KAHIBARO
Table of Contents
What Quantum Numbers Describe
In quantum mechanics, an electron in an atom is not described by a simple orbit like a planet around the Sun. Instead, it is described by a wavefunction, and the allowed states of that wavefunction are labeled by a small set of numbers called quantum numbers. These numbers tell us which state the electron is in and what properties that state has.
Quantum numbers are especially important for atoms, because they organize the allowed electron states in a clear and systematic way. They help answer questions like where an electron is likely to be found, how much angular momentum it has, and how it behaves in a magnetic field.
For an electron in an atom, the main quantum numbers are the principal quantum number, the orbital angular momentum quantum number, the magnetic quantum number, and the spin quantum number.
The Four Quantum Numbers
Each electron in an atom is described by a set of four quantum numbers.
| Quantum number | Symbol | What it describes |
|---|---|---|
| Principal quantum number | $n$ | Energy level and general size of the state |
| Orbital angular momentum quantum number | $\ell$ | Shape of the orbital |
| Magnetic quantum number | $m_\ell$ | Orientation of the orbital angular momentum |
| Spin quantum number | $m_s$ | Orientation of the electron's spin |
These numbers are not arbitrary. Only certain values are allowed.
Principal Quantum Number
The principal quantum number is written as $n$. It takes positive integer values:
$$
n = 1, 2, 3, \dots
$$
This number mainly determines the energy level of the electron in a hydrogen-like atom. It also relates to the typical size of the orbital. Larger $n$ usually means the electron is, on average, farther from the nucleus.
For example, $n=1$ is the first energy level, $n=2$ is the second, and so on.
For the principal quantum number, the allowed values are
$$
n = 1, 2, 3, \dots
$$
A larger $n$ generally corresponds to a higher energy state and a larger orbital.
Orbital Angular Momentum Quantum Number
The orbital angular momentum quantum number is written as $\ell$. For a given value of $n$, it can take integer values from $0$ up to $n-1$:
$$
\ell = 0, 1, 2, \dots, n-1
$$
This quantum number is related to the orbital angular momentum of the electron and also to the general shape of the orbital.
Different values of $\ell$ are traditionally labeled by letters:
| $\ell$ | Letter |
|---|---|
| 0 | s |
| 1 | p |
| 2 | d |
| 3 | f |
So if $n=3$, then the possible values of $\ell$ are $0$, $1$, and $2$, which correspond to $3s$, $3p$, and $3d$ states.
For a given principal quantum number $n$, the orbital quantum number must satisfy
$$
\ell = 0, 1, 2, \dots, n-1
$$
This means $\ell$ can never be equal to or larger than $n$.
Magnetic Quantum Number
The magnetic quantum number is written as $m_\ell$. For a given value of $\ell$, it can take all integer values from $-\ell$ to $+\ell$:
$$
m_\ell = -\ell, -\ell+1, \dots, 0, \dots, \ell-1, \ell
$$
This quantum number tells us the orientation of the orbital angular momentum relative to a chosen axis, usually called the $z$ axis.
For example, if $\ell=1$, then the possible values are
$$
m_\ell = -1, 0, +1
$$
So a $p$ state has three possible orbital orientations.
For a given value of $\ell$, the magnetic quantum number is
$$
m_\ell = -\ell, \dots, 0, \dots, +\ell
$$
The number of possible $m_\ell$ values is
$$
2\ell + 1
$$
Spin Quantum Number
Electrons also have an intrinsic angular momentum called spin. The spin quantum number for an electron is written as $m_s$, and it can take only two values:
$$
m_s = +\frac{1}{2}, \quad -\frac{1}{2}
$$
These are often described informally as spin up and spin down with respect to the chosen axis.
For an electron, the spin magnetic quantum number can only be
$$
m_s = +\frac{1}{2} \quad \text{or} \quad -\frac{1}{2}
$$
Allowed Combinations
The four quantum numbers must fit together according to the allowed ranges. A valid electron state in an atom must obey all of the following:
$$
n = 1, 2, 3, \dots
$$
$$
\ell = 0, 1, 2, \dots, n-1
$$
$$
m_\ell = -\ell, \dots, +\ell
$$
$$
m_s = \pm \frac{1}{2}
$$
For example, the set
$$
n=2,\quad \ell=1,\quad m_\ell=0,\quad m_s=+\frac{1}{2}
$$
is allowed.
