The de Broglie wavelength is the wavelength associated with a moving particle, such as an electron, proton, or even a baseball. It matters because it shows that matter can behave like a wave when its wavelength is large enough to observe. This idea helped launch quantum mechanics and explains why tiny particles can diffract, interfere, and form wave patterns.
For everyday objects, the wavelength is so extremely small that their wave behavior is not noticeable.
Key Facts
- de Broglie wavelength: λ = h / p
- For nonrelativistic motion, momentum is p = mv, so λ = h / mv
- Planck's constant is h = 6.626 x 10^-34 J s
- A smaller momentum gives a longer de Broglie wavelength.
- Electron diffraction occurs when an electron's wavelength is comparable to atomic spacing, about 10^-10 m.
- For an electron accelerated through voltage V, λ = h / sqrt(2meV), where e is the elementary charge and m is electron mass.
Vocabulary
- de Broglie wavelength
- The wavelength associated with a moving particle, given by λ = h / p.
- matter wave
- A wave-like description of a particle's motion and quantum behavior.
- momentum
- A measure of an object's motion equal to mass times velocity for nonrelativistic speeds.
- diffraction
- The spreading and pattern formation of waves when they pass through small openings or around obstacles.
- Planck's constant
- A fundamental constant, h = 6.626 x 10^-34 J s, that links quantum energy, frequency, momentum, and wavelength.
Common Mistakes to Avoid
- Using λ = h / mv for particles moving near the speed of light. This is wrong because relativistic momentum must be used when the speed is very high.
- Forgetting to convert mass, speed, or energy into SI units. This gives wavelengths with incorrect size because h is written in joule seconds.
- Thinking only electrons have de Broglie wavelengths. All moving objects have them, but massive everyday objects have wavelengths far too small to detect.
- Assuming a matter wave means the particle is physically smeared out like a water wave. The wave describes quantum behavior and probability, not a literal ripple of material.
Practice Questions
- 1 An electron has momentum 7.3 x 10^-24 kg m/s. Calculate its de Broglie wavelength using h = 6.626 x 10^-34 J s.
- 2 A 0.145 kg baseball moves at 40 m/s. Calculate its de Broglie wavelength and compare it with an atom, about 1 x 10^-10 m wide.
- 3 Explain why electron diffraction can be observed in crystals, but diffraction of a thrown baseball through an ordinary doorway is not observed.