Quantum Electrodynamics, usually called QED, is the quantum theory of light and electrically charged matter. It explains how photons, electrons, and other charged particles interact at the smallest scales. In classical electricity and magnetism, electric and magnetic fields are treated as continuous fields. In QED, those interactions are described through quantum fields, where light comes in photons and charged particles exchange energy and momentum in precise, measurable ways.
QED is one of the great bridge theories in modern physics. It connects the familiar ideas of charge, force, fields, and electromagnetic waves with the less familiar world of quantum mechanics. It helps explain how light is emitted and absorbed, how electrons scatter, why atoms interact with radiation, and why extremely small corrections can be measured with astonishing accuracy.
Students meeting QED for the first time should not think of it only as a difficult mathematical theory. Its central question is beautifully simple: what really happens when light and matter interact? The answer leads to photons, probability amplitudes, virtual particle exchange, Feynman diagrams, precision measurements, and the deeper idea that forces can be understood through quantum fields rather than ordinary mechanical pushes and pulls.
This page introduces QED as a conceptual gateway rather than a full technical course. It builds naturally from electrodynamics, electromagnetic waves, quantum optics, and quantum mechanics. It also prepares students to understand why QED became a foundation for quantum field theory and a key part of the Standard Model of particle physics.
This vivid, space-themed illustration portrays a QED interaction as an energetic exchange between a bright orange “photon” region and a blue “electron” region. A central interaction point links the two with a wavy, light-like path, while sweeping wave curves suggest fields and quantum amplitudes. Across the background are gridlines, waveform plots, handwritten-style equations, and a simplified Feynman-diagram motif, emphasizing the mathematical language used to predict particle behavior. The overall composition communicates QED’s core idea: electromagnetic interactions arise from photon exchange, captured through diagrams, wave-like patterns, and precise quantitative models.
From Classical Electromagnetism to Quantum Fields
Quantum Electrodynamics sits within the wider Electricity and Magnetism cluster, but it takes the subject into the quantum world. Use this pathway to move from classical charges, currents, fields, and waves toward the quantum theory of light–matter interaction.
Electricity and Magnetism – Overview
Introduces the larger field of electric charge, electric fields, magnetic fields, circuits, waves, induction, and electromagnetic interactions.Electrical Circuits
Builds the practical foundation for understanding current, voltage, resistance, power, and energy transfer in electrical systems.Electrostatics
Explores electric charge, electric fields, potential, and the forces between charged particles at rest.Magnetic Fields
Examines how magnetic fields act on moving charges, current-carrying conductors, and magnetic materials.Magnetostatics
Focuses on magnetic fields produced by steady currents and simple current configurations.Electromagnetic Induction
Shows how changing magnetic fields produce electric effects, leading to generators, transformers, and many power technologies.Electromagnetic Waves
Connects changing electric and magnetic fields to travelling waves such as radio waves, microwaves, light, and X-rays.Electrodynamics
Studies changing electric and magnetic fields, moving charges, radiation, and the deeper structure behind electromagnetic phenomena.Plasma Physics
Applies electromagnetic ideas to ionised matter, where free electrons and ions respond collectively to fields and waves.Magnetohydrodynamics (MHD)
Combines plasma motion with magnetic fields, especially in fusion systems, solar plasma, and astrophysical environments.Superconductivity
Introduces quantum behaviour in materials where electrical resistance vanishes and magnetic fields are deeply affected.Quantum Electrodynamics (QED)
Extends electromagnetism into quantum field theory, explaining how photons and charged particles interact at the smallest scales.
Historical Development of Quantum Electrodynamics
Quantum Electrodynamics did not appear suddenly. It grew from several major steps in physics: classical electromagnetism, early quantum theory, special relativity, the Dirac equation, and the development of quantum field theory. Each step changed how physicists understood light, charge, fields, and matter.
Maxwell’s Equations
In the 1860s, James Clerk Maxwell unified electricity, magnetism, and light into one classical theory of electromagnetic fields. QED later preserved this achievement, but placed it inside a quantum framework.Planck and Einstein
Between 1900 and 1905, Max Planck and Albert Einstein introduced key quantum ideas. Einstein’s explanation of the photoelectric effect suggested that light can behave as packets of energy, later called photons.The Dirac Equation
In 1928, Paul Dirac combined quantum mechanics with special relativity to describe the electron. His theory also predicted the positron, the electron’s antimatter partner.Feynman, Schwinger, and Tomonaga
In the 1940s, Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga developed modern QED, giving physicists powerful tools to calculate light–matter interactions.Renormalisation
Renormalisation made QED predictive by handling infinities that appeared in intermediate calculations. It allowed the theory to produce finite results that could be compared with experiment.Core Concepts in Quantum Electrodynamics
QED describes electromagnetic interaction using quantum fields. Instead of treating light only as a classical wave or force only as an invisible pull, QED describes particles and fields as parts of one quantum framework.
