spin up.

Example: An electron is in a 4f orbital. What are n and l for this electron?

Solution. n is the first number in the designation: 4 in this case. An f sublevel has l =3, so the electron has n=4, l=3


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