Quantum Physics & Relativity

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### Special Theory of Relativity In classical (Newtonian) mechanics, space, time, and mass are absolute and invariant. Time flows uniformly, and simultaneity is absolute. Mass is independent of velocity. Einstein's Special Theory of Relativity (1905) is based on two postulates: 1. All physical laws are the same in all inertial frames of reference. 2. The speed of light in vacuum ($c$) is the same in every inertial frame. ### Frame of Reference A **frame of reference** is a coordinate system defining a particle's position. - **Inertial frames:** Unaccelerated reference frames in uniform translational motion relative to one another (e.g., Galilean frames). - **Non-inertial frames:** Accelerated frames. ### Galilean Transformation For two inertial frames S and S' (S' moves with uniform velocity $v$ along the positive x-direction relative to S), when $v \ll c$: - Position: $x' = x - vt$, $y' = y$, $z' = z$ - Time: $t' = t$ - Inverse Transformation: $x = x' + vt$, $y = y'$, $z = z'$, $t = t'$ - Velocity Transformation: $u_x' = u_x - v$, $u_y' = u_y$, $u_z' = u_z$ - Acceleration: $a_x' = a_x$, $a_y' = a_y$, $a_z' = a_z$ (Acceleration is invariant under Galilean transformation). ### Lorentz Transformation Consistent with the invariance of light velocity. For S' moving at velocity $v$ relative to S along the x-axis: - $x' = \gamma (x - vt)$ - $y' = y$ - $z' = z$ - $t' = \gamma (t - \frac{vx}{c^2})$ Where $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is the Lorentz factor. **Inverse Lorentz Transformation:** - $x = \gamma (x' + vt')$ - $y = y'$ - $z = z'$ - $t = \gamma (t' + \frac{vx'}{c^2})$ #### Low Speed Approximation ($v \ll c$) In this limit, $\gamma \approx 1$. The Lorentz transformation reduces to the Galilean transformation: - $x' \approx x - vt$ - $t' \approx t$ ### Relativistic Velocity Transformation For a particle moving with velocity $u = (u_x, u_y, u_z)$ in frame S, its velocity $u' = (u_x', u_y', u_z')$ in frame S' (moving with velocity $V$ along x-axis relative to S) is: - $U_x' = \frac{U_x - V}{1 - \frac{U_x V}{c^2}}$ - $U_y' = \frac{U_y}{\gamma(1 - \frac{U_x V}{c^2})}$ - $U_z' = \frac{U_z}{\gamma(1 - \frac{U_x V}{c^2})}$ #### Relativistic Law of Addition of Velocities $U = \frac{u' + V}{1 + \frac{u' V}{c^2}}$ If $V = c$ or $u' = c$, then $U = c$. The speed of light is the ultimate speed limit. ### Quantisation of Electromagnetic Radiation **Quantum Theory (QTH)** - Describes matter and energy in discrete, indivisible units called **quanta**. - Differs from classical physics, which approximates QTH at macroscopic scales. - Describes behavior at atomic and subatomic scales. - Predicts probabilities of particle properties (position, momentum). - Explains fundamental forces (electrical, magnetic, weak, strong) but not gravity. #### The Theory of Photons - Electromagnetic radiation consists of particles called **photons** (quanta). - Energy of a photon: $E = hf = h\nu$ (where $f$ or $\nu$ is frequency, $h$ is Planck's constant). - Since $\nu = \frac{c}{\lambda}$, $E = \frac{hc}{\lambda}$. - $h = 6.626 \times 10^{-34} \text{ Js}$ - Photons have zero rest mass and travel at speed $c$. #### Photon Momentum - Momentum of a photon: $p = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda}$. ### Photoelectric Effect The emission of electrons from a substance under light action. #### Experimental Observations 1. **Current & Intensity:** Photoelectric current increases with increasing intensity of radiation (for constant potential difference and frequency). 2. **Current & Potential Difference:** Current varies with potential difference and reaches a constant value. A negative potential (retarding potential) can stop the current (stopping potential $V_0$). - Maximum kinetic energy ($K_{max}$) of photoelectrons: $K_{max} = eV_0$. 3. **Kinetic Energy & Frequency:** $K_{max}$ (and $V_0$) is independent of intensity but depends only on the frequency of radiation. 