⚛️Atomic Physics Unit 7 – Relativistic Quantum Mechanics

Key Concepts and Foundations

• Relativistic quantum mechanics combines the principles of special relativity and quantum mechanics to describe the behavior of particles at high energies and velocities approaching the speed of light
• Fundamental concepts include the wave-particle duality, where particles exhibit both wave-like and particle-like properties depending on the experimental setup
  • Electrons, for example, can behave as waves in the double-slit experiment, producing interference patterns
• The Heisenberg uncertainty principle states that the product of the uncertainties in the position and momentum of a particle is always greater than or equal to $\frac{\hbar}{2}$, where $\hbar$ is the reduced Planck's constant
• The Schrödinger equation, $i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$, describes the time evolution of the wave function $\Psi(\mathbf{r},t)$ of a quantum system
  • The Hamiltonian operator $\hat{H}$ represents the total energy of the system
• The Born interpretation of the wave function states that the probability of finding a particle at a given position is proportional to the square of the absolute value of the wave function at that position, $P(\mathbf{r},t) = |\Psi(\mathbf{r},t)|^2$
• The correspondence principle asserts that quantum mechanics must reduce to classical mechanics in the limit of large quantum numbers or when Planck's constant can be considered negligibly small compared to the action of the system

Special Relativity Refresher

• Special relativity is based on two postulates: the principle of relativity (physical laws are the same in all inertial reference frames) and the invariance of the speed of light in vacuum ($c$) for all observers
• The Lorentz transformations describe how space and time coordinates change between inertial reference frames moving at constant velocity relative to each other
  • The Lorentz factor, $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, appears in these transformations and depends on the relative velocity $v$ between the frames
• Length contraction occurs along the direction of motion, with the proper length $L_0$ observed in the rest frame of the object being contracted to $L = \frac{L_0}{\gamma}$ in a frame moving relative to the object
• Time dilation causes moving clocks to tick more slowly than stationary ones, with the proper time $\tau$ elapsed in the rest frame of the clock being dilated to $t = \gamma\tau$ in a frame moving relative to the clock
• The relativistic energy-momentum relation, $E^2 = (pc)^2 + (mc^2)^2$, connects the total energy $E$, momentum $p$, and rest mass $m$ of a particle
  • The famous equation $E = mc^2$ is a special case of this relation for a particle at rest ($p = 0$)
• The addition of velocities in special relativity is described by the relativistic velocity addition formula, $u = \frac{v+u'}{1+\frac{vu'}{c^2}}$, where $u$ is the velocity of an object in the original frame, $v$ is the relative velocity between the frames, and $u'$ is the velocity of the object in the second frame

Quantum Mechanics Review

• The state of a quantum system is described by a complex-valued wave function $\Psi(\mathbf{r},t)$, which contains all the information about the system
• Observables in quantum mechanics are represented by Hermitian operators acting on the wave function, with the eigenvalues of these operators corresponding to the possible measurement outcomes
  • The position operator $\hat{x}$ and momentum operator $\hat{p} = -i\hbar\frac{\partial}{\partial x}$ are examples of observables
• The expectation value of an observable $\hat{A}$ in a state $\Psi$ is given by $\langle\hat{A}\rangle = \langle\Psi|\hat{A}|\Psi\rangle$, where $\langle\Psi|$ is the complex conjugate of $|\Psi\rangle$
• The commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as $[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$, and its value determines whether the observables represented by these operators can be simultaneously measured with arbitrary precision
  • The position and momentum operators satisfy the canonical commutation relation $[\hat{x},\hat{p}] = i\hbar$
• The Pauli exclusion principle states that no two identical fermions (particles with half-integer spin) can occupy the same quantum state simultaneously
• The superposition principle allows quantum systems to exist in a linear combination of different eigenstates, leading to phenomena such as quantum interference and entanglement

