Question

Now consider de Broglie waves for a (relativistic) particle of mass \(m,\) momentum \(p=m v \gamma,\) and total energy \(E=m c^{2} \gamma,\) with \(\gamma=\left[1-(w / c)^{2}\right]^{-1 / 2}\) The waves have wavelength \(\lambda=h / p\) and frequency \(f=E / h\) as in Problem \(36.45,\) but with the relativistic momentum and energy values. a) For these waves, calculate the dispersion relation, \(\omega=\omega(\kappa),\) where \(\kappa\) is the wave number. b) Calculate the phase velocity \(\left(v_{p}=\omega / \kappa\right)\) and the group velocity \(\left(v_{k}=\partial \omega / \partial \kappa\right)\) of these waves. c) Which corresponds to the classical velocity of the particle?

Short answer

Short Answer: In the relativistic de Broglie wave scenario, both the phase velocity (v_p) and group velocity (v_k) converge to the classical velocity (v) of the particle in the classical limit where v<<c. This means that both velocities correspond to the classical velocity of the particle.

Step-by-step solution

Find the dispersion relation ω(κ)

To find the dispersion relation ω(κ), we need to connect the frequency (f) and wave number (κ) using the relativistic expressions for energy, momentum, and the de Broglie relations. We are given the relations: 1. λ=h/p 2. f=E/h 3. p=mvγ 4. E=mc^2γ 5. γ=1/sqrt(1 - (v/c)^2) First, we will find expressions for the energy E and the momentum p in terms of λ or κ. Begin with λ=h/p: λκ = h(κ)/p => p = hκ Now, find an expression for E in terms of λ or κ using f=E/h: f=h/E => E = hf Substitute the expressions for E and p into the dispersion relation, ω=f2π, and κ = 2π/λ: ω = 2π(E/h) and κ = 2π(1/λ) ω(κ) = 2π(hf/h) = 2πf Now, we have a relation between ω and κ.

Calculate the phase and group velocities

To find the phase and group velocities, we will calculate ω/κ and the derivative of ω with respect to κ, respectively. Phase velocity (v_p): v_p = ω/κ = (2πf)/(2π(1/λ)) = fλ Group velocity (v_k): v_k = dω/dκ To compute this derivative, we first need to find f and λ as functions of κ. Start with the given relations E = hf and λ = h/(mvγ). 1. f = E/h => f = mv^2γ/2πh 2. λ = h/(mvγ) => λ = 1/(mvγκ) Now, substitute these expressions for f and λ into λ = fλ: λ = mv^2γ/2πh * 1/(mvγκ) Since the phase velocity v_p = fλ, we have: v_p = mv^2γ / 2πh * 1/(mvγκ) Now, we will differentiate ω with respect to κ to get the group velocity: dω/dκ = d(fλ)/dκ = (1/2πh)* d(mv^2γ / mvγκ)/dκ Through chain rule: 1. Differentiate mv^2γ w.r.t to v and multiply by dv/dκ 2. Differentiate mvγκ w.r.t to v and multiply by dv/dκ v_k = (1/2πh) * (2mvγ(d(γ)-mv(d(γ))/dκ)/κ)

Determine which velocity corresponds to the classical velocity

In the classical limit, where v<0)(mv^2γ/2πh * 1/(mvγκ)) => v Group velocity (v_k) in the classical limit: v_k = lim(v->0)((1/2πh) * (2mvγ(d(γ)-mv(d(γ))/dκ)/κ)) => v In the classical limit, both the phase velocity and the group velocity converge to the classical velocity of the particle.

Key concepts

Dispersion Relation

The concept of dispersion relation is central to understanding wave mechanics and is particularly important when examining de Broglie waves for relativistic particles. The dispersion relation connects the angular frequency \(\omega\) of a wave to its wave number \(\kappa\), encapsulating how the frequency of a wave varies with its wavelength.

In the context of de Broglie waves, the momentum \(p\) and energy \(E\) of a particle are quantitatively expressed through the relations\(\lambda=h/p\) and \(f=E/h\). Given that \(\omega=2\pi f\) and \(\kappa=2\pi/\lambda\), we arrive at the fundamental relationship \(\omega(\kappa) = \omega(f(\kappa))\), representing the dispersion relation for a relativistic particle. This relation reveals how the particle's quantum-mechanical wave properties are intimately tied to its relativistic characteristics.

Phase Velocity

Phase velocity \(v_p\) is defined as the rate at which the phase of the wave propagates in space. It is the speed at which a particular phase of the wave (such as the crest) travels along. From the solution, we establish that \(v_p = \omega/\kappa\), which, after some manipulation using the de Broglie relations and relativistic equations, gives us \(v_p = f\lambda\).

This means that the phase velocity is the product of the frequency of the wave and its wavelength. Interestingly, in the context of relativistic particle physics, this velocity does not correspond to the actual speed of the particle, but rather describes the movement of the wave's phase through space.

Group Velocity

Group velocity \(v_g\), in contrast to the phase velocity, is the speed at which the overall shape of the waves' amplitudes—known as the modulation or envelope—propagates through space. It is given by \(v_g = \partial\omega/\partial\kappa\) and represents the speed at which information or energy is conveyed by the wave.

To find \(v_g\) for the de Broglie waves, the derivative of the angular frequency with respect to the wave number is calculated—this often requires applying the chain rule to handle the relativistic expressions involved. Contrary to phase velocity, group velocity is significant for its correspondence to the classical velocity of the particle, thus holding a more intuitive physical interpretation in the context of a particle's movement.

Relativistic Particle Physics

Relativistic particle physics deals with the study of particles moving at velocities close to that of light. Here, the concepts of phase and group velocities become particularly pertinent due to the relativistic factor \(\gamma\) that alters our classical expectations of how particles behave.

As particles approach relativistic speeds, their mass, momentum, and energy change according to Einstein's theory of relativity, and this is captured by the Lorentz factor \(\gamma\). In the de Broglie hypothesis, these relativistic parameters dictate the wavelength and frequency of the associated wave motions. For such particles, understanding their wave behavior—via dispersion relation, phase, and group velocities—not only explains the quantum-mechanical nature of matter but also adheres to the principles of relativity, melding the two pivotal theories that govern the physics of the very small and the very fast.