Fields and Particles
In QED, electrons, positrons, and photons arise from underlying quantum fields. Particles may be understood as quantised excitations of these fields.Photon as Force Carrier
The photon is the quantum carrier of the electromagnetic interaction. Charged particles interact by emitting, absorbing, or exchanging photons.Virtual Photons
Virtual photons appear inside QED calculations as internal exchange processes. They are not directly observed like real photons, but they help calculate how charged particles influence one another.Feynman Diagrams
Feynman diagrams provide a visual shorthand for particle interactions. Straight lines usually represent electrons or positrons, wavy lines represent photons, and vertices show interaction points.Probability Amplitudes
QED predicts probabilities, not classical certainties. It calculates probability amplitudes for different possible interaction paths and combines them according to quantum rules.Precision Corrections
QED includes tiny quantum corrections to classical predictions. These corrections explain measurements such as the Lamb shift and the electron’s anomalous magnetic moment.
Mathematical Foundation of QED
The full mathematics of QED is advanced, but students can still understand the role of its central equation. The QED Lagrangian summarises how the electron field, the electromagnetic field, and their interaction are represented in quantum field theory.
The QED Lagrangian
A common compact form of the QED Lagrangian density is:
\[
\mathcal{L}
=
\bar{\psi}(i\gamma^\mu D_\mu – m)\psi
–
\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
\]
Here, \( \psi \) represents the electron field, \( \bar{\psi} \) is its Dirac adjoint, \( \gamma^\mu \) are Dirac gamma matrices, \( D_\mu \) is the covariant derivative, and \( F_{\mu\nu} \) is the electromagnetic field tensor.
The covariant derivative may be written as:
\[
D_\mu=\partial_\mu+i e A_\mu
\]
The electromagnetic field tensor is:
\[
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu
\]
Gauge Invariance
QED is based on a symmetry called U(1) gauge invariance. In simple terms, the theory allows the phase of the electron field to be changed locally without changing the observable physics:
\[
\psi(x)\rightarrow e^{i\alpha(x)}\psi(x)
\]
This symmetry is deeply connected to charge conservation and to the existence of the photon as the quantum field associated with electromagnetic interaction.
Renormalisation
Early QED calculations produced infinities in intermediate steps. Renormalisation is the systematic method that absorbs these infinities into measurable quantities such as physical mass and charge. The final predictions become finite, testable, and extraordinarily accurate.
Key Interactions in QED
QED describes several fundamental light–matter processes. These interactions may look different experimentally, but they are built from the same basic idea: charged particles interact with photons.
Electron–Photon Interaction
An electron can emit or absorb a photon, changing its energy and momentum. This is the basic interaction vertex in QED.Electron–Electron Scattering
Two electrons can repel each other through the exchange of a virtual photon. This process is one quantum-field version of electromagnetic repulsion.Electron–Positron Annihilation
An electron and a positron can annihilate and produce photons: \[ e^-+e^+\rightarrow \gamma+\gamma \]Compton Scattering
A photon can scatter from an electron, transferring energy and momentum: \[ \gamma+e^-\rightarrow \gamma+e^- \]Vacuum Polarisation
The quantum vacuum can affect electromagnetic interactions. In QED, this appears as virtual particle–antiparticle effects that slightly modify how fields behave.Pair Production
Under suitable conditions, high-energy photons can produce an electron–positron pair, provided energy and momentum are conserved.Feynman Diagrams in QED
Feynman diagrams are not literal pictures of particles travelling through space like tiny balls. They are compact visual tools that help physicists organise calculations. Each line and vertex corresponds to a mathematical contribution to a probability amplitude.
External Lines
External lines represent incoming or outgoing particles that can be prepared or detected in an experiment.Internal Lines
Internal lines represent virtual particles inside the calculation. They are part of the interaction process but are not directly observed as free particles.Vertices
A vertex is an interaction point where charged particles and photons meet. In QED, the basic vertex connects an electron line, another electron line, and a photon line.Higher-Order Diagrams
More complicated diagrams represent quantum corrections. These corrections are often small but can be measured with extreme precision.Experimental Verifications of QED
QED is famous because it does not merely sound elegant. It makes numerical predictions that agree with extremely precise experiments. This agreement is one reason QED is often regarded as one of the most successful theories in physics.
