Question

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 w = 2πf and wave number k = 2π/λ instead of frequency and wavelength. By knowing the dispersion relation w(k), one can find the phase and group velocities of a wave:

Vgroup = dw/dk
vphase = w/k

Consider a nonrelativistic free particle of mass m moving with velocity v. Find a dispersion relation w(k) for the de Broglie wave of this particle.

Answer 1

The dispersion relation w(k) for de Broglie waves of a nonrelativistic free particle with mass m and velocity v is ω(k) = (ħk^2)/(2m), where ħ is the reduced Planck's constant and k is the wave number.

Step-by-step explanation:

According to Louis de Broglie's hypothesis, both massless photons and massive particles like electrons share the common characteristic of wave-particle duality. The de Broglie wavelength (λ) of a particle with momentum (p) is given by the equation λ = h/p, where h is Planck's constant. To derive the dispersion relation w(k) for de Broglie waves of a nonrelativistic free particle of mass m moving with velocity v, we first relate the angular frequency (ω) to the particle's energy (E) and the wave number (k) to the particle's momentum (p).

Considering E = 1/2 m v^2 and using the relationship E = ħω, where ħ is the reduced Planck's constant (h/2π), we get ω = E/ħ = (1/2 m v^2)/ħ. Furthermore, since p = mv and k = p/ħ, we derive k = mv/ħ. Now, by substituting m v^2 from the k expression into the ω equation, we get ω = (ħk^2)/(2m). Therefore, the dispersion relation for the de Broglie wave of a nonrelativistic free particle is ω(k) = (ħk^2)/(2m).