Modern Physics Concepts

Cheatsheet Content

1. Special Relativity Postulates: The laws of physics are the same for all observers in uniform motion (inertial frames). The speed of light $c$ in a vacuum is the same for all inertial observers, regardless of the motion of the light source or observer. Key Phenomena: Time Dilation: Moving clocks run slower: $\Delta t = \gamma \Delta t_0$, where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$. Length Contraction: Moving objects appear shorter in the direction of motion: $L = L_0 / \gamma$. Relativistic Mass: $m = \gamma m_0$. Mass-Energy Equivalence: $E = mc^2 = \gamma m_0 c^2$. For a particle at rest, $E_0 = m_0 c^2$. Relativistic Momentum: $p = \gamma m_0 v$. 2. Quantum Mechanics 2.1. Wave-Particle Duality Planck's Hypothesis: Energy is quantized. $E = hf = \hbar \omega$. $h$: Planck's constant ($6.626 \times 10^{-34} \text{ J} \cdot \text{s}$) $\hbar = h / (2\pi)$ Photon Energy: Light quanta (photons) have energy $E = hf$. De Broglie Wavelength: Particles exhibit wave-like properties with wavelength $\lambda = h/p$, where $p$ is momentum. Photoelectric Effect: Electrons emitted from metal surface when light shines on it. $KE_{max} = hf - \phi$, where $\phi$ is the work function. 2.2. Heisenberg's Uncertainty Principle Cannot precisely know both position and momentum: $\Delta x \Delta p \ge \hbar/2$. Cannot precisely know both energy and time: $\Delta E \Delta t \ge \hbar/2$. 2.3. Schrödinger Equation Time-Dependent: $i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}, t) \right) \Psi(\mathbf{r}, t)$ Time-Independent (for stationary states): $-\frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}) + V(\mathbf{r}) \psi(\mathbf{r}) = E \psi(\mathbf{r})$ $\Psi(\mathbf{r}, t)$: Wave function, $|\Psi|^2$ gives probability density. 2.4. Atomic Structure (Bohr Model & Quantum Numbers) Bohr Model (for Hydrogen-like atoms): Quantized orbits and energy levels: $E_n = -\frac{13.6 \text{ eV}}{n^2} Z^2$. Angular momentum quantization: $L = n\hbar$. Quantum Numbers: Principal ($n$): $1, 2, 3, ...$ (energy level, shell) Azimuthal ($l$): $0, 1, ..., n-1$ (orbital shape, subshell) Magnetic ($m_l$): $-l, ..., 0, ..., +l$ (orbital orientation) Spin ($m_s$): $+1/2, -1/2$ (electron spin) Pauli Exclusion Principle: No two identical fermions can occupy the same quantum state simultaneously. 3. Nuclear Physics Nucleus Composition: Protons (Z) and Neutrons (N). Mass number $A = Z+N$. Nuclear Force: Strong force, short-range, attractive, binds nucleons. Binding Energy: Energy required to separate a nucleus into its constituent nucleons. $E_b = (\Delta m) c^2$, where $\Delta m = (Z m_p + N m_n) - m_{nucleus}$. Radioactive Decay: Alpha Decay ($\alpha$): Emission of $^4_2\text{He}$ nucleus. $^A_Z\text{X} \to ^{A-4}_{Z-2}\text{Y} + ^4_2\text{He}$. Beta Decay ($\beta^-$): Neutron converts to proton, emits electron and antineutrino. $^A_Z\text{X} \to ^A_{Z+1}\text{Y} + e^- + \bar{\nu}_e$. Beta Decay ($\beta^+$): Proton converts to neutron, emits positron and neutrino. $^A_Z\text{X} \to ^A_{Z-1}\text{Y} + e^+ + \nu_e$. Gamma Decay ($\gamma$): Emission of high-energy photon from excited nucleus. $^A_Z\text{X}^* \to ^A_Z\text{X} + \gamma$. Half-Life ($T_{1/2}$): Time for half of radioactive nuclei to decay. $N(t) = N_0 e^{-\lambda t}$, where $\lambda = \ln(2)/T_{1/2}$. Nuclear Fission: Heavy nucleus splits into lighter nuclei (e.g., uranium). Nuclear Fusion: Light nuclei combine to form heavier nucleus (e.g., in stars). 4. Particle Physics (Standard Model) Fundamental Particles: Fermions (matter particles, spin 1/2): Quarks (6 flavors): Up, Down, Charm, Strange, Top, Bottom. Combine to form hadrons (e.g., protons, neutrons). Leptons (6 flavors): Electron, Muon, Tau, and their corresponding neutrinos ($\nu_e, \nu_\mu, \nu_\tau$). Bosons (force carriers, integer spin): Photon ($\gamma$): Electromagnetic force. Gluon ($g$): Strong nuclear force. W$^\pm$, Z$^0$: Weak nuclear force. Higgs Boson ($H$): Gives mass to other particles. Fundamental Forces: Strong: Binds quarks, holds nuclei together. (Mediated by gluons) Electromagnetic: Between charged particles. (Mediated by photons) Weak: Radioactive decay. (Mediated by W, Z bosons) Gravitational: Between masses. (Hypothesized graviton, not in Standard Model) 5. Cosmology Big Bang Theory: Universe originated from an extremely hot, dense state. Hubble's Law: Galaxies are receding from us, velocity proportional to distance. $v = H_0 d$. $H_0$: Hubble constant. Cosmic Microwave Background (CMB): Relic radiation from early universe, strong evidence for Big Bang. Dark Matter: Non-luminous matter, inferred from gravitational effects (e.g., galaxy rotation curves). Dark Energy: Mysterious force causing accelerated expansion of the universe.