Wave Nature of Matter

Light shows both wave behaviour and particle behaviour. Interference, diffraction, and polarisation support the wave model, while the photoelectric effect and Compton effect support the particle model.

Louis de Broglie extended this idea to matter and proposed that moving particles such as electrons and protons also possess wave character. This is called the wave nature of matter or wave-particle duality of matter.

The property by which light or matter can show both wave-like and particle-like behaviour depending on the experiment is called wave-particle duality.

The waves associated with a moving material particle are called matter waves or de Broglie waves.

According to de Broglie, every moving particle is associated with a wave whose wavelength depends on its momentum.

For a particle of momentum p, the associated wavelength is:

For a particle of mass m moving with speed v:

• de Broglie proposed that a moving material particle of total energy E and momentum p has a wave associated with it, analogous to a photon.
• He related the wave quantities frequency and wavelength with particle quantities energy and momentum.
• Using the photon relation shown in the content,
  p = \[\frac {E}{c}\] = \[\frac {hν}{c}\] = \[\frac {h}{λ}\]
• Rearranging this gives, λ = \[\frac {h}{p}\]
• For a material particle of mass mmm moving with velocity v, momentum is p = mv, so
  λ = \[\frac {h}{p}\] = \[\frac {h}{mv}\]
• Meaning of the result
• This wavelength is called the de Broglie wavelength.
• The waves associated with moving material particles are called matter waves.
• The idea stated in your content is that if radiation exhibits particle behaviour, then particles of matter should also exhibit wave behaviour.
• The same content also implies that these waves are associated with all moving matter particles.

A photon has momentum and wavelength, so de Broglie proposed that a moving particle should also have an associated wavelength. For very small particles like electrons, this wavelength can be large enough to be experimentally detected.

For large objects such as a ball or a car, the momentum is very high, so the wavelength becomes extremely small. Hence, the wave nature of macroscopic objects is not noticeable in ordinary life.

• Matter waves are associated with moving particles.
• Their wavelength is inversely proportional to momentum.
• They are significant for microscopic particles such as electrons.
• They are not ordinarily observable for macroscopic bodies because the wavelength is extremely small.
• The concept helps explain electron diffraction and the wave behaviour of subatomic particles.

The de Broglie wavelength is calculated using:

λ = \[\frac {h}{mv}\]​

where:

• h = 6.63 × 10⁻³⁴ J s
• m = mass
• v = velocity

(a) Electron

• Mass = 9.11 × 10⁻³¹ kg
• Speed = 5.4 × 10⁶ m/s
• de Broglie wavelength: λ = 0.135 nm
• This wavelength is comparable to X-ray wavelengths, so the wave nature of electrons can be observed experimentally.

(b) Ball

• Mass = 0.150 kg
• Speed = 30.0 m/s
• de Broglie wavelength: λ = 1.47 × 10⁻³⁴ m
• This wavelength is extremely small, making the wave nature of the ball impossible to detect.

Conclusion:
Small particles like electrons have measurable de Broglie wavelengths and exhibit wave behaviour, whereas large objects like balls have extremely tiny wavelengths, so their wave nature is not observable.