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: Understanding the Question:
The question asks for the mathematical expression that gives the total number of permissible values for the magnetic quantum number ($m_{\ell}$) for any given value of the azimuthal quantum number ($\ell$).

Step 2: Key Formula or Approach:
In atomic quantum mechanics, the azimuthal quantum number ($\ell$) defines the shape of an orbital subshell. For a specific subshell value $\ell$, the magnetic quantum number ($m_{\ell}$) denotes the possible spatial orientations of those orbitals. The values of $m_{\ell}$ are integers restricted to the symmetric range: $$m_{\ell} = -\ell, \dots, 0, \dots, +\ell$$

Step 3: Detailed Explanation:
To count the total number of possible integer values in the set ranging from $-\ell$ to $+\ell$:

• There are $\ell$ negative integer values ($-\ell, -(\ell-1), \dots, -1$).

• There is exactly $1$ zero value ($0$).

• There are $\ell$ positive integer values ($+1, \dots, \ell-1, \ell$).
Summing these up gives the total count of orientations: $$\text{Total values} = \ell + 1 + \ell = 2\ell + 1$$ For instance, if $\ell = 2$ (d-subshell), the possible orientations are $2(2) + 1 = 5$ values (namely: $-2, -1, 0, +1, +2$). This matches option (B).

Step 4: Final Answer:
The number of possible values of the magnetic quantum number is $2\ell + 1$, corresponding to option (B).