Classical counterpart of the quantum |nlm> states

[wdlang]

what are the classical counterparts of the quantum |nlm> states?

in a isotropic potential.

i am reading some books on Redberg atoms and i find this question not so trivial.

 

[Bob_for_short]

what are the classical counterparts of the quantum |nlm> states?

in a isotropic potential.

i am reading some books on Redberg atoms and i find this question not so trivial.

Roughly speaking, a classical trajectory with the same (big) angular momentum L = l and the same (big) energy E = E_n corresponds to the quantum state |nlm> with the projection m = l=L.

Stricltly speaking, a classical trajectory is a superposition of the states |nlm> with different n, l, and the projections m with E=<E_n>, L = <l>, and the average value of <m>=L.


Bob.

 

[Entropia]

The classical counterparts of the quantum |nlm> states are known as the Bohr-Sommerfeld states. These states are characterized by the same quantum numbers as the corresponding quantum states, but they describe the motion of a classical particle in an isotropic potential. In other words, they represent the allowed orbits of a classical particle around the nucleus of an atom in an isotropic potential.

The Bohr-Sommerfeld states were originally proposed by Niels Bohr and Arnold Sommerfeld in the early 20th century as a way to bridge the gap between classical and quantum mechanics. They are based on the assumption that the angular momentum of the particle is quantized, just like in quantum mechanics.

One of the key differences between the classical and quantum states is that in the classical states, the energy levels are continuous, while in the quantum states they are discrete. This is due to the wave-like nature of quantum particles, which allows them to only occupy certain energy levels.

Furthermore, the classical states also do not take into account the uncertainty principle, which is a fundamental principle in quantum mechanics that states that the position and momentum of a particle cannot be known simultaneously with absolute certainty.

Overall, the classical counterparts of the quantum |nlm> states provide a useful tool for understanding the behavior of atoms in an isotropic potential, but they do not fully capture the complexities of quantum mechanics.