Question

Consider de Broglie waves for a Newtonian particle of mass \(m\), momentum \(p=m v\), and energy \(E=p^{2} /(2 m)\), that is, waves with wavelength \(\lambda=h / p\) and frequency \(f=E / h\) a) For these waves, calculate the dispersion relation, \(\omega=\omega(\kappa),\) where \(K\) is the wave number. b) Calculate the phase velocity \(\left(v_{p}=\omega / \kappa\right)\) and the group velocity \(\left(v_{g}=\partial \omega / \partial \kappa\right)\) of these waves.

Short answer

Answer: The expressions for the phase velocity (v_p) and group velocity (v_g) of de Broglie waves for a classical Newtonian particle are as follows: v_p = (4π^3m^2 h)/κ^3 v_g = -2(4π^3m^2 h)/κ^3

Step-by-step solution

Express ω and κ in terms of λ and f

We are given λ = h/p and f = E/h, so we can express ω (angular frequency) and κ (wave number) in terms of λ and f using the relationships ω = 2πf and κ = 2π/λ.

Eliminate λ and f in ω and κ expressions

Since we ultimately want to find ω = ω(κ), we need to eliminate λ and f from the expressions for ω and κ. We can do this by using the given equations: 1. λ = h/p 2. f = E/h We rewrite the expressions for ω and κ in terms of λ and f: 1. ω = 2πf 2. κ = 2π/λ Now we substitute the expressions for λ and f into these equations: 1. ω = 2π(E/h) 2. κ = 2π(h/p) We can now solve for ω as a function of κ.

Solve for ω in terms of κ

To solve for ω in terms of κ, we first eliminate p from the equation for κ: κ = 2π(h/(mv)) Now we solve for p: p = 2πm(h/κ) Then we substitute this expression for p into the equation for ω: ω = 2π(E/h) = 2π(p^2/(2mh)) Now we substitute the expression for p: ω = 2π((2πm(h/κ))^2/(2mh)) Simplify the equation: ω = 2\pi((4\pi^2 m^2 h^2/κ^2)/(2mh)) ω = (4π^3m^2 h)/κ^2 Thus, we found the dispersion relation ω = ω(κ): ω(κ) = (4π^3m^2 h)/κ^2 #b) Calculate the phase velocity and group velocity of these waves #

Calculate the phase velocity

The phase velocity v_p is given by the expression: v_p = ω/κ Using the dispersion relation ω(κ) we found earlier, we can calculate v_p: v_p = ((4π^3m^2 h)/κ^2) / κ v_p = (4π^3m^2 h)/κ^3

Calculate the group velocity

The group velocity v_g is given by the expression: v_g = ∂ω/∂κ To find v_g, we need to find the derivative of ω(κ) with respect to κ: ω(κ) = (4π^3m^2 h)/κ^2 dω/dκ = -2(4π^3m^2 h)/κ^3 Then, v_g is given by: v_g = -2(4π^3m^2 h)/κ^3 Since both the phase and group velocities have the same algebraic expression for a classical Newtonian particle, we find that: v_p = (4π^3m^2 h)/κ^3 v_g = -2(4π^3m^2 h)/κ^3