In 1924, the French physicist Louis de Broglie resolved the long-standing problem of wave-particle duality in nature by postulating that a free particle has a characteristic wavelength determined by its momentum; that is λ = h/p,
where h is Planck's constant.
Hence, if a particle is a wave, it must have a characteristic dispersion relation. A dispersion relation is simply a representation of a wave's frequency as a function of wavelength, i. e., f(λ). However, most often scientists use angular frequency ω = 2πf and wave number k = 2π/λ instead of frequency and wavelength. By knowing the dispersion relation ω(k), one can find the phase and group velocities of a wave.
Use the dispersion relation to find the group velocity v_group and phase velocity v_phase. Express your answer in terms of velocity v only.