doPhysics · Modern Physics
Matter Waves and Special Relativity
Every moving particle carries a wave of wavelength \(h/p\), and electrons diffracted by a nickel crystal prove it. Then, beyond the Class 12 syllabus, the two postulates of special relativity, slow clocks and short rods, and the energy of mass, \(E = mc^2\).
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Why this chapter
Why this chapter exists
The last chapter found that light, a wave, also comes in particles. This chapter turns the idea round: electrons, particles, also travel as waves, with a wavelength set by their momentum. Lessons 1 and 2, in the Class 12 syllabus, find that wavelength and watch electrons diffract from a crystal. Lessons 3 to 5 go beyond the Class 12 syllabus, into Einstein's special relativity. They are here because the photon's momentum, the energy of mass \(E = mc^2\) and the energy of the nucleus all rest on it. A short last lesson joins the two halves.
You will learn
By the end of this chapter you will
- Find the de Broglie wavelength of a particle from its momentum, its kinetic energy or its accelerating voltage, and compare particles.
- Explain why a cricket ball never shows its wave, and why electrons and thermal neutrons diffract from crystals.
- Describe the Davisson and Germer experiment, and find the wavelength from the peak at \(54\,\text{V}\) with \(d\sin\phi = \lambda\) or Bragg's rule.
- Beyond the Class 12 syllabus: state the two postulates of special relativity, and explain why simultaneity depends on the frame.
- Beyond the Class 12 syllabus: use \(\Delta t = \gamma\Delta t_0\) and \(L = L_0/\gamma\), and explain how cosmic-ray muons reach the ground.
- Beyond the Class 12 syllabus: use \(p = \gamma m_0v\), \(E = \gamma m_0c^2\) and \(E^2 = (pc)^2 + (m_0c^2)^2\), and find rest energies.
- Say when \(h/\sqrt{2mK}\) fails for fast electrons, and compare a photon and an electron of the same wavelength.
Before you begin
Before you begin
From the chapter on photons you should know what an electron volt is. You should know that a photon of wavelength \(\lambda\) has energy \(hc/\lambda\) and momentum \(h/\lambda\). From mechanics you should know momentum, kinetic energy \(\tfrac12 mv^2\), relative velocity on a line and the conservation of momentum. From the chapters on waves you should know wavelength and how two waves add in step or out of step.
Lessons
1
De Broglie's Idea
Every moving particle carries a wave of wavelength \(\lambda = h/p\). The wavelength from the kinetic energy and from an accelerating voltage, and why an electron shows its wave while a cricket ball never does.
De broglie hypothesisWavelength from kinetic energyComparing particles at one voltage
2
Electrons Diffract: Davisson and Germer
Electrons scattered from a nickel crystal come off strongly at one angle, as a wave would. The peak at 54 V and 50° gives a wavelength that matches h/p, and wave and particle become two descriptions of one thing.
The davisson germer experimentThe peak at 54 voltsWave particle duality
3
The Two Postulates
Beyond the Class 12 syllabus. Inertial frames and the principle of relativity, the speed of light that every observer measures as c, and two flashes that are simultaneous on the platform but not in the train.
Inertial frames and the principle of relativityThe speed of light is the same for allSimultaneity is relative
4
Moving Clocks Run Slow, Moving Rods Are Short
Beyond the Class 12 syllabus. A light clock shows that a moving clock ticks slower by the Lorentz factor \(\gamma\). A moving rod is shorter along its motion by the same factor, and cosmic-ray muons reach the ground because of both.
The light clock and time dilationThe lorentz factorLength contraction
5
Mass, Momentum and Energy
Beyond the Class 12 syllabus. Momentum becomes \(\gamma m_0 v\), energy becomes \(\gamma m_0 c^2\), and a body at rest holds the energy \(m_0 c^2\). The energy-momentum relation joins them and gives the photon as the particle with no rest mass.
Relativistic momentumMass energy equivalenceEnergy momentum relation
6
Waves, Particles and Speeds Together
A short practice lesson joining the two halves of the chapter. When the simple formula \(h/\sqrt{2mK}\) fails for fast electrons, and why a photon and an electron with the same wavelength carry very different energies.