4. **Threshold Frequency:** For each substance, a characteristic threshold frequency $f_0$ exists below which no photoelectrons are ejected, regardless of intensity. The corresponding wavelength is the threshold wavelength $\lambda_0$. 5. **No Time Lag:** Electron ejection is instantaneous upon light incidence. #### Failure of Classical Physics Classical wave theory predicted: - $K_{max}$ should depend on intensity, not frequency. - No threshold frequency. - A time lag for electron emission. These predictions contradict experimental observations. #### Einstein's Explanation - Light consists of photons, each with energy $hf$. - **Work function ($\phi$):** Minimum energy required to free an electron from the material. - **Photoelectric Equation:** $K_{max} = hf - \phi$. - **Threshold condition:** If $hf_0 = \phi$, then $K_{max} = 0$. So, $f_0 = \frac{\phi}{h}$ and $\lambda_0 = \frac{hc}{\phi}$. - Increasing intensity increases the number of photons, thus increasing the number of ejected electrons (current), not their energy. - Energy transfer is instantaneous due to the particle nature of photons. ### Blackbody Radiation A **blackbody** absorbs all incident radiation and appears black. - **Kirchhoff's Theorem:** For a body in thermal equilibrium with radiation, the emitted power is proportional to the absorbed power. A blackbody has an absorptivity $A_{\nu} = 1$ for all frequencies. - Power emitted per unit area per unit frequency: $e_{\nu} = J(\nu, T)$. #### Stefan-Boltzmann Law - The total power per unit area emitted by a hot solid ($e_{total}$) is proportional to the fourth power of its absolute temperature $T$. - $e_{total} = \int_0^\infty e_\nu d\nu = \sigma T^4$ (for a blackbody) - For a non-ideal radiator: $e_{total} = \alpha \sigma T^4$, where $0 ### Compton Effect Evidence for the particle nature of radiation from scattering experiments. - When X-rays scatter off electrons, the scattered X-rays have a longer wavelength ($\lambda'$) than the incident X-rays ($\lambda$). - **Compton shift ($\Delta \lambda$):** The change in wavelength, $\Delta \lambda = \lambda' - \lambda$. - The Compton shift depends only on the scattering angle $\theta$ and is given by: $\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)$ - $\frac{h}{m_e c}$ is the **Compton wavelength** (for electron $m_e = 9.11 \times 10^{-31} \text{ kg}$), which is $2.43 \times 10^{-12} \text{ m}$ or $0.00243 \text{ nm}$. - The effect is more pronounced for X-rays and gamma rays and for lighter particles. ### Introductions to Quantum Mechanics - Describes particles and their interactions with energy. - More general than classical mechanics; classical mechanics is an approximation at large scales. - **Quanta:** Discrete individual units of matter and energy. - Particles do not have definite locations, speeds, or paths; quantum theory describes probabilities. - Essential for understanding atoms, subatomic particles, origins of the universe, and forces (electrical, magnetic, weak, strong). ### Basic Postulates of Quantum Mechanics 1. **Operators:** Each dynamical variable (observable) in classical physics corresponds to a linear operator in quantum mechanics. - Position: $(x, y, z)$ - Momentum: $p_x = -i \hbar \frac{\partial}{\partial x}$ - Total energy (Hamiltonian): $H = -\frac{\hbar^2}{2m} \nabla^2 + V$ 2. **Eigenvalue Equation:** A linear operator can be linked to an Eigenvalue equation. - For a total energy operator $H$: $H \Psi = E \Psi$ - $\Psi$ is the **Eigenfunction**, $E$ is the **Eigenvalue** (corresponding energy). 