Definition of Wave Number
A wave number (often denoted by κ or k, or ṽ for spectroscopic wave number) is a measure of the spatial frequency of a wave, indicating the number of waves per unit distance. It is inversely proportional to wavelength, meaning shorter wavelengths have larger wave numbers. In physics, it can represent the magnitude of the wave vector, which also specifies the direction of wave propagation.
Units and Relationship to Wavelength
The SI unit for wave number is reciprocal meters (m⁻¹), although reciprocal centimeters (cm⁻¹) are commonly used in spectroscopy. The relationship is straightforward: wave number = 1 / wavelength (κ = 1/λ). For electromagnetic waves, it's also related to frequency (f) and the speed of light (c) by κ = f/c.
Application in Spectroscopy
In spectroscopy, the wave number (often expressed in cm⁻¹) is particularly useful because it is directly proportional to the energy of a photon (E = hcκ, where h is Planck's constant and c is the speed of light). This makes it convenient for interpreting vibrational and rotational spectra, as energy level differences are directly mapped to wave number differences.
Importance in Quantum Mechanics
Wave number plays a crucial role in quantum mechanics as part of the wave vector, describing the momentum of a particle (p = ħk, where ħ is the reduced Planck constant). It connects the wave-like properties of matter (de Broglie wavelength) to its particle-like momentum, forming a fundamental concept in quantum physics.