De broglie wavelength of a fast electronPhoton and electron of equal wavelength
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- angstrom
- A unit of length used for atoms and X-rays, \(1\,\text{Å} = 10^{-10}\,\text{m} = 0.1\,\text{nm}\).
- atomic mass unit
- One twelfth of the mass of a carbon-12 atom, written u, about \(1.66 \times 10^{-27}\) kg. Its rest energy is \(931.5\) MeV.
- de Broglie wavelength
- The wavelength of the matter wave of a particle of momentum \(p\), \(\lambda = h/p\). For an electron accelerated through \(V\) volts it is \(1.227/\sqrt{V}\) nm.
- diffraction
- The spreading of a wave round an obstacle or out of an opening. It is strong only when the opening is about as wide as the wavelength.
- electron gun
- A device that gives a narrow beam of electrons of one energy. A hot filament releases electrons, and a voltage \(V\) accelerates them to kinetic energy \(eV\).
- energy-momentum relation
- The relation \(E^2 = (pc)^2 + (m_0c^2)^2\) between the total energy, momentum and rest energy of any free particle. For a photon it gives \(E = pc\).
- ether
- A medium once supposed to fill space and carry light waves, relative to which light would move at \(c\). The Michelson and Morley experiment found no sign of it.
- event
- Something that happens at one place at one instant, such as a flash or the decay of a particle. It is given by a position and a time.
- glancing angle
- The angle between a beam and a set of planes of atoms, used in Bragg's rule \(2d\sin\theta = \lambda\). For the peak of the Davisson and Germer experiment, \(\theta = 90° - \phi/2\).
- inertial frame
- A frame of reference in which Newton's first law holds: a body with no force on it stays at rest or keeps a constant velocity.
- length contraction
- The shortening of a moving body along its direction of motion, \(L = L_0/\gamma\). Lengths across the motion do not change.
- light clock
- A thought-experiment clock: a flash of light bouncing between two mirrors a distance \(d\) apart, one crossing being one tick of time \(d/c\).
- Lorentz factor
- The number \(\gamma = 1/\sqrt{1 - v^2/c^2}\), which is 1 at rest and grows without limit as \(v\) approaches \(c\). It sets the size of every effect of special relativity.
- mass-energy equivalence
- Einstein's result that mass is a form of energy, \(E = mc^2\). A change of mass \(\Delta m\) goes with a change of energy \(\Delta m\,c^2\).
- matter wave
- The wave associated with a moving particle, proposed by de Broglie in 1924. Its wavelength is \(h/p\).
- muon
- An unstable particle like a heavy electron, made by cosmic rays in the upper atmosphere. At rest it lives \(2.2\) μs on average.
- proper length
- The length of a body measured in the frame in which it is at rest. It is the longest length any frame measures for it.
- proper time
- The time between two events that happen at the same place in some frame, read on a clock at that place. It is the shortest time any frame measures between them.
- rest energy
- The energy a body has because of its mass, even at rest, \(E_0 = m_0c^2\). For an electron it is \(0.51\) MeV.
- rest mass
- The mass of a body measured by an observer at rest with it, \(m_0\). It is the same in every frame.
- scattering angle
- In the Davisson and Germer experiment, the angle \(\phi\) between the direction the beam came from and the direction in which scattered electrons are counted.
- simultaneous
- Happening at the same time in a given frame. Two events at different places that are simultaneous in one frame are not simultaneous in a frame moving along the line joining them.
- thermal neutron
- A neutron slowed by collisions until it shares the temperature of its surroundings. At room temperature it has about \(0.04\,\text{eV}\) and a wavelength near \(0.15\) nm.
- time dilation
- The slowing of a moving clock. The time between two events measured in a frame where they happen at different places is \(\gamma\) times the proper time.
- total energy
- The energy of a free body including its rest energy, \(E = \gamma m_0c^2\). The kinetic energy is \(E - m_0c^2\).
- wave-particle duality
- The fact that light and matter each behave as waves in some experiments and as particles in others. Which face shows depends on the experiment.