Find pairs of two hydrogen atom principle quantum numbers n for which the difference in energy eigenvalue is the same and hence they give rise to coincident spectral lines.

Question

Find pairs of two hydrogen atom principle quantum numbers [latex]n[/latex] for which the difference in energy eigenvalue is the same and hence they give rise to coincident spectral lines.

Accepted Answer

We wish to find pairs of quantum numbers $(n_{1},n_{2})$ and $(n_{3},n_{4})$ such that $\frac{1}{n^{2}_{1} } -\frac{1}{n^{2}_{2} }=\frac{1}{n^{2}_{3} }-\frac{1}{n^{2}_{4} }.$

For example, within the first 100 principle quantum numbers

The way to find these numbers is to write a computer program.

For example, within the first 100 principle quantum numbers

The way to find these numbers is to write a computer program.

$n_{1}$	$n_{2}$	$n_{3}$	$n_{4}$
5	6	9	90
5	7	7	35
5	9	6	90
6	8	9	72
6	9	8	72
7	8	14	56
7	14	8	56
10	11	22	55
10	14	14	70
10	22	11	55
14	18	21	63
14	21	18	63
30	34	51	85
30	51	34	85
36	45	48	80
36	48	45	80

Question 11.3: Find pairs of two hydrogen atom principle quantum numbers n ......

Related Answered Questions

Verified Answer:

Use first-order perturbation theory to estimate the energy shifts of the hydrogen 2s and 2p states due to the fact that the proton is not a point charge, treating it (for simplicity) as a uniformly charged hollow spherical shell of radius b = 5×10^-14 cm. Comment on these results. Compare the ...

Verified Answer:

(a) What is the effect of applying a uniform electric field on the energy spectrum of an atom? (b) If spin effects are neglected, the four states of the hydrogen atom with quantum number n = 2 have the same energy, E^0. Show that when a uniform z-directed electric field E is applied to hydrogen ...

Verified Answer:

Verified Answer: