Calculate matter wave properties and explore wave-particle duality
Use scientific notation (e.g., 9.109e-31)
Speed of light c = 2.998 × 10⁸ m/s
Wave-Particle Duality: All matter exhibits both wave and particle properties. The de Broglie wavelength is the wavelength associated with a moving particle.
Formula: λ = h/p = h/(mv), where h is Planck's constant (6.626 × 10⁻³⁴ J·s)
Significance: For macroscopic objects, λ is negligibly small. For subatomic particles like electrons, λ is significant and leads to observable quantum effects like diffraction and interference.
Applications: Electron microscopy, quantum mechanics, particle accelerators, and understanding atomic structure.
The de Broglie wavelength is the wavelength associated with a moving particle. In 1924, Louis de Broglie proposed that all matter exhibits wave-like properties, just as light exhibits both wave and particle properties.
This revolutionary idea extended Einstein's photon theory to all matter, suggesting that particles like electrons, protons, and even macroscopic objects have an associated wavelength determined by their momentum.
Key Concept - Wave-Particle Duality:
All matter has both particle and wave characteristics. For large objects, the wavelength is so small it's undetectable. For subatomic particles like electrons, the wavelength is significant enough to observe quantum effects like diffraction and interference.
De Broglie derived his famous equation by combining Einstein's energy-mass relation with Planck's quantum theory:
λ = h / p
General form
λ = h / (m × v)
Expanded form (non-relativistic)
Where:
For momentum p = mv:
Important Note:
The de Broglie equation is valid for all particles, but the wavelength is only observable when it's comparable to the size of the object or aperture it encounters. For macroscopic objects, λ is so small (typically < 10⁻³⁰ m) that wave properties are undetectable.
Calculate the de Broglie wavelength of an electron moving at 1.0 × 10⁶ m/s.
(Electron mass = 9.109 × 10⁻³¹ kg)
Step 1: Identify Given Information
Step 2: Calculate Momentum
p = m × v
= (9.109 × 10⁻³¹ kg) × (1.0 × 10⁶ m/s)
= 9.109 × 10⁻²⁵ kg·m/s
Step 3: Apply de Broglie Equation
λ = h / p
= (6.626 × 10⁻³⁴ J·s) / (9.109 × 10⁻²⁵ kg·m/s)
= 7.27 × 10⁻¹⁰ m
Step 4: Convert to More Convenient Units
λ = 7.27 × 10⁻¹⁰ m = 0.727 nm = 7.27 Å
(1 nm = 10⁻⁹ m, 1 Å = 10⁻¹⁰ m)
Answer:
The de Broglie wavelength is 7.27 × 10⁻¹⁰ m (0.727 nm). This is comparable to the size of atoms (~1 Å) and X-ray wavelengths, which is why electrons can be diffracted by crystal lattices in electron diffraction experiments. This wavelength is the reason electron microscopes can achieve much higher resolution than optical microscopes.
The de Broglie wavelength varies dramatically depending on the mass and velocity of the object:
| Object | Mass (kg) | Velocity (m/s) | Wavelength (m) | Observable? |
|---|---|---|---|---|
| Electron (slow) | 9.11 × 10⁻³¹ | 10⁶ | 7.3 × 10⁻¹⁰ | Yes ✓ |
| Electron (fast) | 9.11 × 10⁻³¹ | 10⁷ | 7.3 × 10⁻¹¹ | Yes ✓ |
| Proton (thermal) | 1.67 × 10⁻²⁷ | 10⁴ | 4.0 × 10⁻¹¹ | Yes ✓ |
| Neutron (slow) | 1.67 × 10⁻²⁷ | 2.2 × 10³ | 1.8 × 10⁻¹⁰ | Yes ✓ |
| Dust particle | 10⁻⁹ | 0.01 | 6.6 × 10⁻²³ | No ✗ |
| Baseball | 0.145 | 40 | 1.1 × 10⁻³⁴ | No ✗ |
| Car | 1000 | 25 | 2.7 × 10⁻³⁸ | No ✗ |
1. Subatomic Particles: Wavelengths are in the range of 10⁻¹⁰ to 10⁻¹¹ m (angstroms), comparable to atomic dimensions. Wave properties are easily observable through diffraction experiments.
2. Macroscopic Objects: Wavelengths are incredibly small (< 10⁻³⁰ m), far smaller than any possible aperture or obstacle. Wave properties are completely negligible.
3. General Trend: λ decreases as mass or velocity increases. This is why quantum effects are only noticeable for very light, slow-moving particles.
Electron microscopes use the wave nature of electrons to achieve resolutions much higher than optical microscopes. With wavelengths ~0.1 Å, they can image individual atoms and molecules, crucial for materials science, biology, and nanotechnology.
Slow neutrons have wavelengths comparable to atomic spacings in crystals. Neutron diffraction is used to determine crystal structures, study magnetic materials, and investigate biological molecules. Unlike X-rays, neutrons can locate hydrogen atoms easily.
The wave nature of particles allows quantum tunneling, where particles pass through energy barriers. This enables scanning tunneling microscopes (STM), nuclear fusion in stars, and modern electronics like flash memory and tunnel diodes.
De Broglie's hypothesis led to the development of quantum mechanics and the wave equation (Schrödinger equation). Understanding electron wavelengths explains atomic orbitals, chemical bonding, and the periodic table.
In nanoscale transistors and quantum dots, electron wavelengths become comparable to device dimensions. Wave effects determine conductivity, energy levels, and optical properties, enabling modern computing and optoelectronics.
High-energy particle physics relies on de Broglie wavelength calculations. To probe smaller distances (study quarks, etc.), particles must be accelerated to very high energies to achieve shorter wavelengths, as demonstrated at facilities like the Large Hadron Collider.
h = 6.626 × 10⁻³⁴ J·s, not 6.63 × 10⁻³⁴ eV·s or other units without conversion.
Correct: Always use h = 6.626 × 10⁻³⁴ J·s in SI calculations
When given mass and velocity, must calculate p = mv before using λ = h/p.
Correct: Use λ = h/(mv) or calculate p first, then λ = h/p
Mass must be in kg, velocity in m/s. Converting grams to kg or km/h to m/s incorrectly is common.
Correct: 1 g = 10⁻³ kg, 1 km/h = 1/3.6 m/s, always use SI units
For photons: λ = c/f or E = hf. For matter: λ = h/p. Don't mix these equations.
Correct: De Broglie equation applies to matter particles with mass, not massless photons
h = 6.626 × 10⁻³⁴ J·s
me = 9.109 × 10⁻³¹ kg
mp = 1.673 × 10⁻²⁷ kg
mn = 1.675 × 10⁻²⁷ kg
c = 2.998 × 10⁸ m/s
λ = h / p
λ = h / (m × v)
p = m × v
p = h / λ
KE = p² / (2m) = (1/2)mv²
1 nm = 10⁻⁹ m
1 Å = 10⁻¹⁰ m
1 eV = 1.602 × 10⁻¹⁹ J
1 pm = 10⁻¹² m
Electron (thermal): ~1-10 Å
Electron (fast): ~0.1 Å
Proton (slow): ~0.1-1 Å
Neutron (thermal): ~1-2 Å