Anomalous Magnetic Moment of the Electron
QED predicts tiny corrections to the electron’s magnetic moment. Measurements of this quantity provide one of the most precise tests of quantum field theory.Lamb Shift
QED explains a small energy difference between certain hydrogen energy levels. This shift cannot be fully explained by simpler non-relativistic quantum mechanics.Scattering Experiments
Compton scattering, electron–electron scattering, and electron–positron annihilation provide experimental tests of QED predictions.Precision Spectroscopy
High-resolution measurements of atomic and optical transitions allow scientists to test the small corrections predicted by QED.Applications of QED
QED is a fundamental theory, but its influence reaches many areas of science and technology. It helps explain how light and matter interact in atoms, particles, optical devices, and high-energy environments.
Particle Physics
QED forms the electromagnetic part of the Standard Model and provides a model for how quantum field theories describe fundamental interactions.Quantum Optics
QED helps explain how light interacts with atoms, how photons are emitted and absorbed, and how quantum states of light can be studied.Precision Measurement
QED corrections are essential in high-precision measurements involving atoms, electrons, photons, and fundamental constants.Semiconductor and Photonic Devices
Many modern devices rely on quantum descriptions of electrons and photons, including lasers, photodetectors, quantum dots, and optoelectronic systems.Astrophysics
QED ideas become important in extreme environments involving intense fields, high-energy particles, neutron stars, and compact astrophysical systems.Gateway to Quantum Field Theory
QED prepares students for more advanced theories such as electroweak theory, quantum chromodynamics, and quantum field theory in curved spacetime.
This artist’s-impression infographic presents six broad applications of Quantum Electrodynamics. It shows how QED helps explain electromagnetic interactions in particle physics, light–matter behaviour in quantum optics, tiny corrections needed for precision measurement, quantum processes in semiconductor and photonic devices, extreme environments in astrophysics, and the wider development of quantum field theory. The image is intended as a visual overview rather than a technical diagram, helping students see QED as both a fundamental theory and a gateway to advanced physics and modern technologies.
Worked Examples in QED
Example 1: Energy of a Photon
Problem: Calculate the energy of a photon with wavelength \( 500\ \text{nm} \).
Solution:
Use:
\[
E=\frac{hc}{\lambda}
\]
Substitute \( h=6.626\times10^{-34}\ \text{J·s} \), \( c=3.00\times10^8\ \text{m/s} \), and \( \lambda=500\times10^{-9}\ \text{m} \):
\[
E=\frac{(6.626\times10^{-34})(3.00\times10^8)}{500\times10^{-9}}
\]
\[
E\approx3.97\times10^{-19}\ \text{J}
\]
Answer: The photon energy is approximately \( 3.97\times10^{-19}\ \text{J} \).
Example 2: Compton Wavelength of an Electron
Problem: Find the Compton wavelength of an electron.
Solution:
Use:
\[
\lambda_C=\frac{h}{m_e c}
\]
Substitute \( h=6.626\times10^{-34}\ \text{J·s} \), \( m_e=9.11\times10^{-31}\ \text{kg} \), and \( c=3.00\times10^8\ \text{m/s} \):
\[
\lambda_C=\frac{6.626\times10^{-34}}{(9.11\times10^{-31})(3.00\times10^8)}
\]
\[
\lambda_C\approx2.43\times10^{-12}\ \text{m}
\]
Answer: The Compton wavelength of the electron is approximately \( 2.43\times10^{-12}\ \text{m} \).
Example 3: Pair Production Threshold Energy
Problem: Estimate the minimum photon energy needed to produce an electron–positron pair.
Solution:
At minimum, the photon must provide the rest energy of an electron and a positron:
\[
E_{\text{min}}=2m_e c^2
\]
Substitute \( m_e=9.11\times10^{-31}\ \text{kg} \) and \( c=3.00\times10^8\ \text{m/s} \):
\[
E_{\text{min}}=2(9.11\times10^{-31})(3.00\times10^8)^2
\]
\[
E_{\text{min}}\approx1.64\times10^{-13}\ \text{J}
\]
Convert to electron volts:
\[
E_{\text{min}}=\frac{1.64\times10^{-13}}{1.60\times10^{-19}}
\approx1.02\times10^6\ \text{eV}
\]
\[
E_{\text{min}}\approx1.02\ \text{MeV}
\]
Answer: The minimum photon energy is approximately \( 1.02\ \text{MeV} \), before considering the momentum requirements of a real physical situation.
