Quantum Mechanical Model & Quantum Numbers

The Quantum Mechanical Model of the Atom

The failures of Bohr's model, combined with de Broglie's wave-particle duality and Heisenberg's uncertainty principle, led to a fundamentally new approach to atomic structure — quantum mechanics.

Schrödinger Wave Equation (1926)

Erwin Schrödinger developed a mathematical equation that describes the wave-like behaviour of the electron in an atom:

$$\hat{H}\psi = E\psi$$

where:

• $\hat{H}$ is the Hamiltonian operator (represents total energy)
• $\psi$ (psi) is the wave function
• $E$ is the energy of the electron

You don't need to solve this equation at this level, but you need to understand what it tells us.

What is $\psi$ (Wave Function)?

The wave function $\psi$ is a mathematical function that describes the quantum state of an electron. By itself, $\psi$ has no direct physical meaning.

What is $|\psi|^2$ (Probability Density)?

$|\psi|^2$ gives the probability density — the probability of finding the electron per unit volume at a given point in space.

Key Point: We can never say "the electron IS here." We can only say "the probability of finding the electron here is X%."

Orbital vs Orbit

Feature	Orbit (Bohr)	Orbital (QM)
Definition	Circular path of electron	3D region of space with high probability of finding electron
Shape	Circle	s, p, d, f shapes
Electron position	Well-defined path	Probability distribution
Based on	Classical mechanics	Quantum mechanics
Max electrons	$2n^2$ per shell	2 per orbital

Quantum Numbers

The solution of the Schrödinger equation for hydrogen gives a set of functions (orbitals), each characterised by three quantum numbers: $n$, $l$, and $m_l$. A fourth quantum number ($m_s$) describes the spin of the electron.

1. Principal Quantum Number ($n$)

$$n = 1, 2, 3, 4, \ldots$$

What it determines:

• The shell (energy level) of the electron: $n = 1$ (K), $n = 2$ (L), $n = 3$ (M), $n = 4$ (N)
• The size of the orbital: larger $n$ → larger orbital → electron is farther from the nucleus
• The energy of the electron (in hydrogen): $E_n = -13.6/n^2$ eV
• The maximum number of orbitals in a shell: $n^2$
• The maximum number of electrons in a shell: $2n^2$

Shell	$n$	Orbitals ($n^2$)	Max electrons ($2n^2$)
K	1	1	2
L	2	4	8
M	3	9	18
N	4	16	32

2. Azimuthal Quantum Number ($l$) — also called Angular Momentum or Orbital Quantum Number

$$l = 0, 1, 2, \ldots, (n-1)$$

What it determines:

• The subshell (shape of the orbital)
• The orbital angular momentum: $L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi}$
• The number of subshells in a shell = $n$

$l$ value	0	1	2	3
Subshell	s	p	d	f
Shape	Spherical	Dumbbell	Double dumbbell	Complex
Orbitals in subshell	1	3	5	7

3. Magnetic Quantum Number ($m_l$)

$$m_l = -l, -(l-1), \ldots, 0, \ldots, (l-1), l$$

Total values: $2l + 1$

What it determines:

• The orientation of the orbital in space
• The number of orbitals in a subshell

Subshell	$l$	$m_l$ values	Number of orbitals
s	0	0	1
p	1	$-1, 0, +1$	3 ($p_x, p_y, p_z$)
d	2	$-2, -1, 0, +1, +2$	5
f	3	$-3, -2, -1, 0, +1, +2, +3$	7

4. Spin Quantum Number ($m_s$)

$$m_s = +\frac{1}{2} \text{ or } -\frac{1}{2}$$

What it determines:

• The spin of the electron: clockwise ($+1/2$, ↑) or anticlockwise ($-1/2$, ↓)
• Each orbital can hold a maximum of 2 electrons with opposite spins

The spin quantum number was proposed by Uhlenbeck and Goudsmit (1925) to explain certain features of atomic spectra.

Key Point: The four quantum numbers ($n, l, m_l, m_s$) together completely describe the state of an electron in an atom. No two electrons in an atom can have the same set of all four quantum numbers — this is the Pauli Exclusion Principle (covered in Section 11).

Allowed Combinations of Quantum Numbers

Not every combination of quantum numbers is valid. The rules are:

1. $n$ = any positive integer (1, 2, 3, …)
2. $l$ = 0 to $(n - 1)$
3. $m_l$ = $-l$ to $+l$
4. $m_s$ = $+1/2$ or $-1/2$

Example: All Orbitals for $n = 3$

$n$	$l$	Subshell	$m_l$ values	Orbitals
3	0	3s	0	1
3	1	3p	$-1, 0, +1$	3
3	2	3d	$-2, -1, 0, +1, +2$	5

Total orbitals = $1 + 3 + 5 = 9 = n^2$ ✓ Total electrons = $9 \times 2 = 18 = 2n^2$ ✓

Invalid Combinations

These are NOT allowed:

• $n = 1, l = 1$ (because $l$ must be $< n$; for $n = 1$, only $l = 0$)
• $n = 2, l = 2$ (because $l$ can only be 0 or 1 for $n = 2$)
• $l = 1, m_l = 2$ (because $m_l$ must be between $-l$ and $+l$)

[JEE Tip] To quickly check validity: $l < n$ and $|m_l| \leq l$. If either condition fails, the combination is invalid.

