Question

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

Explanation:

Step1: Recall quantum number rules

The four quantum numbers follow these rules:

1. Principal quantum number $n \geq 1$ (integer)
2. Azimuthal quantum number $l$: $0 \leq l \leq n-1$ (integer)
3. Magnetic quantum number $m_l$: $-l \leq m_l \leq +l$ (integer)
4. Spin quantum number $m_s = +\frac{1}{2}$ or $-\frac{1}{2}$

Step2: Check Option 1 (4,2,-1,-½)

$n=4$, so $l$ can be 0,1,2,3 (valid, $l=2$). $m_l$ ranges -2 to +2 (valid, $m_l=-1$). $m_s=-\frac{1}{2}$ is valid.

Step3: Check Option 2 (1,0,0,+½)

$n=1$, so $l$ can only be 0 (valid). $m_l$ ranges 0 to 0 (valid, $m_l=0$). $m_s=+\frac{1}{2}$ is valid.

Step4: Check Option 3 (4,2,3,-½)

$n=4$, $l=2$ (valid). $m_l$ must be between -2 and +2. Here $m_l=3$, which is outside the allowed range. This set is invalid.

Step5: Check Option 4 (3,1,-1,+½)

$n=3$, so $l$ can be 0,1,2 (valid, $l=1$). $m_l$ ranges -1 to +1 (valid, $m_l=-1$). $m_s=+\frac{1}{2}$ is valid.

Answer:

4, 2, 3, -½