How to Calculate Magnetic Quantum Number?

The magnetic quantum number (m_l) is calculated based on the value of the azimuthal quantum number (l). The magnetic quantum number describes the orientation of an atomic orbital in space.

Understanding the Relationship

• Azimuthal Quantum Number (l): This number describes the shape of the atomic orbital and has values ranging from 0 to n-1, where n is the principal quantum number. For example:
  • l = 0 corresponds to an s orbital (spherical)
  • l = 1 corresponds to a p orbital (dumbbell-shaped)
  • l = 2 corresponds to a d orbital (more complex shape)
• Magnetic Quantum Number (m_l): For a given value of l, m_l can take on integer values ranging from -l to +l, including 0.

Calculation

The possible values for m_l are determined by the following rule:

m_l = -l, -l+1, ..., -1, 0, 1, ..., l-1, l

This means that for a given l, there are a total of (2l + 1) possible values for m_l.

Examples

Let's illustrate with some examples:

• If l = 0 (s orbital):

  • m_l = 0
  • There is only one possible orientation for an s orbital (it's spherical).

• If l = 1 (p orbital):

  • m_l = -1, 0, +1
  • There are three possible orientations for a p orbital, corresponding to the p_x, p_y, and p_z orbitals.

• If l = 2 (d orbital):

  • m_l = -2, -1, 0, +1, +2
  • There are five possible orientations for a d orbital.

Summary

To calculate the magnetic quantum number:

1. Determine the azimuthal quantum number (l).
2. Apply the rule: m_l = -l to +l (including 0).
3. List all the possible integer values within that range. The number of values you list will always be (2l + 1).