# 6.4 Quantum Numbers and Atomic Orbitals ## Atomic Orbitals ## Quantum Numbers

The following table summarizes the name, meaning, and range of possible values for each of the quantum numbers.

NAME MEANING RANGE
n Principal Shell Number [1...∞]
l Azimuthal Subshell (Type of Orbital) [0...(n-1)]
ml Magnetic Orientation in Space [-l...+l]
ms Spin Spin Up or Spin Down +1/2 or -1/2

### The Principal Quantum Number (n)

The principal quantum number, n, gives the electronic shell in which an electron resides and can have values from 1 to ∞.

For an electron in the 1s orbital n=1.

For an electron in a 2s or 2p orbital n=2.

For an electron in a 3s, 3p, or 3d orbital n=3, etc.

In the hydrogen atom (or any one electron atom/ion) all the orbitals with the same principal quantum number are degenerate, that is they have the same energy.  But in any atom with more than one electron this degeneracy disappears due to electronic repulsions.

### The Azimuthal Quantum Number (l)

The azimuthal quantum number, l, (a.k.a. the orbital angular momentum quantum number) gives an indication regarding the angular momentum of the orbital in which an electron resides and can have values from 0 to (n-1).  Ultimately it indicates the number of angular nodes for an orbital, but students will recognize that this tells them the 'type' of orbital or subshell in which an electron resides.

The azimuthal quantum number of an s orbital is zero (l=0) indicating that an s orbital has no angular nodes.

The azimuthal quantum number of a p orbital is 1 (l=1) indicating that a p orbital has 1 angular node.

The azimuthal quantum number of a d orbital is 2 (l=2) indicating that a d orbital has 2 angular nodes.

The azimuthal quantum number of an f orbital is 3 (l=3) indicating that an f orbital has 3 angular nodes.

### The Magnetic Quantum Number (ml)

The magnetic quantum number, ml, gives the orientation in space of the orbital in which an electron resides and can have values of -l to +l (determined by the value of the azimuthal quantum number).

An s orbital is spherical.  Being the only possible orientation in space and ml=0 for an s orbital.

A p orbital is 'dumbbell' shaped.  It can be oriented along either the x-, y-, or z-axis and ml can have values of -1, 0, and +1 for a p orbital.

Most of the d orbitals look like a four-leaf clover.  There are 5 different orientations in space and ml can have values of -2, -1, 0, +1, and +2 for a d orbital.

The shapes of the f orbitals are complex.  There are 7 different orientations in space and ml can have values of -3, -2, -1, 0, +1, +2, and +3 for an f orbital.

### The Spin Quantum Number (ms)

The spin quantum number, ms, gives an indication regarding how an electron will interact with a magnetic field.  It can have a value of either -1/2 or +1/2.  This is the only one of the four quantum numbers that is not derived from the Schroedinger equation.

## Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers.  Ultimately, this means that you can't have two electrons in the same orbital with the same spin.