Question

an electron in an atom is known to be in a state with magnetic quantum number m_l = 1. what is the smallest possible value of the principal quantum number n of the state?

Explanation:

Step1: Recall quantum - number relationship

The magnetic quantum number $m_l$ ranges from $-l$ to $l$, and the orbital - angular - momentum quantum number $l$ ranges from $0$ to $n - 1$. Given $m_l=1$, the minimum value of $l$ for which $m_l = 1$ is possible is $l = 1$.

Step2: Find minimum $n$ from $l$

Since $l$ ranges from $0$ to $n - 1$, and $l = 1$, then solving the inequality $l\leq n - 1$ for the minimum $n$ gives $n=l + 1$. Substituting $l = 1$ into the equation, we get $n=2$.

Answer:

2