Question

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

Explanation:

Step1: Recall quantum number rules

The rules are:

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 $m_s = \pm\frac{1}{2}$

Step2: Check Option 1 (1,0,0,+½)

$n=1$, $l=0$ (valid, $0\leq0\leq0$), $m_l=0$ (valid, $-0\leq0\leq0$), $m_s=+\frac{1}{2}$ (valid). Set is possible.

Step3: Check Option 2 (3,1,-1,+½)

$n=3$, $l=1$ (valid, $0\leq1\leq2$), $m_l=-1$ (valid, $-1\leq-1\leq1$), $m_s=+\frac{1}{2}$ (valid). Set is possible.

Step4: Check Option 3 (4,2,3,-½)

$n=4$, $l=2$ (valid, $0\leq2\leq3$), $m_l=3$ (invalid, since $-2\leq m_l\leq2$; 3 is outside this range).

Step5: Check Option 4 (4,2,-1,-½)

$n=4$, $l=2$ (valid, $0\leq2\leq3$), $m_l=-1$ (valid, $-2\leq-1\leq2$), $m_s=-\frac{1}{2}$ (valid). Set is possible.

Answer:

4,2,3,-½