But the set
$$
n=2,\quad \ell=2,\quad m_\ell=0,\quad m_s=+\frac{1}{2}
$$
is not allowed, because for $n=2$, the largest possible $\ell$ is $1$.
Visual Picture of Quantum Number Structure
The quantum numbers form a hierarchy. First choose $n$, then $\ell$, then $m_\ell$, and finally $m_s$.
Example States
It helps to list the possible quantum numbers for small values of $n$.
For $n=1$, only $\ell=0$ is allowed. Then only $m_\ell=0$ is allowed. Each such state can still have two spin values.
| $n$ | $\ell$ | $m_\ell$ | $m_s$ |
|---|---|---|---|
| 1 | 0 | 0 | $+\frac{1}{2}$ or $-\frac{1}{2}$ |
For $n=2$, there are more possibilities.
| $n$ | $\ell$ | Possible $m_\ell$ values |
|---|---|---|
| 2 | 0 | $0$ |
| 2 | 1 | $-1, 0, +1$ |
Including spin, each of these orbital states splits into two electron states.
Number of States in a Shell
For a given $\ell$, there are $2\ell+1$ possible values of $m_\ell$. Since each of these can also have two spin values, the number of electron states for a given $\ell$ is
$$
2(2\ell+1)
$$
If all allowed $\ell$ values for a fixed $n$ are included, the total number of electron states in the shell turns out to be
$$
2n^2
$$
For example:
| Shell | $n$ | Total number of electron states |
|---|---|---|
| First | 1 | 2 |
| Second | 2 | 8 |
| Third | 3 | 18 |
Important counting formulas:
For a given $\ell$,
$$
\text{number of orbital states} = 2\ell + 1
$$
Including spin,
$$
\text{number of electron states} = 2(2\ell + 1)
$$
For a given shell $n$,
$$
\text{total number of electron states} = 2n^2
$$
Connection to Angular Momentum
Quantum numbers are closely related to angular momentum. The orbital angular momentum magnitude is determined by $\ell$, and its component along the chosen $z$ axis is determined by $m_\ell$.
The allowed values are
$$
|\mathbf{L}| = \sqrt{\ell(\ell+1)}\,\hbar
$$
and
$$
L_z = m_\ell \hbar
$$
Similarly, the electron spin has a fixed intrinsic angular momentum, while $m_s$ gives its component along the chosen axis.
For orbital angular momentum,
$$
|\mathbf{L}| = \sqrt{\ell(\ell+1)}\,\hbar
$$
and
$$
L_z = m_\ell \hbar
$$
This shows that angular momentum in quantum mechanics is quantized.
Why Quantum Numbers Matter
Quantum numbers are the labels of atomic states. They make it possible to distinguish one allowed electron state from another. They also explain why atomic structure is organized into shells and subshells such as $1s$, $2s$, and $2p$.
Later, quantum numbers become essential for understanding atomic spectra, electron configurations, and why no two electrons in the same atom can occupy exactly the same quantum state.
A Simple Orbital Orientation Sketch
The magnetic quantum number changes the orientation of the orbital angular momentum relative to the chosen axis.
This sketch is only a simple way to picture quantized orientations. The real quantum state is described by a wavefunction, not by a classical arrow moving in space.
Summary
Quantum numbers are the labels that specify allowed electron states in atoms. The principal quantum number $n$ gives the energy level, $\ell$ gives the orbital type, $m_\ell$ gives the orientation of orbital angular momentum, and $m_s$ gives the spin orientation.
Together they obey strict rules:
A valid set of quantum numbers for an electron in an atom must satisfy
$$
n = 1,2,3,\dots
$$
$$
\ell = 0,1,2,\dots,n-1
$$
$$
m_\ell = -\ell,\dots,+\ell
$$
$$
m_s = \pm \frac{1}{2}
$$
These simple rules are one of the foundations of atomic quantum physics.