Why an orbital needs an address
A very large number of orbitals are possible in an atom, and qualitatively they differ in three respects: their size, their shape and their orientation in space. A smaller orbital implies a higher chance of finding the electron close to the nucleus, while shape and orientation tell us along which directions the electron is more likely to be found. To label these orbitals precisely, the quantum-mechanical model assigns each one a unique set of integers known as quantum numbers.
Each orbital is fully described by three quantum numbers — $n$, $l$ and $m_l$. These emerge naturally as the permitted solutions of the Schrödinger equation for the hydrogen atom; only certain integer combinations yield acceptable wave functions. A fourth quantum number, the spin $m_s$, is needed to describe an electron rather than the orbital it occupies. The distinction is precise and frequently examined: an orbital is specified by three quantum numbers, an electron by four.
It is worth fixing the vocabulary before the rules. An orbital is not an orbit. A Bohr orbit is a definite circular path, an idea that the Heisenberg uncertainty principle renders meaningless. An atomic orbital is instead a one-electron wave function $\psi$, characterised by $n$, $l$ and $m_l$, whose square $|\psi|^2$ gives the probability density of the electron at a point.
Principal quantum number (n)
The principal quantum number $n$ is a positive integer, $n = 1, 2, 3, \dots$. It determines the size of the orbital and, to a large extent, its energy. For the hydrogen atom and hydrogen-like species such as $\ce{He+}$ and $\ce{Li^2+}$, the energy and size of the orbital depend on $n$ alone. As $n$ increases the electron is, on average, located further from the nucleus; since energy is required to move a negatively charged electron away from the positive nucleus, the orbital energy rises with $n$.
The principal quantum number also identifies the shell. All orbitals sharing a value of $n$ constitute one shell, lettered K, L, M, N for $n=1,2,3,4$ respectively. The number of orbitals permitted within a shell increases as $n^2$.
| n | Shell letter | No. of subshells | No. of orbitals ($n^2$) | Max electrons ($2n^2$) |
|---|---|---|---|---|
1 | K | 1 | 1 | 2 |
2 | L | 2 | 4 | 8 |
3 | M | 3 | 9 | 18 |
4 | N | 4 | 16 | 32 |
Azimuthal quantum number (l)
The azimuthal quantum number $l$, also called the orbital angular momentum or subsidiary quantum number, defines the three-dimensional shape of the orbital. For a given $n$, it can take $n$ integer values from $0$ to $(n-1)$:
$l = 0, 1, 2, \dots, (n-1)$
Each value of $l$ corresponds to one subshell, and the number of subshells in the $n$th shell equals $n$. Thus $n=1$ has only $l=0$; $n=2$ has $l=0,1$; and $n=3$ has $l=0,1,2$. The subshells are denoted by the spectroscopic letters $s, p, d, f, g, h$ for $l=0,1,2,3,4,5$. The "total orbital angular momentum" of an electron in an $s$ orbital is zero precisely because $l=0$ — a fact NEET has tested directly.
| l value | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Subshell notation | s | p | d | f | g | h |
| Orbitals in subshell ($2l+1$) | 1 | 3 | 5 | 7 | 9 | 11 |
A subshell is named by combining the shell number with the letter for $l$: an electron with $n=2,\ l=1$ lies in the $2p$ subshell, while $n=4,\ l=0$ describes a $4s$ subshell. Because $l$ is derived from $n$, the existence of a subshell is constrained by $n$ — there is no $2d$ or $1p$ subshell.
The value of $l$ fixes the geometry of the orbital. See how $s$, $p$ and $d$ orbitals actually look in Shapes of Orbitals.
Magnetic quantum number (m_l)
The magnetic orbital quantum number $m_l$ gives the spatial orientation of the orbital relative to a standard set of coordinate axes. For any subshell defined by $l$, there are $(2l+1)$ permitted values of $m_l$, given by:
$m_l = -l,\ -(l-1),\ \dots,\ 0,\ \dots,\ +(l-1),\ +l$
Each permitted value labels one orbital, so the number of orbitals in a subshell equals $(2l+1)$. For $l=0$ the only value is $m_l=0$ — a single $s$ orbital. For $l=1$, $m_l = -1, 0, +1$ — three $p$ orbitals. For $l=2$, $m_l = -2, -1, 0, +1, +2$ — five $d$ orbitals. The values of $m_l$ are derived from $l$, just as $l$ is derived from $n$; this nested dependence is the source of most valid-versus-invalid set questions.
Spin quantum number (m_s)
The three quantum numbers above describe an orbital completely, yet they cannot account for the fine structure of spectral lines in multi-electron atoms — lines that appear as closely spaced doublets or triplets. In 1925, George Uhlenbeck and Samuel Goudsmit introduced a fourth quantum number, the electron spin quantum number $m_s$. An electron behaves as if it spins about its own axis, much as the Earth spins while orbiting the Sun, giving it an intrinsic spin angular momentum in addition to charge and mass.
This spin can adopt only two orientations relative to a chosen axis, distinguished by $m_s = +\tfrac{1}{2}$ and $m_s = -\tfrac{1}{2}$, often drawn as $\uparrow$ (spin up) and $\downarrow$ (spin down). Two electrons with opposite $m_s$ are said to have opposite spins. Consequently, a single orbital can hold at most two electrons, and those two must have opposite spins — the basis of NEET 2016's question that two electrons in the same orbital are distinguished only by their spin quantum number.
