Question

For hydrogen atom the difference between any two consecutive energy levels (where $n$ is the principal quantum number)

• A. is always the same
• B. decreases inversely with $n$
• C. decreases inversely with $n^2$
• D. decreases inversely with $n^3$

Answer 1

The correct option is D decreases inversely with $n^3$

The difference between any two consecutive energy levels.
$E_n[\frac{1}{(n-1{)}^2}-\frac{1}{n^2}]$$=E_n\left[\frac{2n-1}{n^2(n-1{)}^2}\right]$

So, approximating as $n\to \infty$
Difference $=E_n\left[\frac{2}{n^3}\right]=\frac{2E_n}{n^3}$

So, the difference decreases inversely with $n^3$.