Question

which of the following is an impossible set of quantum numbers?
1, 0, 0, -½
2, 2, -1, +½
4, 3, -2, +½
3, 2, 0, -½

Explanation:

Step1: Recall quantum number rules

For a principal quantum number $n$:

• Azimuthal quantum number $l$ must satisfy $0 \leq l \leq n-1$
• Magnetic quantum number $m_l$ must satisfy $-l \leq m_l \leq l$
• Spin quantum number $m_s$ is either $+\frac{1}{2}$ or $-\frac{1}{2}$

Step2: Check Option 1 ($1,0,0,-\frac{1}{2}$)

$n=1$, so $l$ can only be $0$. $m_l=0$ (valid for $l=0$), $m_s=-\frac{1}{2}$ (valid). This set is possible.

Step3: Check Option 2 ($2,2,-1,+\frac{1}{2}$)

$n=2$, so $l$ must be $0$ or $1$. Here $l=2$, which violates $l \leq n-1$. This set is impossible.

Step4: Verify remaining options (optional)

• Option3: $n=4$, $l=3$ (valid, $3 \leq 4-1$), $m_l=-2$ (valid, $-3 \leq -2 \leq 3$), $m_s=+\frac{1}{2}$ (valid).
• Option4: $n=3$, $l=2$ (valid, $2 \leq 3-1$), $m_l=0$ (valid, $-2 \leq 0 \leq 2$), $m_s=-\frac{1}{2}$ (valid).

Answer:

2, 2, -1, +½