# Why have $n$, $\ell$, $m_\ell$, $m_s$ been picked as quantum number symbols $\mathbf{\text{in this order}}$?

I’m learning about electron configurations and don’t quite understand why $n$, $\ell$, $m_\ell$, $m_s$ have been picked as symbols for the quantum numbers. As far as I understand it, the principal quantum number $n$ is the most “general” or “characteristic” one, then comes the azimuthal quantum number $\ell$, then the magnetic quantum number $m_\ell$ and then the spin projection quantum number $m_s$.

This order disturbs me – I think $n$, $m$, $\ell$ (reverse alphabetic order) or $\ell$, $m$, $n$ (alphabetic order) would have been more natural (and more commonly used) choices. I assume there is a reason that the symbols have been picked in that specific order – what is the reason?

• The solutions to electron orbitals can be expressed in terms of equations that take integer parameters. So all valid ψ(r) can be expressed as ψ(n,l,m_l,m_s). The math works out to require that "l" be 0 to n-1, "m_l" be -l to l and "m_s" be -1/2 or 1/2. It makes sense to order them based on the fact that their range depends on the previous number's value (except m_s)
– Nick
Commented Mar 23, 2014 at 16:24
• There is the same convention for spherical harmonics, you should check whether the chicken or the egg came first. Commented Mar 23, 2014 at 16:26
• @Nick The order of the quantum numbers themselves is clear – I just want to understand why, for example, the azimuthal quantum number hasn’t gotten the symbol $m$, the magnetic quantum number the symbol $\ell_m$ and the spin projection quantum number the symbol $\ell_s$? That way, they would be sorted in reverse alphabetical order. Commented Mar 23, 2014 at 16:32
• Related meta post: meta.physics.stackexchange.com/q/1083/2451 Commented Mar 23, 2014 at 17:01
• It helps to realize that $\ell$ represents angular momentum, and L is a standard classical mechanics choice for angular momentum. Then $\ell,m$ make sense by the principle of alphabetic ordering. The quantum number $n$ is a separate non angular thing measuring the total energy. The choice of $n$ here matches the choice made for the analogous quantity in the quantum harmonic oscillator, for example. Commented Mar 23, 2014 at 17:21

Such an ordering arises from the fact that they are arranged chronologically, i.e., according to the dates they were "discovered".

The principle quantum number $n$ entered the picture with Bohr's theory of the Hydrogen atom in 1913.Bohr introduced $n$ in his quantization of angular momentum postulate where $n$ is the allowed orbit. Mathematically, $L = n{h \over 2\pi} = n\hbar$ where $n=1,2,3..$ was called the principle quantum number. To answer why he settled on the letter $n$, one can only speculate and comment that earlier Planck had used $n$ to denote quantization of energy for light ($E=n\hbar\nu$) and hence $n$ was selected to drawn an analogy.

The azimuthal quantum number $l$ was introduced by Sommerfeld in his relativistic atom model (with elliptical orbits) in 1916. The derivation involved finding the solution to Legendre's differential equation.See this for more information.

The magnetic quantum $m_l$ number came into the picture with the Vector Atom model which was based on the concept of space quantization.The Vector Model of the Atom was an extension of the Rutherford-Bohr-Sommerfeld Model. Classically, the orbits of the atomic electrons can orient in all possible directions in space but quantum theory allows certain discrete orientations in space out of all the infinite possibilities and $m_l$ was introduced in this context. (I can not find an exact date and it was due to several physicists, notably Bohr, Somerfeld, Uhlenbeck, Goudsmith, Pauli, Stern and Gerlach.)

Finally the spin quantum numbers $s$ were introduced to strengthen the Vector Atom Model and explain a host of physical phenomena like the Zeeman Effect and also the fact that investigation of alkali spectra using a high resolving power spectroscope revealed that many spectral lines consist of a group of lines very close to each other.Actually to explain this multiple structure of the spectral lines, G.E. Uhlenbeck and S.A Goudsmit put forward the 'spinning electron hypothesis' in 1925 which introduced $s$. Note that the magnetic spin quantum number $m_s$ is actually a secondary quantum number and is related to spin quantum number $s$ by

$s_z = m_s \, \hbar$ where $m_s = \sqrt{s \, (s+1)} \, \hbar$.

However it is easier to write $\Vert \mathbf{s} \Vert = \sqrt{s \, (s+1)} \, \hbar$. The symbols have their usual meanings but you may see this more details.

To sum up :

• The principle quantum number got $n$ because it was introduced in the context of quantization of angular momentum and $n$ was used already used to denote the quantization of energy in light.
• The azimuthal quantum number got $l$ because the constant in Legendre's equation is written using $l$ by convention. See this for more details.
• Magnetic quantum number $m$ gets it from the word "magnetic".
• The spin quantum number $s$ gets its name from the word "spin".

Also, the fact that each quantum number depends on the previous one justifies its ordering.

• So Sommerfeld found out about the azimuthal quantum number, looked at Bohr’s model which already featured the principle quantum number $n$, and decided that his new number should be named $\ell$? Wouldn’t $m$ be a more natural choice for Sommerfeld? Commented Mar 23, 2014 at 16:48
• I think it was due to the fact that the constant in Legendre's eqaution is conventionally as written as $l(l+1)$. But this is merely a speculation. Commented Mar 23, 2014 at 16:54
• A small clarification, Bohr introduces, the $n$ in quantising angular momentum. This $n$ is not the same as the $n$ that comes out when you solve Schrodinger equation for hydrogen atom. Commented Mar 24, 2014 at 1:22
• The $n$ we get by solving the Schrodinger's equation for Hydrogen atom can be interpreted as the allowed Energy State.Actually it is one of the constants we use conventionally while solving the azimuthal equation. Historically, Bohr introduced n in his quantization of angular momentum postulate where n is the allowed orbit. Mathematically,$L = n{h \over 2\pi} = n\hbar$ where $n=1,2,3...$ was called the principle quantum number.Obviously, starting from Bohr's postulate we can also arrive at the expression for energy.The two thus become equivalent, in retrospect.I will make the necessary edits. Commented Mar 24, 2014 at 5:31