Relativistic Wave Equations

• The Klein-Gordon equation, $(\Box + \frac{m^2c^2}{\hbar^2})\phi = 0$, where $\Box = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2$ is the d'Alembert operator, is a relativistic wave equation describing spin-0 particles
  • However, it suffers from issues such as negative probabilities and the lack of a consistent single-particle interpretation
• The Dirac equation, $(i\hbar\gamma^\mu\partial_\mu - mc)\psi = 0$, where $\gamma^\mu$ are the Dirac matrices and $\psi$ is a four-component spinor, is a relativistic wave equation describing spin-1/2 particles like electrons and quarks
  • It naturally incorporates the concept of spin and predicts the existence of antimatter
• The Dirac equation leads to the Dirac Hamiltonian, $\hat{H}_D = c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2$, where $\boldsymbol{\alpha}$ and $\beta$ are related to the Dirac matrices
• Solutions to the Dirac equation include positive-energy states (particles) and negative-energy states (antiparticles), with the Dirac sea interpretation suggesting that all negative-energy states are filled in the vacuum
• The Proca equation, $\partial_\mu F^{\mu\nu} + \frac{m^2c^2}{\hbar^2}A^\nu = 0$, where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$ is the electromagnetic field tensor and $A^\nu$ is the four-potential, describes massive spin-1 particles like the W and Z bosons
• The Rarita-Schwinger equation, $(\gamma^\mu\partial_\mu + \frac{mc}{\hbar})\psi_\nu = 0$, where $\psi_\nu$ is a vector-spinor field, describes massive spin-3/2 particles like the hypothetical gravitino in supergravity theories

Spin and Angular Momentum

• Spin is an intrinsic angular momentum possessed by elementary particles, with fermions having half-integer spin and bosons having integer spin
  • Electrons, protons, and neutrons have spin-1/2, while photons have spin-1
• The spin operator $\hat{\mathbf{S}}$ satisfies the commutation relations $[\hat{S}_i,\hat{S}_j] = i\hbar\epsilon_{ijk}\hat{S}_k$, where $\epsilon_{ijk}$ is the Levi-Civita symbol
• The eigenvalues of the spin angular momentum operator $\hat{S}^2$ are $s(s+1)\hbar^2$, where $s$ is the spin quantum number, and the eigenvalues of the spin projection operator $\hat{S}_z$ are $m_s\hbar$, where $m_s = -s,-s+1,\ldots,s-1,s$
• The Stern-Gerlach experiment demonstrated the quantization of spin by observing the deflection of a beam of silver atoms in an inhomogeneous magnetic field
• Total angular momentum $\hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}$ is the sum of orbital angular momentum $\hat{\mathbf{L}}$ and spin angular momentum $\hat{\mathbf{S}}$
  • The eigenvalues of $\hat{J}^2$ are $j(j+1)\hbar^2$, where $j$ is the total angular momentum quantum number, and the eigenvalues of $\hat{J}_z$ are $m_j\hbar$, where $m_j = -j,-j+1,\ldots,j-1,j$
• The spin-statistics theorem relates the spin of a particle to its statistical behavior: fermions obey Fermi-Dirac statistics and bosons obey Bose-Einstein statistics
• The fine structure of atomic spectra arises from the coupling between the electron's spin and its orbital angular momentum, described by the spin-orbit interaction Hamiltonian $\hat{H}_{SO} = \frac{1}{2m^2c^2}\frac{1}{r}\frac{dV}{dr}\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}$, where $V(r)$ is the electrostatic potential

