Question

How many values of magnetic quantum number are possible for each value of azimuthal quantum number?

• A. $n \ell$
• B. $2\ell + 1$
• C. $n - \ell$
• D. $2\ell$

Hint:

Think of the magnetic quantum number as counting individual orbital boxes in a subshell diagram. An s-subshell ($\ell=0$) has $2(0)+1=1$ box, a p-subshell ($\ell=1$) has $2(1)+1=3$ boxes, and a d-subshell ($\ell=2$) has $2(2)+1=5$ boxes!

Answer 1

The Correct Option is B

Step 1: Recall what each number controls.
The azimuthal number $\ell$ fixes the subshell shape, and the magnetic number $m_\ell$ counts the orientations of that subshell in space.
Step 2: Write the allowed range of $m_\ell$.
For a given $\ell$, the values run as integers from $-\ell$ up to $+\ell$, that is $m_\ell = -\ell, \dots, 0, \dots, +\ell$.
Step 3: Count the negative entries.
The negatives are $-1, -2, \dots, -\ell$, which is exactly $\ell$ values.
Step 4: Count zero and the positives.
There is one zero, and the positives $+1, +2, \dots, +\ell$ give another $\ell$ values.
Step 5: Add them up.
Total $= \ell + 1 + \ell = 2\ell + 1$.
Step 6: Check with a known case.
For the d subshell, $\ell = 2$ gives $2(2)+1 = 5$ orientations, matching the five d orbitals.
\[ \boxed{2\ell + 1 \text{ (option B)}} \]