Structure of Atom - Quantum Mechanical Model

Dual Nature of Matter: Proposed by de Broglie, suggesting that moving particles like electrons exhibit both wave and particle properties. The de Broglie wavelength is given by $\lambda = \frac{h}{mv}$.

Heisenberg's Uncertainty Principle: It states that it is impossible to determine simultaneously the exact position and exact momentum (or velocity) of an electron with absolute accuracy: $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$.

Schrödinger Wave Equation: A fundamental equation in quantum mechanics that describes the behavior of an electron in an atom. The solution gives the wave function $\psi$, where $|\psi|^2$ represents the probability density of finding an electron.

Quantum Numbers: Four numbers used to describe the state of an electron: Principal ($n$), Azimuthal ($l$), Magnetic ($m_l$), and Spin ($m_s$).

Principal Quantum Number ($n$): Defines the main energy shell and size of the orbital. $n = 1, 2, 3, \dots$.

Azimuthal Quantum Number ($l$): Defines the shape of the orbital. Values range from $0$ to $(n-1)$. $l=0 (s)$, $l=1 (p)$, $l=2 (d)$, $l=3 (f)$.

Magnetic Quantum Number ($m_l$): Describes the orientation of orbitals in space. Values range from $-l$ to $+l$ including zero.

Spin Quantum Number ($m_s$): Describes the direction of electron spin, either $+\frac{1}{2}$ or $-\frac{1}{2}$.

Pauli Exclusion Principle: No two electrons in an atom can have the same set of all four quantum numbers. An orbital can hold a maximum of 2 electrons with opposite spins.

Aufbau Principle: Electrons are filled in orbitals in the increasing order of their energies, following the $(n+l)$ rule.

Hund's Rule of Maximum Multiplicity: Pairing of electrons in degenerate orbitals ($p, d, f$) does not occur until each orbital is singly occupied with parallel spins.

Nodes: Regions where the probability of finding an electron is zero. Radial nodes $= n - l - 1$, Angular nodes $= l$, Total nodes $= n - 1$.