Quantum Field Theory Basics

• Quantum field theory (QFT) is the framework that combines quantum mechanics and special relativity to describe the behavior of fields and their associated particles
• In QFT, particles are excitations of underlying quantum fields, with each type of particle corresponding to a specific field (e.g., the electron field, the photon field)
• The Lagrangian formalism is used to derive the equations of motion for the fields, with the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)$ depending on the field $\phi$ and its space-time derivatives $\partial_\mu\phi$
  • The action is defined as $S = \int d^4x\,\mathcal{L}(\phi,\partial_\mu\phi)$, and the principle of least action leads to the Euler-Lagrange equations of motion
• Canonical quantization promotes the fields to operators satisfying equal-time commutation relations, with the commutator $[\phi(x),\pi(y)] = i\hbar\delta^{(3)}(\mathbf{x}-\mathbf{y})$, where $\pi = \frac{\partial\mathcal{L}}{\partial(\partial_0\phi)}$ is the conjugate momentum field
• The Hamiltonian density $\mathcal{H} = \pi\partial_0\phi - \mathcal{L}$ is used to construct the Hamiltonian operator $\hat{H} = \int d^3x\,\mathcal{H}$, which generates time evolution in the Heisenberg picture
• Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of interacting particles, with lines representing particle propagators and vertices representing interaction points
  • The Feynman rules associate mathematical expressions with each element of a Feynman diagram, allowing the calculation of scattering amplitudes and cross-sections
• Renormalization is the process of dealing with infinities that arise in QFT calculations by absorbing them into the redefinition of physical quantities like mass and charge
  • Renormalizable theories, such as quantum electrodynamics (QED) and the Standard Model, have a finite number of divergent diagrams that can be consistently renormalized

Applications and Experimental Evidence

• The Lamb shift is a small difference in energy between the $2S_{1/2}$ and $2P_{1/2}$ states of the hydrogen atom, arising from the interaction of the electron with virtual photons in the quantum electrodynamic vacuum
  • Its precise measurement and agreement with theoretical predictions provided early evidence for the validity of QED
• The anomalous magnetic moment of the electron, described by the g-factor $g_e \approx 2.00231930436256(35)$, deviates slightly from the Dirac equation prediction of 2 due to QED corrections
  • The extraordinary agreement between theory and experiment (to within a few parts in $10^{13}$) is one of the most precise tests of QED and relativistic quantum mechanics
• The discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 confirmed the existence of the Higgs field, which is responsible for the generation of mass for the W and Z bosons and fermions in the Standard Model
• Neutrino oscillations, where neutrinos change flavor as they propagate through space, provide evidence for neutrino mass and lepton mixing, requiring an extension of the Standard Model
• The observation of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO) in 2015 not only confirmed a key prediction of general relativity but also opened a new window for studying compact objects like black holes and neutron stars
• Precision measurements of the fine structure constant $\alpha \approx 1/137$ and the Rydberg constant $R_\infty \approx 10973731.568160(21)\,\mathrm{m}^{-1}$ test the foundations of QED and provide input for the determination of other fundamental constants

Mind-Bending Implications

• The concept of particle-antiparticle pair creation and annihilation, where particles can be created from pure energy and annihilate into pure energy, challenges our intuitive notions of conservation and the nature of matter
• Quantum entanglement, a phenomenon where the quantum states of two or more particles are correlated even when separated by large distances, leads to seemingly paradoxical effects like the violation of Bell's inequalities and the possibility of quantum teleportation
• The quantum vacuum is not empty but rather a seething sea of virtual particle-antiparticle pairs constantly appearing and disappearing, with observable consequences like the Casimir effect and vacuum polarization
• The possibility of extra dimensions, as suggested by theories like string theory and Kaluza-Klein theory, could provide a framework for unifying gravity with the other fundamental forces and may lead to observable effects at high energies or small length scales
• The holographic principle, inspired by the study of black hole thermodynamics and the AdS/CFT correspondence, suggests that the information content of a region of space can be encoded on its boundary, hinting at a deep connection between gravity and quantum information theory
• The measurement problem in quantum mechanics, exemplified by the Schrödinger's cat thought experiment, highlights the difficulty in reconciling the deterministic evolution of the wave function with the probabilistic nature of measurement outcomes, leading to various interpretations like the Copenhagen interpretation, the many-worlds interpretation, and the objective collapse theories
• The interplay between quantum mechanics and general relativity at the Planck scale ($\sim 10^{-35}\,\mathrm{m}$ or $\sim 10^{19}\,\mathrm{GeV}$) suggests the need for a theory of quantum gravity, which could resolve singularities like the Big Bang and the center of black holes, and provide insights into the origin and fate of the universe
• The arrow of time, or the apparent asymmetry between the forward and backward directions of time, remains a mystery in the context of relativistic quantum mechanics, as the fundamental laws are time-symmetric, and the origin of the thermodynamic arrow of time (entropy increase) is still a topic of active research