Introductory Quantum Mechanics: Postulates, Duality, and Uncertainty Principles

Quantum mechanics is necessary to explain behavior at the atomic level ( $size < 10^{-10}\,m$ ) where classical physics fails.
Phenomena classical physics cannot explain: spectrum of blackbody radiation, photoelectric effect, wave-particle duality, and discrete energy levels.

A blackbody is an object that completely absorbs radiation of all wavelengths incident upon it.
Max Planck's Postulates (1900):
- Oscillators possess only discrete energy values: $E = nh\nu$ (where $n = 0, 1, 2...$ , $h$ is Planck's constant, and $\nu$ is frequency).
- Energy emission or absorption occurs in packets ( $h\nu$ ) rather than continuously: $\Delta E = \Delta nh\nu$ .
Planck's Radiation Law: $u(\lambda) d\lambda = \frac{8\pi hc}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1} d\lambda$

This phenomenon involve the emission of photoelectrons from a metal surface when light of a suitable frequency is incident upon it.
Incident light consists of photons with energy $E = h\nu$ .
Einstein's equation: $h\nu = \phi_0 + K.E.$ (where $h\nu$ is photon energy, $\phi_0$ is the work function, and $K.E.$ is the kinetic energy of emitted electrons).

It is impossible to simultaneously determine the exact position and momentum of a small moving particle.
Mathematical representations:
- $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$
- $\Delta E \cdot \Delta t \ge \frac{h}{4\pi}$
- $\Delta J \cdot \Delta \theta \ge \frac{h}{4\pi}$
Non-existence of electrons in the nucleus: Given a nucleus size $\Delta x \approx 10^{-14}\,m$ , the required kinetic energy for an electron is calculated at approximately $97\,MeV$ , exceeding the observed limit of $4\,MeV$ .
Existence of protons in the nucleus: The calculated kinetic energy for a proton is approximately $52\,keV$ , which is smaller than the energies of emitted particles, allowing for nuclear existence.

Every moving particle exhibits wave nature with a wavelength given by: $\lambda = \frac{h}{mv} = \frac{h}{p}$ .
Matter waves are localized wave packets (finite in space) and are not progressive or single waves.
Alternate forms of De-Broglie wavelength:
- In terms of kinetic energy: $\lambda = \frac{h}{\sqrt{2m(K.E.)}}$
- For a charged particle in potential ( $V$ ): $\lambda = \frac{h}{\sqrt{2mqV}}$
- In thermal equilibrium ( $T$ ): $\lambda = \frac{h}{\sqrt{3mk_B T}}$

Group Velocity ( $v_g$ ): The velocity of a wave packet formed by the superposition of waves: $v_g = \frac{d\omega}{dk}$ .
Phase Velocity ( $v_p$ ): The individual velocity of a wave within the packet: $v_p = \frac{\omega}{k}$ .
Relation between $1v_g $and$ v_p $:$ v_g = v_p - \lambda \frac{dv_p}{d\lambda}.\n\n# Wave Function (\psi)\n\n* In quantum mechanics, \psi(x, y, z, t) is a mathematical function describing the state of a particle.\n* Physical Significance: Purely mathematical on its own, but the modulus square |\psi|^2 (probability density) represents the probability of finding the particle at a specific position and time (interpreted by Max Born).\n* Properties of \psi:\n * Must be finite, single-valued, and continuous.\n * First-order derivatives (\frac{\partial \psi}{\partial x} $,$ \frac{\partial \psi}{\partial y} $,$ \frac{\partial \psi}{\partial z}) must be continuous.\n * Must be normalized: \int |\psi|^2 \, dx \, dy \, dz = 1.\n * Must approach zero as coordinates approach infinity.\n\n# Fundamental Constants\n\n* Planck's constant (h $):$ 6.626 \times 10^{-34}\,J\cdot s\n* Mass of electron (m_e $):$ 9.1 \times 10^{-31}\,kg\n* Mass of proton (m_p $):$ 1.67 \times 10^{-27}\,kg\n* Boltzmann constant (k_B $):$ 1.38 \times 10^{-23}\,J\cdot K^{-1}\n* Elementary charge (e $):$ 1.6 \times 10^{-19}\,C\n* Speed of light (c $):$ 3 \times 10^8\,m/s$$