Degeneracy

This problem was asked in JEE 2015. It states:
If the degeneracy of hydrogen atom in second excited state (n=3n=3) is 9 then find the degeneracy of hydride anion i.e. HX\ce{H^-} ion in second excited state.

Note: electronic spin is not to be considered here.

Note by Dipanjan Chowdhury
3 years, 7 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Before I answer this, let me ask you a question that will be key to understanding what's happening here:
Why is the degeneracy of hydrogen atom 9 when n = 3?

Ameya Daigavane - 3 years, 7 months ago

Log in to reply

I think because of the one 3s three 3p and five 3d orbitals which are not excited as have no electron and as a whole exhibit degeneracy

Dipanjan Chowdhury - 3 years, 7 months ago

Log in to reply

Okay, good! The fact that hydrogen has only one electron means that orbitals with the same value of n have the same energies. n = 3 has 9 orbitals, as you said, so 9 is the degeneracy.
But for H-, there are two electrons. The orbitals' energies now depend on l (the azimuthal quantum number that differentiates between s,p, d, f and so on.) like, 1s<2s<2p<3s<3p... 1s < 2s < 2p < 3s < 3p ...
Now the ground state of H- is 1s21s^2 .
The 1st excited state is 1s12s1 1s^1 2s^1 .
The second excited state is 1s12s02p1 1s^1 2s^0 2p^1 (Compare with H atom for which the 2nd excited state is 3s1 3s^1 .)
But there are 3 p orbitals for the electron to 'choose' between when making the transition from first excited state to second excited state.
So the degeneracy is 3. Hope this helps.

Ameya Daigavane - 3 years, 7 months ago

Log in to reply

@Ameya Daigavane Absolutely clear, Sir. Thank You . :D I was unaware of this sort of splitting.

Dipanjan Chowdhury - 3 years, 7 months ago

Log in to reply

Sir isn't the ground state of an atom 1s¹

Abhinav Rajyakirti - 2 years, 5 months ago

Log in to reply

Yes .

Dipanjan Chowdhury - 2 years, 5 months ago

Log in to reply

1s2^2 is the ground state electronic configuration. So, 1s1^1 2s1^1 is the first excited state and 1s1^1 2p1^1 is the second excited state. Now in 2p, all 3 orientations are degenerate so the degeneracy is 3.

Sai Krishna Attaluri - 2 years, 4 months ago

Log in to reply

3

Qwerty Asdfghjkl - 1 year, 5 months ago

Log in to reply

3

Pratham Pahuja - 1 year, 3 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...