Angular Momentum: l and m_l ranges

l and m_l ranges

• For each azimuthal (orbital) quantum number value $l$, the magnetic quantum number $m_l$ takes a range of integer values from
  -l to +l inclusive.
• General expression for the allowed set of $m<em>l$: $m</em>l \in { -l, -l+1, \ldots, l-1, l }$
• In inequality form, the range is:
  $-l \le m<em>l \le l, \quad m</em>l \in \mathbb{Z}$
• Examples:
  • If $l = 1$, then $m_l \in { -1, 0, 1 }$
  • If $l = 2$, then $m_l \in { -2, -1, 0, 1, 2 }$
• Number of possible $m_l$ values for a given $l$ is:
  $\text{Number of values} = 2l + 1$
• Special case: if $l = 0$, then $m_l = 0$ (one value)
• Physical meaning:
  • $m_l$ labels the projection of the orbital angular momentum along a chosen axis (commonly the z-axis).
  • The set of possible $m_l$ values reflects the different orientation states of the orbital angular momentum for a given $l$ (2l+1 orientations).
• Context in atomic structure:
  • In a central potential (e.g., hydrogen-like atoms), energy levels depend on the principal quantum number $n$ and the azimuthal quantum number $l$, but are independent of the magnetic quantum number $m<em>l$ (degeneracy with respect to $m</em>l$).
  • If an external magnetic field is applied along the z-axis, the different $m_l$ states can split (Zeeman effect) due to field-coupling to the angular momentum projection.
• Quick recap of the transcript specifics:
  • For $l = 1$, $m_l = -1, 0, +1$.
  • For $l = 2$, $m_l = -2, -1, 0, +1, +2$.
  • In general, the allowed $m_l$ are all integers from $-l$ to $+l$ inclusive.
• Related formulas to remember:
  • Allowed set: $m_l \in {-l, -l+1, \ldots, l-1, l}$
  • Range inequality: $-l \le m_l \le l$
  • Count: $2l + 1$ values
• Connections to broader concepts:
  • $l$ is the azimuthal quantum number.
  • The pair ( $l$, $m_l$ ) characterizes angular momentum state and its projection in a given axis.
  • Understanding these values underpins orbital shapes, Hund’s rules, and selection rules in transitions.