QM Questions: Operators, Eigenfunctions, and Hydrogen-like Atoms Explained"

  • Thread starter deep582
  • Start date
  • Tags
    Qm
In summary, this dialogue discusses how the energy levels of an atom depend on its n and l values. The levels are not always degenerate, and there is a symmetry that affects hydrogen.
  • #1
deep582
1
0
I just had a couple quick questions while I'm reviewing for a test...

If two operators commute, what can we say about their eigenfunctions?

x commutes with pz, correct?

I understand that the energy for a hydrogen-like atom depends on n according to the equation ((-z^2 μe^4)/(2(4πϵ_o )^2 ℏ^2 ) 1/n^2 ), but why does it not depend on l also? Isn't a 3p electron supposed to have more energy than a 3s electron? Or is that only in many electron atoms?

Thanks, any help appreciated
 
Physics news on Phys.org
  • #2
deep582 said:
If two operators commute, what can we say about their eigenfunctions?
If two [hermitian] operators commute, then there exists a common eigenbasis for the two operators.
deep582 said:
x commutes with pz, correct?
Indeed.
deep582 said:
I understand that the energy for a hydrogen-like atom depends on n according to the equation ((-z^2 μe^4)/(2(4πϵ_o )^2 ℏ^2 ) 1/n^2 ), but why does it not depend on l also? Isn't a 3p electron supposed to have more energy than a 3s electron? Or is that only in many electron atoms?
You are correct. Since the Hydrogen atom has a purely Coulomb potential, the energy levels are indeed degenerate with respect to l. However, as you correctly note, this is not the case with many-electron atoms.
 
  • #3
Actually, even in hydrogen the 3p and 3s (or 2p and 2s) levels are not quite degenerate because of QED effects: the Lamb shift of 2p vs 2s being the most famous example.

But leaving out QED effects the levels with the same n and different l are indeed degenerate in H: this has to do with a symmetry that the H atom has, but no other: the Laplace-Runge-Lenz vector is conserved for an exact 1/r potential.
 
  • #4
Attention mna skt!

mna skt, I've moved your homework question to a new thread in one of the homework forums... click the following link to go to it:

https://www.physicsforums.com/showthread.php?t=271329

Everybody else, please carry on. I'll try to remember to delete this post in a few days.
 

1. What are operators in quantum mechanics?

In quantum mechanics, operators are mathematical tools used to represent physical quantities, such as position and momentum, within the framework of the theory. They are represented by symbols and act on quantum states to produce measurable values.

2. What are eigenfunctions in quantum mechanics?

Eigenfunctions are solutions to the Schrödinger equation in quantum mechanics. They represent the possible states of a quantum system and are associated with specific eigenvalues, which correspond to the measurable values of physical quantities.

3. How do eigenfunctions and eigenvalues relate to each other?

Eigenfunctions and eigenvalues are closely related in quantum mechanics. The eigenvalues correspond to the measurable values of physical quantities, while the eigenfunctions represent the possible states of a system that can have those values.

4. What is a hydrogen-like atom in quantum mechanics?

In quantum mechanics, a hydrogen-like atom is a simplified model used to describe the electronic structure of atoms with only one electron, such as hydrogen, helium, and lithium. It assumes that the nucleus is a point charge and the electron moves in a spherically symmetric potential.

5. How are operators used to describe hydrogen-like atoms in quantum mechanics?

In quantum mechanics, operators are used to describe the physical properties of hydrogen-like atoms. For example, the position operator describes the location of the electron in the atom, while the kinetic energy operator describes the energy associated with the electron's motion. These operators act on the eigenfunctions of the system to produce measurable values, such as the electron's energy levels.

Similar threads

  • Quantum Physics
Replies
31
Views
2K
Replies
18
Views
1K
Replies
2
Views
770
  • Quantum Physics
Replies
24
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Quantum Physics
2
Replies
38
Views
3K
  • Quantum Physics
Replies
2
Views
734
Replies
3
Views
1K
Replies
16
Views
2K
Back
Top