22.3.2Quantum and nuclear physics (A Level)
The de Broglie Wavelength
5 min read 9 parts Lesson 10 of 23 1 quiz
Lesson summary
22.3.2 The de Broglie Wavelength — AS PHYSICS 9702 notes: Define the de Broglie wavelength and calculate it from momentum, speed, kinetic energy and accelerating voltage.De Broglie wavelength = the wavelength associated with a moving particle.Idea — If light waves can behave as p…
- What the symbols mean
- Confirmation
- Link to photons
- Inverse dependence
- Everyday objects
- Condition for diffraction
Section 1 of 9The de Broglie hypothesis
De Broglie wavelength
De Broglie wavelength the wavelength associated with a moving particle.
Idea
- If light waves can behave as particles carrying momentum, then particles such as electrons should have a wavelength.
Universal
- All particles show wave-like properties, not only electrons.
Equation
- λ = h/p = h/(mv).What the symbols mean: λ = de Broglie wavelength (m) · h = 6.63 × 10−34 J s · p = momentum (kg m s−1) · m = mass (kg) · v = speed (m s−1)
- λ is in metres, h is the Planck constant, p is the momentum in kg m s−1, m is the mass and v is the speed.
Confirmation
- Electron diffraction measurements gave wavelengths that match h/p.
Link to photons
- The same relation applies to photons, whose momentum is p = h/λ.
Section 2 of 9Observing wave effects
1Inverse dependence
- Wavelength is inversely proportional to momentum.
2Everyday objects
- Their momentum is so large that the wavelength is far too small to observe any quantum effect.
3Electrons
- A typical electron in a metal has a de Broglie wavelength of about 10 nm.
- Wave effects show only when the sample width is around this size.
- Fast electrons have wavelengths similar to atomic spacings, so diffraction is noticeable.
4Condition for diffraction
- Diffraction is appreciable only when the wavelength is comparable with the gap or lattice spacing.
- This is why electrons, neutrons and atoms probe crystal structure but large objects do not diffract.
Section 3 of 9Wavelength from kinetic energy
Kinetic energy
- E = ½mv2 = p2/2m.
Momentum
- Momentum p = √(2mE).
Wavelength
- Substituting gives λ = h/√(2mE).
- E is the kinetic energy in joules and m is the particle mass.
Mass dependence
- For the same kinetic energy, λ is proportional to 1/√m, so a heavier particle has a shorter wavelength.
Same speed
- A heavier particle has greater momentum and so a shorter wavelength.
Section 4 of 9Electrons accelerated through a p.d.
Energy gained
- An electron accelerated from rest through a p.d.
- V gains kinetic energy eV, so ½mv2 = eV.
Wavelength
- λ = h/√(2meV).
Voltage dependence
- λ is proportional to 1/√V, so quadrupling the voltage halves the wavelength.
Electron microscope
- Fast electrons have wavelengths much shorter than visible light, so they resolve finer detail.
Units
- V is in volts and e in coulombs, giving kinetic energy in joules.
Section 5 of 9Graphs of de Broglie wavelength
λ against p
- A reciprocal curve (hyperbola).
λ against 1/p
- A straight line through the origin with gradient h.
λ against 1/√V
- A straight line through the origin with gradient h/√(2me).
Ring radii
- Smaller wavelength gives smaller ring radii in electron diffraction by graphite.
- Measuring ring radii at different voltages can be used to find h.
Section 6 of 9Calculation routes
1
Given mass and speed, find p = mv and then λ = h/p.
2
Given an accelerating voltage, use λ = h/√(2meV), or find the kinetic energy eV, then p, then λ.
3
Comparing two particles, use ratios with λ proportional to 1/p.
4
Use the non-relativistic formula E = ½mv2, since speeds close to c are outside the syllabus.
Section 7 of 9More from the specification
More from the specification
Absorbed
- Taken in by an object, with light energy transferred to thermal energy.
Wave-particle duality
- The property that radiation and matter show both wave and particle behaviour.
Potential difference
- Work done by a unit charge passing through a component.
Proton
- Positive particle in the nucleus, relative charge +1 and relative mass 1.
Theory
- A well-tested, wide-ranging explanation supported by evidence.
Frequency
- Revolutions per second, f = 1/T.What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)
Threshold
- The output value at which a switching circuit changes state.
Intensity
- Power per unit area perpendicular to the wave direction.
Photoelectron
- An electron emitted from a metal surface by the photoelectric effect.
Section 8 of 9Key terms
| Term | Meaning |
|---|---|
| de Broglie wavelength | The wavelength associated with a moving particle, λ = h/p. |
| Momentum | The product of mass and velocity, p = mv. |
| Planck constant | 6.63 × 10−34 J s, the constant linking wavelength and momentum. |
| Matter wave | The wave associated with a moving particle such as an electron. |
| Accelerating potential difference | The p.d. through which a charged particle gains kinetic energy qV. |
Section 9 of 9Exam technique
1
Using the speed of light for an electron — use the electron's own speed in p = mv, since c is only for photons.
2
Forgetting the square root when finding p from kinetic energy — p = √(2mE), not 2mE.
3
Thinking a faster particle has a longer wavelength — wavelength is inversely proportional to momentum, so faster means shorter.
4
Using eV as joules in λ = h/√(2meV) — V is in volts and e in coulombs, giving kinetic energy in joules.
5
Believing large objects have no wavelength — they have one, but it is far too small to observe.