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What is the group velocity of a de Broglie wave?

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The group velocity of a de Broglie wave refers to the velocity at which the **envelope** of a wave packet propagates through space. This concept is especially relevant in the context of wave-particle duality and quantum mechanics, where particles such as electrons are described not just as points but as wave-like entities with associated de Broglie waves.

### Key Concepts:
1. **de Broglie Waves:**
   According to Louis de Broglie, particles like electrons, protons, and even larger objects, can be associated with waves. This wave has a wavelength \( \lambda \) related to the particle's momentum \( p \) by the de Broglie relation:

   \[
   \lambda = \frac{h}{p}
   \]
   
   where \( h \) is Planck’s constant, and \( p \) is the momentum of the particle. This is a fundamental result from wave-particle duality, which suggests that particles can exhibit wave-like behavior.

2. **Wave Packets:**
   A single particle is not described by a single, pure de Broglie wave, but by a **wave packet**. A wave packet is a superposition of many waves with slightly different wavelengths (or momenta). This packet has a well-defined **group velocity**, which is the velocity at which the entire wave packet (and thus the particle) moves. The wave packet is essential because real particles are often not perfectly represented by a single frequency or wavelength but by a combination of many waves.

3. **Group Velocity:**
   The **group velocity** of a wave packet is defined as the velocity at which the **envelope** of the wave packet moves. Mathematically, it is given by the derivative of the angular frequency \( \omega \) with respect to the wave number \( k \):

   \[
   v_{\text{group}} = \frac{d\omega}{dk}
   \]

   where \( \omega \) is the angular frequency, and \( k \) is the wave number. For a particle, this is the velocity at which the probability distribution of the particle (represented by the wave packet) propagates.

4. **Relation to Particle's Velocity:**
   For a non-relativistic particle with energy \( E \) and momentum \( p \), the angular frequency \( \omega \) and wave number \( k \) are related to the particle's energy and momentum as follows:

   \[
   \omega = \frac{E}{\hbar}, \quad k = \frac{p}{\hbar}
   \]

   where \( \hbar \) is the reduced Planck's constant. The energy \( E \) is typically related to the particle’s kinetic energy \( E_k \) (for non-relativistic particles):

   \[
   E_k = \frac{p^2}{2m}
   \]

   Using this, the group velocity can be expressed as:

   \[
   v_{\text{group}} = \frac{d\omega}{dk} = \frac{d}{dk}\left(\frac{E}{\hbar}\right)
   \]

   Since \( E = \frac{p^2}{2m} \) and \( p = \hbar k \), we get:

   \[
   v_{\text{group}} = \frac{d}{dk}\left( \frac{\hbar^2 k^2}{2m\hbar} \right) = \frac{p}{m} = v
   \]

   So, the group velocity of the de Broglie wave is **the same as the velocity of the particle**. In simpler terms, the group velocity corresponds to the speed at which the particle (associated with the wave packet) moves.

### Summary:
- The **de Broglie wave** is a wave associated with a particle's motion, where the wavelength is related to the particle’s momentum.
- A **wave packet** describes the particle more accurately than a single wavelength.
- The **group velocity** of the wave packet is the velocity at which the **particle** moves, and it is mathematically found to be equal to the particle's velocity, \( v_{\text{group}} = v \).
- The **phase velocity**, which refers to the velocity of a single wave component in the packet, is generally different from the group velocity and is not directly related to the particle's velocity.

Thus, the group velocity of a de Broglie wave is an essential concept in understanding how particles move in quantum mechanics and wave-particle duality.
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