Question

Write de-Broglie formula for the wavelength of matter waves.

Hint:

For small particles (like electrons), the de-Broglie wavelength is large enough to produce noticeable effects like diffraction. For larger objects, the wavelength is negligible and cannot be observed. Always use the formula \(\lambda = \frac{h}{mv}\) for finding the wavelength of matter waves.

Answer 1

The de-Broglie equation relates the wavelength of a particle to its momentum. It shows that matter exhibits wave-like properties, just like light. The equation is given by: \[ \lambda = \frac{h}{p} \] Where:
- \(\lambda\) is the de-Broglie wavelength, which is the wavelength associated with the particle,
- \(h\) is Planck's constant, with a value of \(6.626 \times 10^{-34}~\text{J} \cdot \text{s}\),
- \(p\) is the momentum of the particle. Momentum \(p\) is given by the product of mass and velocity (\(p = mv\)), so the equation becomes: \[ \lambda = \frac{h}{mv} \] This shows that the wavelength is inversely proportional to both the mass and the velocity of the particle. The de-Broglie wavelength is especially significant for subatomic particles, such as electrons, where the wavelength becomes comparable to atomic distances.
For macroscopic objects, the de-Broglie wavelength is incredibly small, so it does not have noticeable effects. However, for very small particles like electrons, neutrons, and protons, the wave-like nature is significant, and it explains phenomena like electron diffraction.
In conclusion, de-Broglie postulated that all matter has an associated wavelength, and the smaller the mass of the particle, the larger its wavelength at a given velocity. The wavelength of an object moving at high speeds (close to the speed of light) becomes comparable to the wavelength of light itself. This wave-particle duality is a fundamental concept in quantum mechanics.