Example 4: Electron–Positron Annihilation Photon Energy
Problem: What is the energy of each photon produced when an electron and a positron annihilate at rest into two photons?
Solution:
Each photon receives energy equal to the rest energy of one electron:
\[
E=m_e c^2
\]
\[
E=(9.11\times10^{-31})(3.00\times10^8)^2
\]
\[
E\approx8.19\times10^{-14}\ \text{J}
\]
Convert to electron volts:
\[
E=\frac{8.19\times10^{-14}}{1.60\times10^{-19}}
\approx5.11\times10^5\ \text{eV}
\]
\[
E\approx0.511\ \text{MeV}
\]
Answer: Each photon carries approximately \( 0.511\ \text{MeV} \) of energy.
Example 5: Fine-Structure Constant
Problem: Estimate the value of the fine-structure constant \( \alpha \).
Solution:
Use:
\[
\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}
\]
Substituting standard constants gives:
\[
\alpha\approx\frac{1}{137}
\]
Answer: The fine-structure constant is approximately \( 1/137 \).
Cosmic Alchemy: Accelerators, Matter Creation, and the Death of Certainty
Quantum Electrodynamics completely shatters the classical illusion that the universe is made of permanent, unchanging blocks of matter. In the classical view, an electron is a permanent object that can be pushed or pulled, but can never cease to exist. QED introduces a wild, high-energy reality: matter is completely fluid, and subatomic particles are continuously created and destroyed out of thin air through the shifting vibrations of underlying quantum field matrices.
This capability is not merely theoretical speculation; it is the physical engine driving heavy industrial particle colliders and high-energy cosmic events. When intense energetic fields collide, the local field ripples become so violent that real physical matter materializes directly out of pure kinetic energy. QED provides the exact mathematical blueprint tracking these sensational, transformative subatomic phenomena:
The Violence of Electron–Positron Annihilation
The Phenomenon: When an electron collides head-on with its antimatter twin, the positron, both particles vanish instantly from the universe.
The Quantum Reality: Their entire rest masses and kinetic energies are completely wiped out, collapsing back into the underlying fields to produce an explosive burst of pure electromagnetic radiation in the form of high-energy gamma-ray photons.
Materialization: Spontaneous Pair Production
The Phenomenon: High-energy light waves can spontaneously manifest physical, massive material particles straight out of a vacuum void.
The Quantum Reality: When a gamma-ray photon holding sufficient threshold energy passes near an atomic nucleus, the photon self-destructs and vanishes. In its place, a real, solid electron and a positron are instantly created, transforming immaterial light into physical matter before your eyes.
The Quantum Vacuum: A Boiling Sea of Virtual Ghosts
The Phenomenon: Even in absolute deep space, a vacuum is never truly empty; it is a chaotic, boiling cauldron of constant creation and destruction.
The Quantum Reality: Due to quantum uncertainty, virtual electron-positron pairs are constantly tearing themselves into existence out of nothingness, interacting for a fraction of a picosecond, and then violently annihilating one another before they can be directly seen, leaving a measurable imprint on atomic energy levels.
Why Study Quantum Electrodynamics?
The Quantum Theory of Electromagnetic Interaction
QED explains how light and charged matter interact at the quantum level. Students learn how photons mediate electromagnetic interactions and why classical ideas of force and field must be refined at atomic and subatomic scales.
A Model of Precision in Physics
QED is famous for predictions that agree with experiment to extraordinary accuracy. Studying it helps students understand how a theory can be both deeply mathematical and experimentally testable.
A Gateway to Feynman Diagrams and Quantum Field Theory
Feynman diagrams give students a visual entry point into quantum field theory. They show how complex interactions can be organised into simpler pieces that correspond to mathematical terms.
A Bridge to Modern Technologies
QED supports the deeper understanding of light–matter interaction behind lasers, spectroscopy, photonics, quantum optics, semiconductor devices, and high-precision measurements.
A Foundation for Advanced Physics
QED prepares students for electroweak theory, particle physics, quantum chromodynamics, and wider studies of quantum field theory. It is one of the clearest examples of how symmetry, fields, particles, and measurement come together in modern physics.
Summary
Quantum Electrodynamics is the quantum field theory of light and charged matter. It explains how photons and charged particles interact, how electromagnetic force appears at the quantum level, and why tiny corrections to classical predictions can be measured with remarkable accuracy.
