Difference between shells, subshells and orbitals

What are the definitions of these three things and how are they related? I've tried looking online but there is no concrete answer online for this question.




Here's a graphic I use to explain the difference in my general chemistry courses:


![Shells, sub-shells, and orbitals](https://i.stack.imgur.com/i8V5L.png)


* All electrons that have the same value for **$n$** (the principle quantum number) are in the same **shell**
* Within a shell (same $n$), all electrons that share the same **$l$** (the angular momentum quantum number, or orbital shape) are in the same **sub-shell**
* When electrons share the same **$n$**, **$l$**, and **$m\_l$**, we say they are in the same **orbital** (they have the same energy level, shape, and orientation)


So to summarize:


* same $n$ - shell
* same $n$ and $l$ - sub-shell
* same $n$, $l$, and $m\_l$ - orbital


Now, in the other answer, there is some discussion about spin-orbitals, meaning that each electron would exist in its own orbital. For practical purposes, you don't need to worry about that - by the time those sorts of distinctions matter to you, there won't be any confusion about what people mean by "shells" and "sub-shells." For you, for now, orbital means "place where up to two electrons can exist," and they will both share the same $n$, $l$, and $m\_l$ values, but have opposite spins ($m\_s$).


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Have a look [here](http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html#shells):



> 
> Orbitals that have the same value of the principal quantum number $n$ form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number $l$. Chemists describe the shell and subshell in which an orbital belongs with a two-character code such as 2p or 4f. The first character indicates the shell (n = 2 or n = 4). The second character identifies the subshell. By convention, the following lowercase letters are used to indicate different subshells.
> 
> 
> * s: l = 0
> * p: l = 1
> * d: l = 2
> * f: l = 3
> 
> 
> 


What is called an orbital might differ according to the context.
With orbitals in the context of shells and subshells one usually means atomic orbitals, i.e. two-electron eigenstates of an atom's Hamilton operator which are characterized by the three quantum numbers: the principal quantum number $n$, the angular quantum number $l$ and the magnetic quantum number $m$.
But often the word orbital is also used for spin-orbitals, i.e. one-electron eigenstates of the system's one-electron Hamilton operator which are characterized not only by $n$, $l$ and $m$ but also by the spin quantum number $m\_{\mathrm{s}}$ which can be either $+\frac{1}{2}$ or $-\frac{1}{2}$.


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