# Quantum mechanical model of atom and the quantum numbers

I studied Bohr's model of atom and then the drawbacks of it and then quantum mechanical model of atom. Now quantum model is according to uncertainty principal and dual nature of matter and it says we can not talk about fixed orbits (uncertainty principle) and can only talk of an area of maximum probability of finding an electron known as "orbital". Now when I'm studying principal quantum number, it says it is the "shell number". So how come we are talking about fixed shells again and returning to what Bohr said?

• The Bohr model is completely wrong and really shouldn't be used for anything other than a historical account of how Quantum Mechanics came about. It did okay for explaining the atomic emission line spectrum for the Hydrogen atom but that is about it. Related: chemistry.stackexchange.com/questions/31639/… – LordStryker Jun 13 '15 at 18:52
• There's no fixed in the quantum theory! Unless, if you consider the energies as fixed. FWIW, I think if you study a little bit more, you simply wouldn't have this question, as it rises from a misunderstanding about what shells are. – M.A.R. Jun 13 '15 at 19:03

So how come we are talking about fixed shells again and returning to what Bohr said?

Nope, we are not returning to the old quantum theory. We do not talk about orbits anymore, we rather have orbitals. Note that it is important to bear in mind that orbitals also aren't real, i.e. this description is still approximate, but it is a much better way of thinking about atoms (and molecules) than Bohr model.

Now, to understand the concept of orbitals you have to learn the basics of quantum chemistry first, namely, the Hartree-Fock method. Until then just realize that:

• We do use hydrogen-like one-electron wave functions (orbitals, or, spin-orbitals, to be more precise) to describe the state of each and every electron in any atom.
• These hydrogen-like orbitals are described by a set of quantum numbers: the principal quantum number, the azimuthal quantum number, the magnetic quantum number, and the spin quantum number.
• The set of orbitals belonging to a given principal quantum number $n$ constitutes what we call a shell in the orbital model.
• When you say that orbitals aren't real, do you mean to say that orbitals are not an exact description of the system? One might argue that (not in the complex number sense) wave functions are a very realistic aspect of any quantum system. – Ian Jun 15 '15 at 5:20
• Also, the hydrogenic orbitals are exact in one-electron atoms like H or He+. – Ian Jun 15 '15 at 5:21
• @Ian, the reason to say that orbitals aren't real is that the electronic wave function for a many-electron system could not be written as a product (antisymmetrized product, to be more precise) of one-electron wave functions (say, hydrogenic orbitals). As a result, one-electron wave functions (orbitals) arise in the description of a many-electron only as a part of an approximation (the HF one). It is in this sense they aren't real. – Wildcat Jun 15 '15 at 6:31
• @Ian, well, strictly speaking, the hydrogenic orbitals are not exact even for one-electron atoms. They are exact only in the non-relativistic approximation, which is, well, an approximation. :D So even in this case one can argue that the hydrogenic orbitals aren't real, though, in this case it is a sort of nitpicking. – Wildcat Jun 15 '15 at 6:36

We are not talking about fixed shells. Rather, we are observing that for any given atomic orbital you can find an orbital of very similar shape but of a higher energy. Examples:

• The simplest orbital is the 1s orbital of hydrogen. We can find an orbital of higher energy that has the same shape (namely: spherical symmetry): The 2s orbital.

• There is an orbital called $3\mathrm{d}_{z^2}$ which is rather important in transition metal chemistry. Its ‘shape’ consists of two lobes of identical phase with a nodal plane in the middle plus a ‘ring’ of the opposite phase around the middle. Again, an orbital of similar shape but higher energy can be found ($4\mathrm{d}_{z^2}$).

This fitted in so nicely with the previous models, that they were called ‘same orbitals of different shells’. They’re still orbitals, so wave functions, but we grouped them into shells.

As Ian correctly mentioned in a comment, when dealing with a one-electron system (such as the basic hydrogen atom) all orbitals of the same shell will have identical energies. (That’s just what Schrödinger’s maths tells us.) Another reason to still use the word shell without referring to it’s meaning of the fixed Bohr orbits.

• Also, we organize the orbitals by energy. In one-electron atoms like hydrogen, any orbitals with the same principal quantum number will have the same energy. This fact is not quite true for multi-electron atoms, but there is a sufficiently reasonable direct mapping from the orbitals in a one-electon atom to the "orbitals" in a Hartree-Fock multi-electron atom such that it still makes sense to assign orbitals with principal quantum numbers and to organize them into groups by principal quantum number. – Ian Jun 15 '15 at 5:25