38-5 Electrons and Matter Waves#
Prompts
de Broglie’s symmetry argument: Light is a wave but transfers energy at points (photons). Why might a beam of electrons behave the same way? What is the de Broglie wavelength \(\lambda\) in terms of momentum \(p\)?
For a nonrelativistic electron with kinetic energy \(K\), write \(\lambda\) in terms of \(m\) and \(K\). For an electron with \(K = 120\) eV, estimate \(\lambda\). (Is it comparable to atomic size?)
Describe the electron double-slit experiment. Electrons arrive one at a time yet build an interference pattern. How is this interpreted? What is a matter wave?
The same diffraction pattern appears for x rays and electrons when both have wavelength \(\lambda\) and scatter from crystals. What does this imply?
For an electron and a proton with the same (a) kinetic energy, (b) momentum, or (c) speed—which has the shorter de Broglie wavelength?
Lecture Notes#
Overview#
de Broglie (1924) proposed that particles such as electrons have an associated matter wave with wavelength \(\lambda = h/p\), the same relation used for photons.
Electron double-slit: Electrons sent one at a time build an interference pattern—each electron behaves as a wave in transit and as a particle when detected.
Particle vs wave: When an electron interacts with matter (absorption, scattering), it behaves like a particle. When in transit, it is described as a probability wave.
The same two-slit and diffraction equations (Ch. 35–36) apply to matter waves: \(d\sin\theta = m\lambda\) for bright fringes, etc.
de Broglie wavelength#
Louis de Broglie (1924) argued by symmetry: Light is a wave yet transfers energy and momentum at points via photons (\(p = h/\lambda\)). Why can’t a beam of particles have the same dual character—a wave that determines where the particle may be detected?
He proposed that a particle with momentum of magnitude \(p\) has an associated wavelength
called the de Broglie wavelength. This relation applies to electrons, protons, neutrons, atoms, and molecules—any moving particle.
Nonrelativistic case: For kinetic energy \(K\) and mass \(m\), \(p = \sqrt{2mK}\), so
Relativistic case: Use \(p\) from \(E^2 = (pc)^2 + (mc^2)^2\) and the particle’s total energy.
Example
Electron with \(K = 120\) eV: \(p = \sqrt{2m_e K} = \sqrt{2(9.11\times 10^{-31}\,\text{kg})(120\,\text{eV})(1.60\times 10^{-19}\,\text{J/eV})} \approx 5.91\times 10^{-24}\) kg·m/s. Then \(\lambda = h/p \approx 6.63\times 10^{-34}/(5.91\times 10^{-24}) \approx 1.12\times 10^{-10}\) m \(\approx 112\) pm—comparable to atomic dimensions. Electron diffraction from crystals is feasible.
Electron double-slit and matter waves#
In the electron double-slit experiment, electrons are sent one at a time through a two-slit apparatus. Each electron produces a single flash on the screen. Initially the flashes seem random, but after many electrons, an interference pattern emerges—bright fringes where many electrons hit, dark fringes where few hit.
Interpretation: Each electron travels through the apparatus as a matter wave. The portion through one slit interferes with the portion through the other. The interference determines the probability that the electron will be detected at each point. We cannot say which slit a given electron “went through”; we only know it was emitted and later detected somewhere. The pattern is the same as for light waves.
Similar interference has been observed with protons, neutrons, atoms, and even molecules (e.g., C\(_{60}\) fullerenes). For larger objects (e.g., a cat), the de Broglie wavelength is immeasurably small—we recover classical behavior.
Particle and wave: complementary descriptions#
Situation |
Description |
|---|---|
In transit |
Matter wave (probability wave); interference, diffraction |
Interaction |
Particle-like; localized detection, energy and momentum transfer at a point |
A bubble chamber shows curved tracks—electrons and positrons leave particle-like paths. Yet between detection points, the electron is a wave exploring many paths; only the straight-line path and nearby paths constructively interfere, so the track forms along that direction. Both descriptions are needed.
Applying optical equations to matter waves#
The same interference and diffraction formulas apply. For matter waves with de Broglie wavelength \(\lambda\):
Double-slit (Module 35-2): Bright fringes at \(d\sin\theta = m\lambda\) (\(m = 0, 1, 2, \ldots\)); dark fringes at \(d\sin\theta = (m + \tfrac{1}{2})\lambda\).
Single-slit diffraction (Module 36-1): Minima at \(a\sin\theta = m\lambda\) (\(m = 1, 2, 3, \ldots\)).
Crystal diffraction: X rays and electrons with the same \(\lambda\) produce identical diffraction patterns when scattered by crystals—both are waves.
Electron vs proton
For the same kinetic energy \(K\): \(p = \sqrt{2mK}\), so the heavier proton has larger \(p\) and thus shorter \(\lambda\). For the same momentum \(p\): \(\lambda = h/p\) is the same. For the same speed \(v\): \(p = mv\), so the heavier proton has larger \(p\) and shorter \(\lambda\).
Summary#
de Broglie wavelength \(\lambda = h/p\); for nonrelativistic particles, \(\lambda = h/\sqrt{2mK}\).
Matter waves: Electrons (and other particles) exhibit interference and diffraction—wave behavior in transit.
Particle detection: When an electron is detected, it is localized—particle-like behavior.
Optical equations (two-slit, diffraction) apply to matter waves: use \(\lambda = h/p\).