Quantum Mechanical Model and Quantum Numbers

Classical mechanics — Newton's laws — works brilliantly for planets, cricket balls, and cars. But it has three fatal flaws when applied to electrons:

• It ignores the wave nature of matter (de Broglie)
• It assumes you can know an electron's exact position and velocity simultaneously (violates Heisenberg)
• It can't explain why atomic energy levels are quantized

Quantum mechanics is the branch of science built to handle exactly these situations. It was developed independently in 1926 by Werner Heisenberg (matrix mechanics) and Erwin Schrödinger (wave mechanics). Although they looked completely different, Schrödinger and others soon proved they are mathematically equivalent. We'll focus on Schrödinger's wave formulation — it is far more visual and intuitive.

The fundamental equation of quantum mechanics is the Schrödinger equation.

For a system whose energy does not change with time (like an electron in an atom), it is written as:

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

Where:

• $\hat{H}$ = the Hamiltonian operator — a mathematical recipe for calculating the total energy(kinetic + potential) of all particles in the system
• $\psi$ (psi) = the wave function — the solution we are looking for
• $E$ = the allowed energy values corresponding to each solution $\psi$

Don't worry about solving this equation — that requires advanced mathematics you'll cover later.What matters for JEE/NEET is understanding what the solutions mean.

When you solve the Schrödinger equation mathematically, you get many possible solutions — but not all of them are physically meaningful. You must apply three strict conditions to filter out the valid ones:

1. Single-valued: At any point in space, $\psi$ must have only one value. If a wave function gave two different probabilities at the same location, it would be physically absurd. Think of it like a map — a location can only have one altitude, never two simultaneously.

2. Finite: $\psi$ must not blow up to infinity at any point. An infinite probability density is physically impossible — the electron cannot be "infinitely likely" to be anywhere. Any solution where $\psi \to \infty$ is rejected.

3. Continuous: $\psi$ and its slope ($d\psi/dr$) must be smooth and continuous everywhere. Abrupt jumps in the wave function would imply discontinuous physical properties, which nature does not permit.

Only solutions that pass all three tests are the acceptable energy levels an electron can occupy. These filters are why atomic energy levels are quantized — the three conditions are satisfied only for specific values of $E$, not any arbitrary value.

When the Schrödinger equation is solved for the hydrogen atom, each valid solution gives:

1. An energy value $E$ — the energy level the electron occupies
2. A wave function $\psi$ — a mathematical function that varies across space

Important: $\psi$ itself has no direct physical meaning. You cannot measure $\psi$. It's like the birth chart — the chart itself doesn't do anything, but it encodes the information.

What has physical meaning is $|\psi|^2$ — the square of the wave function:

$|\psi|^2$ at any point = probability density = how likely you are to find the electron near that point

$|\psi|^2$ is always positive (a square is always $\geq 0$), which makes physical sense — probability is never negative.

Each wave function corresponding to an allowed energy level is called an atomic orbital. An orbital is not a path (like Bohr's circular orbit) — it is a region of space described by $\psi$ where there is a high probability of finding the electron.

When the Schrödinger equation is solved, the three boundary conditions force the solutions to depend on three integers: $n$, $l$, and $m_l$. These are called quantum numbers and they arise naturally from the mathematics — they are not invented or assumed. A fourth quantum number, $m_s$ (electron spin), was added later to explain fine details of spectra.

Think of quantum numbers like an address system:

• $n$ = the city (which shell/energy level)
• $l$ = the neighbourhood (which subshell/shape)
• $m_l$ = the street (which orientation in space)
• $m_s$ = the apartment (which spin: up or down)

Every electron in every atom in the universe is completely described by its four quantum numbers. No two electrons in the same atom can have all four quantum numbers identical — that is the Pauli Exclusion Principle (coming soon).

$n$ is a positive integer: $n = 1, 2, 3, 4, \ldots$

It determines:

• Size of the orbital — larger $n$ = electron farther from nucleus
• Energy of the orbital — larger $n$ = higher energy (electron more loosely held)
• Shell (principal energy level) — identified by letters:

The total number of orbitals in the $n^{\text{th}}$ shell is $n^2$.For example, the M shell ($n = 3$) has $3^2 = 9$ orbitals.

For hydrogen-like species ($\ce{H}$, $\ce{He^{+}}$, $\ce{Li^{2+}}$):energy depends only on $n$. For multi-electron atoms, both $n$ and $l$ determine energy.

$l$ is also called the orbital angular momentum or subsidiary quantum number.

For a given $n$: $l$ can be $0, 1, 2, \ldots, (n-1)$ — so there are $n$ possible values of $l$.

$l$ determines:

• Shape of the orbital (its 3D geometry)
• Subshell (sub-level within a shell)

Example: For $n = 3$, $l$ can be 0, 1, or 2 → three subshells: $3s$, $3p$, $3d$.

In multi-electron atoms: $l$ also influences energy.For the same $n$, energy increases as: $s < p < d < f$.This is because of electron-electron repulsion and shielding — concepts we cover in the orbital energy section.

$m_l$ gives information about the spatial orientation of the orbital — how it ispointed in 3D space relative to a set of co-ordinate axes $(x, y, z)$.

For a given $l$, $m_l$ can take $(2l + 1)$ values:

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

Examples:

• $l = 0$ (s subshell): $m_l = 0$ only → 1 orbital (sphere, one orientation)
• $l = 1$ (p subshell): $m_l = -1, 0, +1$ → 3 orbitals ($p_x$, $p_y$, $p_z$)
• $l = 2$ (d subshell): $m_l = -2, -1, 0, +1, +2$ → 5 orbitals
• $l = 3$ (f subshell): $m_l = -3$ to $+3$ → 7 orbitals

The three quantum numbers $n$, $l$, $m_l$ come directly from Schrödinger's equation. But spectroscopy showed that spectral lines sometimes split into pairs — there seemed to be a few more energy states than the three numbers could explain.

In 1925, Uhlenbeck and Goudsmit proposed a fourth quantum number: the electron spin quantum number $m_s$.

An electron spins around its own axis (just like Earth spins while orbiting the Sun). This spin can point in only two directions relative to an external magnetic field:

• $m_s = +\frac{1}{2}$ → spin up ↑
• $m_s = -\frac{1}{2}$ → spin down ↓

These are the two spin states of every electron.

Key rules that follow:

• An orbital can hold maximum 2 electrons
• Those two electrons must have opposite spins ($+\frac{1}{2}$ and $-\frac{1}{2}$)
• This is why each orbital on an energy diagram is drawn as a box containing at most one ↑ and one ↓