Designation of Orbitals

Orbitals are written as $nl$ notation:

• $1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, \ldots$

The number represents $n$ and the letter represents $l$.

Solved Examples

Example 1: Listing Quantum Numbers

List all possible values of $l$ and $m_l$ for $n = 3$.

Solution: For $n = 3$: $l = 0, 1, 2$

• $l = 0$: $m_l = 0$ (1 orbital → 3s)
• $l = 1$: $m_l = -1, 0, +1$ (3 orbitals → 3p)
• $l = 2$: $m_l = -2, -1, 0, +1, +2$ (5 orbitals → 3d)

Total orbitals = 9.

Example 2: Checking Validity of Quantum Numbers

Which of these sets of quantum numbers is not allowed? (a) $n = 2, l = 1, m_l = 0, m_s = +1/2$ (b) $n = 1, l = 1, m_l = 0, m_s = -1/2$ (c) $n = 3, l = 2, m_l = -2, m_s = +1/2$

Solution: (a) Valid: $l < n$ (1 < 2) ✓, $|m_l| \leq l$ (0 ≤ 1) ✓ (b) Invalid: $l = 1$ but $n = 1$, so $l$ can only be 0. $l$ must be $< n$. (c) Valid: $l < n$ (2 < 3) ✓, $|m_l| \leq l$ (2 ≤ 2) ✓

Answer: (b) is not allowed.

Example 3: Number of Orbitals in a Subshell

How many orbitals are in the 4d subshell?

Solution: For 4d: $n = 4$, $l = 2$. Number of orbitals $= 2l + 1 = 2(2) + 1 = 5$.

Answer: 5 orbitals.

Example 4: Maximum Electrons in a Shell

What is the maximum number of electrons in the M shell?

Solution: M shell: $n = 3$. Maximum electrons $= 2n^2 = 2(9) = 18$.

Breakdown: 3s (2) + 3p (6) + 3d (10) = 18. ✓

Answer: 18 electrons.

Example 5: Orbital Angular Momentum

Calculate the orbital angular momentum of an electron in a 3p orbital.

Solution: For 3p: $l = 1$. $$L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi} = \sqrt{1 \times 2} \cdot \frac{h}{2\pi} = \sqrt{2} \cdot \frac{6.626 \times 10^{-34}}{2\pi}$$ $$= 1.414 \times 1.055 \times 10^{-34} = 1.49 \times 10^{-34} \text{ J s}$$

Answer: $L = \sqrt{2} \cdot \hbar = 1.49 \times 10^{-34}$ J s.

Example 6: Identifying the Orbital

An electron has quantum numbers $n = 4, l = 2, m_l = -1, m_s = +1/2$. Identify the subshell.

Solution: $n = 4$ and $l = 2$ → This is a 4d orbital. $m_l = -1$ indicates one of the five orientations of d orbital. $m_s = +1/2$ means spin-up electron.

Answer: The electron is in a 4d orbital.

Example 7: Total Orbitals for a Given $n$

How many orbitals are present in $n = 4$?

Solution: Total orbitals $= n^2 = 4^2 = 16$.

Subshells: 4s (1) + 4p (3) + 4d (5) + 4f (7) = 16 ✓

Answer: 16 orbitals.

Example 8: Which Subshells Exist?

Which of the following subshells exist: 1p, 2s, 2d, 3f, 3d?

Solution:

• 1p: $n = 1, l = 1$ → Invalid ($l$ must be $< n$, so $l = 0$ only) → Does not exist
• 2s: $n = 2, l = 0$ → Valid → Exists
• 2d: $n = 2, l = 2$ → Invalid ($l$ must be 0 or 1 for $n = 2$) → Does not exist
• 3f: $n = 3, l = 3$ → Invalid ($l$ must be 0, 1, or 2 for $n = 3$) → Does not exist
• 3d: $n = 3, l = 2$ → Valid → Exists

Answer: 2s and 3d exist. 1p, 2d, and 3f do not.

Example 9: Quantum Numbers for the 2p Subshell

Write all possible sets of quantum numbers for electrons in the 2p subshell.

Solution: For 2p: $n = 2, l = 1$. Possible $m_l$ values: $-1, 0, +1$. Each can have $m_s = +1/2$ or $-1/2$.

$n$	$l$	$m_l$	$m_s$
2	1	$-1$	$+1/2$
2	1	$-1$	$-1/2$
2	1	$0$	$+1/2$
2	1	$0$	$-1/2$
2	1	$+1$	$+1/2$
2	1	$+1$	$-1/2$

Total: 6 sets → maximum 6 electrons in 2p subshell.

Example 10: Relating $|\psi|^2$ to Probability

What is the physical significance of $|\psi|^2$?

Solution: $|\psi|^2$ at a point gives the probability density of finding the electron at that point. To find the probability of finding the electron in a small volume $dV$ around a point, we calculate $|\psi|^2 \cdot dV$.

The total probability over all space must equal 1: $$\int |\psi|^2 \, dV = 1$$

This is called the normalisation condition.

Answer: $|\psi|^2$ is the probability density — it gives the probability of finding the electron per unit volume at a given point.