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) Calculate the dispersion relation \(\omega=\omega(k)\) for these waves. b) Calculate the phase and group velocities of these waves. Which of these corresponds to the classical velocity of the particle?

Short answer

Question: Determine the dispersion relation, phase velocities, and group velocities for de Broglie waves of a Newtonian particle, given mass (m), momentum (p), energy (E), wavelength (λ), and frequency (f). Also, identify which velocity corresponds to the classical velocity of the particle. Short Answer: The dispersion relation for the de Broglie waves of a Newtonian particle is given by: $$ \omega(k) = \frac{\hbar k^2}{2m} $$ The phase velocity (v_p) and the group velocity (v_g) are given by: $$ v_p = \frac{\hbar k}{2m} $$ $$ v_g = \frac{\hbar k}{m} $$ The classical velocity of the particle corresponds to the group velocity (v_g).

Step-by-step solution

Write the expressions for momentum and energy in terms of mass and velocity

Note down the expressions for momentum and energy: $$ p= mv $$ $$ E = \frac{p^2}{2m} $$

Write expressions for wavelength and frequency using de Broglie relations

Recall de Broglie's wavelength and frequency expressions: $$ \lambda = \frac{h}{p} $$ $$ f = \frac{E}{h} $$

Calculate the angular frequency and wavenumber

Express ω (angular frequency) and k (wavenumber) in terms of f (frequency) and λ (wavelength): $$ \omega = 2\pi f $$ $$ k = \frac{2\pi}{\lambda} $$

Calculate the dispersion relation ω(k)

Express the energy (E) using the angular frequency (ω), and the momentum (p) using the wavenumber (k): $$ E = \hbar\omega $$ $$ p = \hbar k $$ Now, substitute p and E from the above equations into the energy equation: $$ E = \frac{p^2}{2m} $$ $$ \hbar\omega = \frac{(\hbar k)^2}{2m} $$ Finally, solve for the dispersion relation ω(k): $$ \omega(k) = \frac{\hbar k^2}{2m} $$

Calculate the phase and group velocities

The phase velocity (v_p) and group velocity (v_g) can be calculated using the following equations: $$ v_p = \frac{\omega}{k} $$ $$ v_g = \frac{d\omega}{dk} $$ For phase velocity, substitute the expression for the dispersion relation we found in step 4: $$ v_p = \frac{\hbar k^2}{2m} \cdot \frac{1}{k} = \frac{\hbar k}{2m} $$ For group velocity, differentiate the dispersion relation with respect to k: $$ v_g = \frac{d}{dk}(\frac{\hbar k^2}{2m}) = \frac{\hbar k}{m} $$

Determine which velocity corresponds to the classical velocity of the particle

Recall the expression for the classical velocity of the particle, obtained from the momentum equation: $$ v = \frac{p}{m} $$ Now, substitute the expression for p obtained in step 3 from the de Broglie relations: $$ v = \frac{\hbar k}{m} $$ Comparing this with the expressions obtained earlier for phase and group velocities, we can see that the classical velocity of the particle corresponds to the group velocity (v_g).

Key concepts

Dispersion Relation

The concept of a dispersion relation is fundamental in understanding wave phenomena, and it's no different for de Broglie waves, which describe particles exhibiting wave-like behavior. A dispersion relation connects the angular frequency \( \omega \) of a wave to its wavenumber \( k \), encapsulating how waves of different wavelengths travel through a medium.

In our exercise, the de Broglie waves of a Newtonian particle lead to the discovery of the specific dispersion relation \( \omega(k) = \dfrac{\hbar k^2}{2m} \). This equation is derived from the energy and momentum relations of a quantum particle, where the Planck constant divided by \( 2\pi \) (denoted as \( \hbar \)) appears due to the wave nature of particles at the quantum scale.

• Dispersion relations are crucial because they indicate how the phase velocity of a wave varies with frequency.
• A linear dispersion relation means the phase velocity is constant, indicating no dispersion or spreading of the wave. Conversely, a quadratic relation, as found for de Broglie waves, results in dispersion, which leads to the spreading of wave packets over time.

Understanding the dispersion relation is an essential step in predicting the wave behavior in various mediums and is intensely used in fields like optics, acoustics, and quantum mechanics.

Phase Velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. It is the speed at which any one phase of the wave (for example, the crest) travels. For de Broglie waves, the phase velocity is given by the equation \( v_p = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wavenumber.

In the context of our specific exercise, the phase velocity for de Broglie waves was found to be \( v_p = \frac{\hbar k}{2m} \). Importantly, this is not the speed of the particle itself but rather the velocity of the repeating patterns (phases) within the wave associated with the particle. One interesting property is that for de Broglie waves of free particles, the phase velocity actually exceeds the speed of light. However, this doesn't violate relativity since it doesn't represent the speed of any physical object or information.

• It is important to note that the phase velocity can sometimes be greater than the speed of light, but this does not imply the transmission of information faster than light, as only group velocities carry information.

Group Velocity

The group velocity of a wave is the velocity with which the overall shape of the waves' amplitudes—known as the modulation or envelope—propagates through space. This concept is important when dealing with wave packets or pulses, which can be thought of as a group of waves of different frequencies dispersed over some region.

For de Broglie waves, the group velocity \( v_g \) is determined by differentiating the dispersion relation with respect to the wavenumber \( k \): \( v_g = \frac{d\omega}{dk} \). When we perform this differentiation for the dispersion relation we obtained from the exercise \( \omega(k) = \frac{\hbar k^2}{2m} \), the result is \( v_g = \frac{\hbar k}{m} \).

Interestingly enough, it turns out that the group velocity for de Broglie waves corresponds to the classical velocity of the particle, \( v = \frac{p}{m} \). This highlights a fascinating connection between classical and quantum descriptions of motion: while the quantum wave has both a phase and group velocity, when it comes to physical measurements of the particle's velocity, it's the group velocity we observe, meshing quantum mechanics with classical physics.

• Group velocity is highly relevant when discussing the transfer of energy and information in wave phenomena—it is the speed at which the energy or information conveyed by the wave travels.