3. **Measurement & Probability:** Measurements of a dynamical quantity generally yield different values over trials (in conformity with the uncertainty principle). - The probability density for finding a particle is $|\Psi|^2 = \Psi^* \Psi$. - The expectation value of an observable $A$ is $\langle A \rangle = \int \Psi^* A \Psi \, dv$. ### The Wave Function - Represented by $\Psi$ (Psi), a function of position ($x, y, z$) and time ($t$). - Can have real and imaginary parts. - $|\Psi|^2 = \Psi^* \Psi$ is proportional to the probability per unit volume of finding the particle. - **Normalization:** $\int \Psi^* \Psi \, dv = 1$. If $\Psi$ is normalized, $|\Psi|^2$ is the probability density. - $\Psi$ must be single-valued, and its partial derivatives ($\frac{\partial \Psi}{\partial x}$, etc.) must be continuous. ### Schrödinger Wave Equation (SWE) Describes the wave-like properties of particles. #### Time-Dependent SWE - Three-dimensional: $i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi$ - Where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ is the Laplacian operator. #### Time-Independent SWE For a potential energy $V$ that does not depend on time, we can separate variables $\Psi(\mathbf{r}, t) = \psi(\mathbf{r}) e^{-iEt/\hbar}$. Substituting this into the time-dependent SWE gives: - Three-dimensional: $-\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi$ - One-dimensional: $-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V \psi = E \psi$ ### Matter and Waves #### Wave-Particle Duality - Light exhibits both wave and particle nature. - De Broglie hypothesized that particles like electrons also have wave properties. #### De Broglie Hypothesis - The wavelength associated with a particle is called the De Broglie wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$. - Momentum of a photon: $P = \frac{E}{c}$. - For a particle with zero rest mass (photon): $E = PC$. - **Velocity of De Broglie Waves:** The phase velocity of the De Broglie wave ($W$) is not the same as the particle velocity ($v$). - $W = f\lambda$. - Using relativistic energy $E = \gamma mc^2$ and momentum $P = \gamma mv$, and $E=hf$, $P=h/\lambda$: $W = \frac{c^2}{v}$. - Since $v c$. A particle travels within a wave packet, and the *group velocity* of the wave packet is $v$. ### The Uncertainty Principle - It is impossible to simultaneously measure certain pairs of physical properties, such as position and momentum, with arbitrary precision. #### Heisenberg Uncertainty Principle - **Position-Momentum Uncertainty:** The product of the uncertainties in position ($\Delta x$) and momentum ($\Delta p_x$) is greater than or equal to a minimum value: - $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ (or $\Delta x \Delta p_x \ge \frac{h}{4\pi}$) - If $\Delta x = 0$, then $\Delta p_x \to \infty$ (and vice versa). - **Energy-Time Uncertainty:** The product of the uncertainties in energy ($\Delta E$) and time ($\Delta t$) is greater than or equal to a minimum value: - $\Delta E \Delta t \ge \frac{\hbar}{2}$ (or $\Delta E \Delta t \ge \frac{h}{4\pi}$) - This isn't due to measurement inaccuracies but is fundamental to the wave nature of matter. ### Atom and Elementary Particles - Common particles: electrons, protons, neutrons, photons, neutrinos. Mostly stable (except free neutrons). - **Mesons:** First discovered (1947), rest energy > 100 MeV. Involved in strong nuclear force exchange. - **Quarks:** Fundamental building blocks of protons, neutrons, and other heavy particles. #### Particle Interaction - Forces are mediated by particle exchange. - Electromagnetic force: virtual photons. - Gravitational force: virtual gravitons (undiscovered). - Weak nuclear force: massive virtual W$^\pm$ and Z$^0$ bosons. - Strong nuclear force: gluons. | S/N | Force | Strength | Name | Rest Mass | Charge | Spin | |-----|----------------|------------|-----------|-------------------------|---------|------| | 1 | Strong nuclear | 1 | Gluon | 0 | 0 | 1 | | 2 | E-M | $10^{-2}$ | Photons | 0 | 0 | 1 | | 3 | Weak nuclear | $10^{-13}$ | W$^\pm$, Z$^0$ | 82.95 Gev/$c^2$ | $\pm e, 0$ | 1 | | 4 | Gravitational | $10^{-35}$ | Gravitons | 0 | 0 | 2 | ### Unified Theories - Goal: Single theory encompassing all four fundamental forces. - **Maxwell:** Unified electric and magnetic forces into electromagnetism. - **Electroweak Theory:** Steven Weinberg, Abdus Salam, Sheldon Glashow (1960s). Unified weak nuclear and electromagnetic forces. - **Grand Unified Theories (GUTs):** Aim to unify electroweak and strong nuclear forces. - Predicts proton decay (very long lifetime). - Predicts existence of magnetic monopoles. - Neutrinos should have a small rest mass. - Unification is apparent at very high energies or very small distances due to **symmetry breaking** at lower energies/larger distances. ### Elementary Particle Classification Particles are grouped into three families: 1. **Bosons:** - Interact via electromagnetic and gravitational forces; responsible for electroweak force. - Considered fundamental (not composed of smaller particles). - Examples: Photons, Z$^0$, W$^\pm$, gluons, gravitons. 