QED links classical electromagnetism, quantum mechanics, special relativity, particle physics, and quantum optics. It is not only a difficult mathematical theory; it is also a powerful way of seeing how nature behaves when light, charge, probability, and fields meet at the smallest scales.
Quantum Electrodynamics (QED) — FAQ
What is Quantum Electrodynamics?
Quantum Electrodynamics, or QED, is the quantum field theory that describes how light and charged particles interact. It treats light as photons and describes charged particles such as electrons as excitations of quantum fields.
How does QED relate to classical electromagnetism?
Classical electromagnetism treats electric and magnetic fields as continuous fields. QED extends this picture by quantising the electromagnetic field and describing interactions through probability amplitudes and photons. In ordinary large-scale situations, QED agrees with classical electromagnetism.
What is a photon in QED?
A photon is the quantum of the electromagnetic field. Real photons can be emitted, absorbed, and detected as light or electromagnetic radiation. Virtual photons appear inside calculations describing interactions between charged particles.
What are Feynman diagrams?
Feynman diagrams are visual tools that represent particle interactions in quantum field theory. In QED, they help organise calculations involving electrons, positrons, photons, and interaction vertices.
Why is QED considered so successful?
QED is considered successful because it predicts measurable quantities with extraordinary precision. Its predictions for effects such as the electron’s anomalous magnetic moment and the Lamb shift agree closely with experimental results.
What is renormalisation?
Renormalisation is a method for handling infinities that appear in intermediate steps of QED calculations. By redefining physical quantities such as mass and charge in terms of measured values, physicists obtain finite predictions that can be tested experimentally.
How does QED fit into the Standard Model?
QED describes the electromagnetic interaction between charged particles. In the Standard Model, it appears as the low-energy electromagnetic part of the electroweak theory.
What mathematics is needed to study QED?
University-level QED requires quantum mechanics, special relativity, linear algebra, complex numbers, calculus, differential equations, and later the methods of quantum field theory such as Lagrangians, operators, perturbation theory, and Feynman diagrams.
Is QED only theoretical?
QED is a fundamental theory, but it also supports practical understanding of lasers, spectroscopy, atomic physics, photonics, semiconductor devices, precision measurement, and quantum optics.
How is QED different from ordinary quantum mechanics?
Ordinary non-relativistic quantum mechanics usually describes particles moving in fixed external fields. QED is a relativistic quantum field theory in which both matter and electromagnetic fields are quantised, allowing particle creation, annihilation, and photon exchange.
QED Review Questions and Answers
Review Questions: Quantum Electrodynamics
1. What is Quantum Electrodynamics?
QED is the quantum field theory that describes how photons and charged particles interact. It combines quantum mechanics, special relativity, and electromagnetism.
2. Who were major contributors to modern QED?
Major contributors include Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga, who developed powerful methods for calculating QED processes in the 1940s.
3. What role do Feynman diagrams play in QED?
Feynman diagrams help represent particle interactions visually and organise the mathematical terms used to calculate probability amplitudes.
4. How does QED describe light–matter interaction?
QED describes light–matter interaction through photons and charged-particle fields. Charged particles can emit, absorb, or exchange photons.
5. What is the fine-structure constant?
The fine-structure constant \( \alpha \) is a dimensionless number, approximately \( 1/137 \), that measures the strength of electromagnetic interaction in QED.
6. What is renormalisation in QED?
Renormalisation is the process used to handle infinities in intermediate calculations and produce finite predictions for measurable quantities.
7. What experimental evidence supports QED?
QED is supported by measurements such as the anomalous magnetic moment of the electron, the Lamb shift, Compton scattering, and electron–positron annihilation experiments.
8. How has QED influenced modern physics?
QED shaped the development of quantum field theory and became a model for later theories in the Standard Model, including electroweak theory and quantum chromodynamics.
Thought-Provoking Questions and Answers
Thinking More Deeply About QED
1. How does QED change the classical picture of electromagnetic force?
Classically, electromagnetic force is described by continuous fields. QED replaces this with a quantum-field picture in which charged particles interact through photons and probability amplitudes.
2. Why is the quantum vacuum not simply empty?
In quantum field theory, the vacuum is the lowest-energy state of fields, not absolute nothingness. QED predicts effects such as vacuum polarisation and other measurable consequences of field fluctuations.
3. Why are tiny QED corrections scientifically important?
Tiny corrections matter because they can be measured precisely. Agreement between QED predictions and experiments gives strong evidence that the theory captures deep features of light–matter interaction.
