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First in school I learnt that when supplied enough energy, the electron in a hydrogen atom will jump energy level(s), denoted by $n=1$, $n=2$ and so on for integral values of $n$. Then I learnt about shells, subshells and orbitals and what they have to do with the quantum numbers $n$, $l$ and $m_l$.

  1. I'm confused about the difference between energy level and shell, if there is any. The quantum number $n$ tells us the shell, but we also use $n$ to talk about energy levels in hydrogen atom as well.
  2. When we say that the electron in hydrogen jumps energy levels, does it mean that it is jumping shells? Is it going from the orbital in $1s$ subshell to the orbital in $2s$ subshell? If we supply even more energy, can it go from the orbital in $1s$ subshell to one of the three orbitals in the $2p$ subshell? Do my questions make sense, or am I wrong about some basic fundamentals?
  3. A comment to a different question said:

"s orbitals have lower energy than p orbitals which have lower energy than d orbitals" This statement is only true when there is more than one electron.

Why is it not true in multi-electron atoms? Is the person referring to the Aufbau Principle i.e. the orbitals in the $4s$ subshell have lower energy than the orbitals in $3d$ subshell?

If I have used any terminology incorrectly, please do let me know.

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  • $\begingroup$ In a single-electron atom (such as H or He+), the $l$ and $m$ quantum numbers make no difference, so all $n$ levels are the same energy. In multi-electron atoms that is no longer the case since electron-electron effects come in to play, causing all sorts of headaches in the transition metals. Also, to perhaps add to your confusion, since a photon has spin, the transition from 1$s$ to 2$s$ will be less favored than 1$s$ to 2$p$. $\endgroup$
    – Jon Custer
    Jul 27, 2022 at 12:48
  • $\begingroup$ Could you please explain why the $l$ and $m$ quantum numbers not matter? $\endgroup$
    – AVS
    Jul 27, 2022 at 14:44
  • $\begingroup$ in the single electron case, the energy levels for a given $n$ are the same regardless of $l$ and $m$. $\endgroup$
    – Jon Custer
    Jul 27, 2022 at 14:52
  • $\begingroup$ so if n=2 (i.e. 2nd shell) the energy levels of 2s and 2p are the same? i thought 2p > 2s in terms of energy $\endgroup$
    – AVS
    Jul 27, 2022 at 16:03
  • $\begingroup$ @JonCuster Under certain experimental conditions (in a magnetic field, in a non-homogeneous magnetic field) you can make hydrogen-spectral lines "split", i.e. the states are no longer degenerate, see e.g. here. $\endgroup$
    – Karsten
    Jul 27, 2022 at 16:42

1 Answer 1

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Shell, level and orbital can often be used interchangeably although they do have different meanings.

The level represents an ordinal or numerical representation of the state and corresponding relative energy of an electron in an atom. Electrons occupy particular quantum states described by fixed quantum numbers and with fixed associated energies. Therefore you can refer interchangeably to the state, energy or quantum number(s) of an electron, keeping in mind the possibility of degeneracy, which occurs when various possible states share the same energy.

Shell suggests an onion-like atomic structure, in which electrons are wrapped around the nucleus in shells, with shells of lower energy electrons closer to the nucleus. This gels with Bohr's idea of electrons occupying fixed orbits about the nucleus. Despite being incorrect, the fixed orbit idea is useful when trying to visualize how energy levels and electron distributions (such as represented with orbitals, see below) in a hydrogen atom (or a hydrogenic atom, one with a single electron) are related. A particular shell is associated with a value of the principal atomic quantum number and may contain subshells, each representing different possible values of the angular momentum quantum numbers.

An orbital describes the spatial distribution of an individual electron and is typically depicted as a boundary that encompasses a volume in which an electron with particular quantum numbers (and corresponding energy) is found with a certain high probability. Higher energy electrons will generally have orbitals with probability extending further away from the nucleus, which is consistent with the idea of a shell.

As an example, when we refer to an electron as being in the 1s subshell (or equivalently in a 1s orbital), we are specifying its energy and angular momentum (but not spin) quantum numbers ($\ce{n=1, l=0, m_l=0}$), and thereby its energy and angular momenta (but not spin angular momentum). The energy of an electron in the 1s shell can be computed exactly as -13.6 eV (relative to a stationary free electron).

Electrons may undergo transitions ("jumps") between shells corresponding to different energy levels and quantum numbers, subject to transition rules. In other words, not all imaginable transitions are allowed, because various properties (specifically angular momentum) must be conserved, just like energy.

When an electron in a hydrogen atom is excited into a higher energy state (or level), its main quantum number must change, since for hydrogen atoms states with the same main quantum number but differing in angular momentum are degenerate in energy (see above).

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