2. **Leptons:** - Lightest particles. - Not affected by the strong nuclear force. Interact via weak and electromagnetic forces. - Considered fundamental. - Examples: Electrons ($\text{e}^-$), muons ($\mu^-$), taus ($\tau^-$), and their corresponding neutrinos ($\nu_e, \nu_\mu, \nu_\tau$). - **Lepton number (L):** $+1$ for leptons, $-1$ for anti-leptons. Lepton numbers (electron, muon, tau flavors) are conserved. 3. **Hadrons:** - Largest family, interact strongly, weakly, electromagnetically, and gravitationally. - Divided into: - **Baryons:** Obey baryon number conservation ($B=+1$ for baryons, $-1$ for anti-baryons, $0$ for others). Examples: Protons ($p$), neutrons ($n$). - **Mesons:** Do not have baryon number. Examples: Pions ($\pi$), Kaons ($K$), eta ($\eta$). #### Strangeness Number - Another quantum number assigned to particles, particularly "strange particles." - Arises from specific creation/decay patterns and longer-than-expected lifetimes for certain particles. - A particle and its antiparticle have opposite charge and strangeness (but same rest energy, mean life, and spin). ### Conservation Laws Elementary particle reactions and decays obey: - Mass-energy conservation - Linear momentum conservation - Angular momentum (spin) conservation - Charge conservation (quantized in units of $e$) - Lepton number conservation (for each flavor) - Baryon number conservation #### Isotopic Spin (Isospin) - Describes the charge independence of the strong interaction. - Particles in a multiplet interact strongly in the same way, with small differences due to electromagnetism. - Isospin $I$ is treated as a vector with magnitude $\sqrt{I(I+1)}$. - Its z-component is $M_I$. - Total isospin $I$ and its z-component $M_I$ are conserved in strong interactions. - Not conserved in weak interactions. ### Rutherford and Bohr's Model of Atoms #### Rutherford's Model (1911) - Based on alpha particle scattering experiments. - Positive charge concentrated in a small central **nucleus** ($r 1$). $E_2 = -3.40 \text{ eV}$ (First excited state); $E_3 = -1.51 \text{ eV}$ (Second excited state). - **Excitation Energy:** Energy required to move an electron from ground state to an excited state (e.g., $E_2 - E_1 = 10.20 \text{ eV}$). - $E_{\infty} = 0$: Electron is free. ### Atomic Spectra - When excited atoms de-excite, electrons jump to lower energy states, emitting photons. - These photons cause discrete spectral lines, not a continuous spectrum. - Frequency $f$ and wavelength $\lambda$ depend on the energy difference between states. - **Rydberg Formula:** - For frequency: $f = Z^2 (\frac{3.29 \times 10^{15}}{\text{Hz}}) \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$ - For wavelength: $\frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$ - $R = 1.097 \times 10^7 \text{ m}^{-1}$ (Rydberg constant). - $n_i$: initial principal quantum number. - $n_f$: final principal quantum number. #### Spectral Series for Hydrogen ($Z=1$) Each series corresponds to a specific final state $n_f$: | Series | Final State ($n_f$) | Initial States ($n_i$) | Region | |-------------|---------------------|------------------------|-------------------| | Lyman Series | 1 | 2, 3, 4, ... | Ultraviolet (UV) | | Balmer Series | 2 | 3, 4, 5, ... | Visible | | Paschen Series| 3 | 4, 5, 6, ... | Infrared (IR) | | Brackett Series| 4 | 5, 6, 7, ... | Infrared (IR) | | Pfund Series | 5 | 6, 7, 8, ... | Infrared (IR) | - Emission spectra correspond to $n_i > n_f$. - Absorption spectra correspond to $n_f > n_i$. Atoms must be hot to absorb at many wavelengths.