What exactly are quantum numbers and how do we obtain them?

Question

While studying atomic orbitals, I came to know about the quantum numbers $n, l, m$ and $s$. I studied that solving the Schrödinger wave equation for atoms gives us the allowed energy levels of the electrons and their corresponding wave functions. But then from where do we obtain the quantum numbers? If solving the Schrödinger wave equation gives us only $E$ and $\Psi$ then from where do we obtain the values of the quantum numbers $n, l, m$ and $s$? And what are the quantum numbers for? What is their significance? Doesn't $E$ and $\Psi$ give us all the information that we need about an electron? How can we define a quantum number and how do we obtain them? Can someone please explain? I am so confused. Please help.

Answer 1

Solving the Schroedinger equation for a hydrogen atom shows you that there is a plethora of solutions $\Psi$ that ALL solve the equation. An electron is in a state that corresponds to one of these or more generally a superposition of several solutions.

The quantum numbers that have been named $n,l,m,s$ kind of 'categorise' these solutions, meaning: A specific set of $n,l,m,s$ corresponds to one specific solution $\Psi_{nlms}$. The first quantum number $n$ corresponds to the overall energy level, meaning independent of $l,m$ and $s$, a solution corresponding to a specific $n$ will give you a specific energy, i.e. $$\hat{H}\Psi_n=E_n\Psi_n,$$ which is different for all $n$. This means, if you measure the energy of the electron in an actual hydrogen atom orbit, for example by observing the photon that is emitted when the electron goes back to the ground state (which is the $\Psi$ solution with the lowest energy level), then you can find out, what the quantum number $n$ of the electron was.

However, most of the energy levels are what we call degenerated, i.e. there are different solutions to the Schroedinger equation that correspond to the same energy level. This is, because the full solution (omitting the spin for a moment) is a product of a function that only depends on $n$ and $l$ (and in real space on the radius $r$), and a factor that only depends on the $l,m$ quantum numbers (and in real space only on the angular coordinates $\theta$ and $\phi$). $$\Psi_{nml}(r,\theta,\phi)=\psi_{n,l}(r)Y_l^m(\theta,\phi).$$ The different spherical solutions $Y_l^m(\theta,\psi)$ are called spherical harmonics. All these different solutions correspond to the same total energy IF $n$ is the same. If you want to find out in which exact state the electron is, you would also need to measure the angular momentum, to which the $l,m$ quantum number correspond to, but I am not sure how this is actually done.

Finally the $s$ quantum number corresponds to the spin of the electron, this does not have a representation in real space (meaning spin does not depend on any coordinates) and can be measured in a magnetic field.

Note that there are constraints on the quantum numbers, for example $l$ has to be smaller than $n$. A good book to understand all of this is Quantum Mechanics by Griffiths.

Answer 2

Actually, solving the Schrodinger equation itself, along with the proper boundary conditions, and an understanding of the probabilistic nature of quantum mechanics, is what gives rise to these quantum numbers. The quantum numbers are a natural result of this mathematical process.

The quantum numbers are simply constants which identify each specific solution to the Schrodinger equation. A set of quantum numbers will specify a particular state of an electron in an atom.

Answer 3

Quantum numbers are just labels that pick out the particular $\Psi$ we are interested in. Moreover, such labeling isn't even unique: e.g. for the hydrogen atom, solutions of the time-independent Schrödinger equation in spherical coordinates will be labeled by quantum numbers

• principal $n$,
• azimuthal $\ell$,
• magnetic $m$,

while the solutions in parabolic coordinates will be labeled by quantum numbers

• two parabolic $n_1$, $n_2$,
• magnetic $m$.

This non-uniqueness is directly related to degeneracy of energy levels.