• Tags: de-broglie, thermal, relativistic
• Difficulty: Advanced

Thermal de Broglie wavelength: For a particle at temperature T with average KE = 3kT/2: λ = h/√(3mkT). For thermal neutrons at room temperature (300 K): λ ≈ 1.45 Å — comparable to interatomic spacing, enabling neutron diffraction.

Relativistic correction: When KE is comparable to rest mass energy ($mc^{2}$): p = √($KE^{2}$ + 2K$Ec^{2}$m)/c... More precisely, for a particle with total energy E and rest mass m: p = √($E^{2}$ - ($mc^{2}$)^{2})/c, and λ = hc/√($E^{2}$ - ($mc^{2}$)^{2}). For non-relativistic particles (KE << $mc^{2}$), this reduces to the standard λ = h/√(2mKE).

Electron in nth Bohr orbit: 2πr = nλ (Bohr's quantization is equivalent to fitting n de Broglie wavelengths in the orbit circumference). This provides a beautiful connection between wave nature and atomic structure.