The four quantum numbers at a glance
The four quantum numbers form a strict hierarchy: $n$ fixes the range of $l$, and $l$ fixes the range of $m_l$, while $m_s$ stands independent. The master table below consolidates the symbol, name, allowed values and physical meaning of each — the single most exam-relevant summary of this subtopic.
| Symbol | Name | Allowed values | What it tells you |
|---|---|---|---|
n |
Principal | $1, 2, 3, \dots$ (positive integers) | Shell; size of the orbital and largely its energy |
l |
Azimuthal (angular momentum) | $0$ to $(n-1)$ — $n$ values | Subshell ($s,p,d,f$); shape of the orbital |
ml |
Magnetic | $-l$ to $+l$ — $(2l+1)$ values | Orientation of the orbital in space |
ms |
Spin | $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$ | Orientation of the electron's intrinsic spin |
Read the table as a chain of dependencies. Choosing $n$ first restricts $l$; choosing $l$ then restricts $m_l$; only $m_s$ is free. NEET 2024 tested this mapping directly by asking which quantum number conveys orbital shape, orientation, size and spin orientation — exactly the four entries in the final column.
Counting orbitals and electrons
Two counting relations follow immediately from the rules and are the workhorse formulae of this subtopic. The number of orbitals in a subshell is $(2l+1)$; summing across all subshells of a shell gives the total number of orbitals in that shell.
$\text{Orbitals in shell } n = \displaystyle\sum_{l=0}^{n-1}(2l+1) = n^2 \qquad \text{Electrons in shell } n = 2n^2$
The factor of two arises because each orbital accommodates two electrons of opposite spin. The relation between the count of magnetic values $n_m$ and $l$, examined in NEET 2023, is simply $n_m = 2l+1$, which rearranges to $l = \dfrac{n_m - 1}{2}$.
Counting subshells vs. orbitals
The number of subshells in shell $n$ equals $n$ (because $l$ runs $0$ to $n-1$). The number of orbitals equals $n^2$, and the maximum number of electrons equals $2n^2$. Candidates routinely confuse these three counts. For $n=3$: 3 subshells, 9 orbitals, 18 electrons — three different answers from the same shell.
Subshells $= n$ · Orbitals $= n^2$ · Electrons $= 2n^2$.
Worked example: mapping n = 3
A complete walk through $n=3$ ties every rule together and is a model answer for the most common NEET stem, "the total number of orbitals associated with $n=3$."
List every subshell, orbital and the maximum electron count for the third shell ($n=3$).
Step 1 — find allowed $l$. For $n=3$, $l$ runs from $0$ to $n-1=2$, giving $l = 0, 1, 2$, i.e. the $3s$, $3p$ and $3d$ subshells. That is 3 subshells, equal to $n$.
Step 2 — find $m_l$ per subshell.
| Subshell | l | m_l values | Orbitals ($2l+1$) | Electrons ($2(2l+1)$) |
|---|---|---|---|---|
3s | 0 | $0$ | 1 | 2 |
3p | 1 | $-1, 0, +1$ | 3 | 6 |
3d | 2 | $-2, -1, 0, +1, +2$ | 5 | 10 |
| Total | — | — | 9 | 18 |
Step 3 — verify with the formulae. Total orbitals $= 1+3+5 = 9 = 3^2 = n^2$. Maximum electrons $= 18 = 2\times 9 = 2n^2$. Both routes agree, confirming the count. This is identical to NCERT Problem 2.17, which obtains $n^2 = 3^2 = 9$.
Impossible quantum-number sets
Many NEET items present four sets of $(n, l, m_l, m_s)$ and ask which is permissible. The screening is mechanical once the dependency chain is internalised: test $l$ against $n$, then $m_l$ against $l$, then $m_s$ against $\pm\tfrac12$.
Spotting a forbidden set
A set fails if any single rule is broken. Common violations: $l \ge n$ (e.g. $n=2,\ l=2$, implying a non-existent $2d$ subshell); $|m_l| > l$ (e.g. $l=1,\ m_l=+2$); or $m_s$ being anything other than $+\tfrac12$ or $-\tfrac12$. Note that $n$ can never be zero or negative.
Check order: is $0 \le l \le n-1$? is $-l \le m_l \le +l$? is $m_s = \pm\tfrac12$? All three must hold.
Which of these sets $(n, l, m_l, m_s)$ is not allowed?
(a) $(3, 2, -2, +\tfrac12)$ (b) $(2, 2, 0, -\tfrac12)$ (c) $(4, 0, 0, +\tfrac12)$ (d) $(3, 1, +1, -\tfrac12)$
Answer: (b). For $n=2$, $l$ can be at most $n-1 = 1$, so $l=2$ is impossible — a $2d$ orbital does not exist. Set (a) is a valid $3d$ orbital, (c) a valid $4s$, and (d) a valid $3p$; all satisfy $0\le l\le n-1$ and $-l\le m_l\le +l$.
Lock these in before the exam
- An orbital needs three quantum numbers ($n, l, m_l$); an electron needs four (add $m_s$).
- $n = 1,2,3,\dots$ sets size and energy; $l = 0$ to $(n-1)$ sets shape/subshell; $m_l = -l$ to $+l$ sets orientation; $m_s = \pm\tfrac12$ sets spin.
- Subshell letters: $l=0,1,2,3 \to s,p,d,f$. Orbitals per subshell $= 2l+1$.
- Orbitals in shell $= n^2$; maximum electrons $= 2n^2$; subshells in shell $= n$.
- A set is forbidden if $l \ge n$, if $|m_l| > l$, or if $m_s \ne \pm\tfrac12$.