Chemistry - What do the quantum numbers actually signify?
Solution 1:
Quantum numbers give information about the location of an electron or set of electrons. A full set of quantum numbers describes a unique electron for a particular atom.
Think about it as the mailing address to your house. It allows one to pinpoint your exact location out of a set of $n$ locations you could possibly be in. We can narrow the scope of this analogy even further. Consider your daily routine. You may begin your day at your home address but if you have an office job, you can be found at a different address during the work week. Therefore we could say that you can be found in either of these locations depending on the time of day. The same goes for electrons. Electrons reside in atomic orbitals (which are very well defined 'locations'). When an atom is in the ground state, these electrons will reside in the lowest energy orbitals possible (e.g. 1$s^2$ 2$s^2$ and 2$p^2$ for carbon). We can write out the physical 'address' of these electrons in a ground-state configuration using quantum numbers as well as the location(s) of these electrons when in some non-ground (i.e. excited) state.
You could describe your home location any number of ways (GPS coordinates, qualitatively describing your surroundings, etc.) but we've adapted to a particular formalism in how we describe it (at least in the case of mailing addresses). The quantum numbers have been laid out in the same way. We could communicate with each other that an electron is "located in the lowest energy, spherical atomic orbital" but it is much easier to say a spin-up electron in the 1$s$ orbital instead. The four quantum numbers allows us to communicate this information numerically without any need for a wordy description.
Of course carbon is not always going to be in the ground state. Given a wavelength of light for example, one can excite carbon in any number of ways. Where will the electron(s) go? Regardless of what wavelength of light we use, we know that we can describe the final location(s) using the four quantum numbers. You can do this by writing out all the possible permutations of the four quantum numbers. Of course, with a little more effort, you could predict the exact location where the electron goes but in my example above, you know for a fact you could describe it using the quantum number formalism.
The quantum numbers also come with a set of restrictions which inherently gives you useful information about where electrons will NOT be. For instance, you could never have the following possible quantum numbers for an atom:
$n$=1; $l$=0; $m_l$=0; $m_s$=1/2
$n$=1; $l$=0; $m_l$=0; $m_s$=-1/2
$n$=1; $l$=0; $m_l$=0; $m_s$=1/2
This set of quantum numbers indicates that three electrons reside in the 1$s$ orbital which is impossible!
As Jan stated in his post, these quantum numbers are derived from the solutions to the Schrodinger equation for the hydrogen atom (or a 1-e$^-$ system). There are any number of solutions to this equation that relate to the possible energy levels of they hydrogen atom. Remember, energy is QUANTIZED (as postulated by Max Planck). That means that an energy level may exist (arbitrarily) at 0 and 1 but NEVER in between. There is a discrete 'jump' in energy levels and not some gradient between them. From these solutions a formalism was constructed to communicate the solutions in a very easy, numerical way just as mailing addresses are purposefully formatted in such a way that is easy that anyone can understand with minimal effort.
In summary, the quantum numbers not only tell you where electrons will be (ground state) and can be (excited state), but also will tell you where electrons cannot be in an atom (due to the restrictions for each quantum number).
Principle quantum number ($n$) - indicates the orbital size. Electrons in atoms reside in atomic orbitals. These are referred to as $s,p,d,f...$ type orbitals. A $1s$ orbital is smaller than a $2s$ orbital. A $2p$ orbital is smaller than a $3p$ orbital. This is because orbitals with a larger $n$ value are getting larger due to the fact that they are further away from the nucleus. The principle quantum number is an integer value where $n$ = 1,2,3... .
Angular quantum number ($l$) - indicates the shape of the orbital. Each type of orbital ($s,p,d,f..$) has a characteristic shape associated with it. $s$-type orbitals are spherical while $p$-type orbitals have 'dumbbell' orientations. The orbitals described by $l$=0,1,2,3... are $s,p,d,f...$ orbitals, respectively. The angular quantum number ranges from 0 to $n$-1. Therefore, if $n$ = 3, then the possible values of $l$ are 0, 1, 2.
Magnetic quantum number ($m_l$) - indicates the orientation of a particular orbital in space. Consider the $p$ orbitals. This is a set of orbitals consisting of three $p$-orbitals that have a unique orientation in space. In Cartesian space, each orbital would like along an axis (x, y, or z) and would be centered around the origin at 0,0. While each orbital is indeed a $p$-orbital, we can describe each orbital uniquely by assigning this third quantum number to indicate its position in space. Therefore, for a set of $p$-orbitals, there would be three $m_l$, each uniquely describing one of these orbitals. The magnetic quantum number can have values of $-l$ to $l$. Therefore, in our example above (where $l$ = 0,1,2) then $m_l$ would be -2, -1, 0, 1, 2.
Spin quantum number ($m_s$) - indicates the 'spin' of the electron residing in some atomic orbital. Thus far we have introduced three quantum numbers that localize a position to an orbital of a particular size, shape and orientation. We now introduce the fourth quantum number that describes the type of electron that can be in that orbital. Recall that two electrons can reside inside one atomic orbital. We can define each one uniquely by indicating the electron's spin. According to the Pauli-exclusion principle, no two electrons can have the exact same four quantum numbers. This means that two electrons in one atomic orbital cannot have the same 'spin'. We generally denote 'spin-up' as $m_s$ =1/2 and spin-down as $m_s$=-1/2.
Solution 2:
The Schrödinger equation for most system has many solutions $\hat{H}\Psi_i=E_i\Psi_i$, where $i=1,2,3,..$. In the case of the hydrogen atom the solutions has a specific notation, which are where the quantum numbers come from.
In the case of the H atom the principal quantum number $n$ refers to solutions with different energy.
For $n>1$ there are several solutions with the same energy, which come in different shapes ($s$, $p$, etc with different angular quantum numbers $l$) that can point in different directions ($p_x$, $p_y$, etc with different magnetic quantum numbers $m$)
These quantum numbers are also applied to multi-electron atoms within the AO approximation.
So the quantum numbers are a way to count (label) the solutions to the Schrödinger equation.