In a hydrogen sample atoms are excited to the state with principal quantum number n. The number of different possible wavelengths emitted is

Options

A. `n(n-1)/2`
B. `n(n+1)/2`
C. `n^2/2`
D. `n^2/2+1`

Answer

A

Explanation

Let the atom be in `n^(th)` state and the possible transitions will be `(n - 1)^(th)` state, `(n - 2)^(th)` state.... And so on to `1^(st)` state.
Thus for electron in `n^(th)` state there are (n-1) possible transitions.
Now, let the atom be in (n-1) the state, then possible transitions will be `(n - 2)^(th)` state, `(n-3)^(th)` state ... so on to `1^(st)` state.
Thus there are again (n-2) possible ways. Similarly for electron in `(n - 2)^(th)` state, there are (n-3) possible transitions. Thus total transitions will be sum total of (n-1) + (n-2)+(n-3)+....+ 3+2+1 .
`rArr (n-1)((n-1+1))/2=(n(n-1))/2`