4. How could future experiments test the limits of QED?
More precise measurements of particle magnetic moments, atomic transitions, scattering processes, or strong-field effects could reveal tiny differences between QED predictions and nature, possibly pointing to physics beyond the Standard Model.
5. Why is gauge invariance important?
Gauge invariance shows that the mathematical description can change while the observable physics remains the same. In QED, this symmetry is connected to charge conservation and the structure of electromagnetic interaction.
6. Why does QED serve as a gateway to other quantum field theories?
QED introduces key ideas such as fields, gauge symmetry, Feynman diagrams, perturbation theory, renormalisation, and particle exchange. These ideas reappear in more advanced theories such as electroweak theory and quantum chromodynamics.
Numerical Practice: QED
Numerical Problems and Solutions
1. Calculate the Compton wavelength of an electron.
Use:
\[
\lambda_C=\frac{h}{m_e c}
\]
\[
\lambda_C=\frac{6.626\times10^{-34}}{(9.11\times10^{-31})(3.00\times10^8)}
\]
\[
\lambda_C\approx2.43\times10^{-12}\ \text{m}
\]
Answer: \( \lambda_C\approx2.43\times10^{-12}\ \text{m} \).
2. Determine the fine-structure constant.
Use:
\[
\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}
\]
Substitution of standard constants gives:
\[
\alpha\approx\frac{1}{137}
\]
Answer: \( \alpha\approx1/137 \).
3. Compute the energy of a photon with wavelength \( 400\ \text{nm} \).
Use:
\[
E=\frac{hc}{\lambda}
\]
\[
E=\frac{(6.626\times10^{-34})(3.00\times10^8)}{400\times10^{-9}}
\]
\[
E\approx4.97\times10^{-19}\ \text{J}
\]
Answer: \( E\approx4.97\times10^{-19}\ \text{J} \).
4. Calculate the rest energy of an electron in MeV.
Use:
\[
E_0=m_e c^2
\]
\[
E_0=(9.11\times10^{-31})(3.00\times10^8)^2
\]
\[
E_0\approx8.19\times10^{-14}\ \text{J}
\]
Convert to electron volts:
\[
E_0=\frac{8.19\times10^{-14}}{1.602\times10^{-19}}
\approx5.11\times10^5\ \text{eV}
\]
\[
E_0\approx0.511\ \text{MeV}
\]
Answer: The electron rest energy is approximately \( 0.511\ \text{MeV} \).
5. A photon has energy \( 2.0\ \text{MeV} \). Find its wavelength.
Convert energy to joules:
\[
E=(2.0\times10^6)(1.602\times10^{-19})
=3.204\times10^{-13}\ \text{J}
\]
Use:
\[
\lambda=\frac{hc}{E}
\]
\[
\lambda=\frac{(6.626\times10^{-34})(3.00\times10^8)}{3.204\times10^{-13}}
\]
\[
\lambda\approx6.20\times10^{-13}\ \text{m}
\]
Answer: \( \lambda\approx6.20\times10^{-13}\ \text{m} \).
6. Calculate the Bohr magneton.
Use:
\[
\mu_B=\frac{e\hbar}{2m_e}
\]
\[
\mu_B=\frac{(1.602\times10^{-19})(1.055\times10^{-34})}{2(9.11\times10^{-31})}
\]
\[
\mu_B\approx9.27\times10^{-24}\ \text{J/T}
\]
Answer: \( \mu_B\approx9.27\times10^{-24}\ \text{J/T} \).
7. A QED correction is \(0.1\%\) of a classical cross-section \(1.0\times10^{-28}\ \text{m}^2\). Find the corrected value.
A \(0.1\%\) correction means:
\[
0.1\%=0.001
\]
\[
\sigma=\sigma_0(1+0.001)
\]
\[
\sigma=(1.0\times10^{-28})(1.001)
\]
\[
\sigma=1.001\times10^{-28}\ \text{m}^2
\]
Answer: The corrected cross-section is \(1.001\times10^{-28}\ \text{m}^2\).
8. The Lamb shift is \(1057\ \text{MHz}\). Convert this frequency to energy in joules.
Use:
\[
E=h\nu
\]
\[
\nu=1057\ \text{MHz}=1.057\times10^9\ \text{Hz}
\]
\[
E=(6.626\times10^{-34})(1.057\times10^9)
\]
\[
E\approx7.01\times10^{-25}\ \text{J}
\]
Answer: \( E\approx7.01\times10^{-25}\ \text{J} \).