de Broglie's Hypothesis: Matter (like electrons) exhibits both particle and wave properties.
de Broglie Wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$, where $h$ is Planck's constant, $p$ is momentum, $m$ is mass, $v$ is velocity.
For an electron accelerated through potential $V$ (in Volts): $\lambda = \frac{12.27}{\sqrt{V}} \text{ Å}$
NCERT Point: This hypothesis was experimentally confirmed by Davisson and Germer.
Relation to Bohr's model: Bohr's condition for quantization of angular momentum ($mvr = n\frac{h}{2\pi}$) can be derived from de Broglie's concept of standing waves. For a stable orbit, the circumference must be an integral multiple of the de Broglie wavelength: $2\pi r = n\lambda = n\frac{h}{mv} \implies mvr = n\frac{h}{2\pi}$.

Q: An electron is accelerated through a potential difference of $100$ V. What is the de Broglie wavelength associated with it?

A: Given $V = 100 \text{ V}$.

Using the formula: $\lambda = \frac{12.27}{\sqrt{V}} \text{ Å}$

$\lambda = \frac{12.27}{\sqrt{100}} = \frac{12.27}{10} = 1.227 \text{ Å}$

It is impossible to determine simultaneously, with absolute precision, both the position and momentum of a subatomic particle (like an electron).
$\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$ or $\Delta x \cdot m \Delta v \ge \frac{h}{4\pi}$
$\Delta E \cdot \Delta t \ge \frac{h}{4\pi}$
NCERT Point: This principle rules out the existence of definite paths or trajectories for electrons and other similar particles. It is significant for microscopic objects, but negligible for macroscopic objects.

Based on wave mechanics (Schrödinger Equation). Describes electrons as waves.
Schrödinger Wave Equation (Time-independent): $\hat{H}\psi = E\psi$
- $\hat{H}$: Hamiltonian operator (total energy operator).
- $\psi$ (psi): Wave function, describes the amplitude of the electron wave.
- $E$: Total energy of the electron.
Significance of $\psi^2$: Represents the probability density of finding an electron at a particular point in space. It is always positive.
Atomic Orbitals: Regions of space around the nucleus where the probability of finding an electron is maximum (typically 90-95% probability contour).
NCERT Point: The quantum mechanical model describes the probability of finding an electron around the nucleus, not its exact position.

These numbers arise naturally from the solution of Schrödinger equation and specify the exact location and energy of an electron in an atom.
1. Principal Quantum Number ($n$):
- $n = 1, 2, 3, ...$ (positive integers)
- Determines the main energy shell and size of the orbital.
- As $n$ increases, energy and average distance from nucleus increase.
- Number of orbitals in a shell = $n^2$. Maximum electrons in a shell = $2n^2$.
- Energy of electron in a multi-electron atom depends primarily on $n$ and $l$.
2. Azimuthal/Angular Momentum Quantum Number ($l$):
- $l = 0, 1, 2, ..., (n-1)$
- Determines the subshell and shape of the orbital.
- $l=0 \implies s$ subshell (spherical)
- $l=1 \implies p$ subshell (dumbbell)
- $l=2 \implies d$ subshell (double dumbbell/complex)
- $l=3 \implies f$ subshe