vg and vp relation

Definition

Group velocity is the velocity with which a group or packet of waves travels, and it represents the speed of energy or information transfer, as well as the speed of the associated particle in quantum mechanics.

Phase velocity is the velocity at which a single wave crest or phase of a wave travels.

For matter waves, these are related by:

$$v_g = \frac{d\omega}{dk}$$

$$v_p = \frac{\omega}{k}$$

For a free particle in non-relativistic quantum mechanics, the relation becomes:

$$v_g = 2v_p$$

and since the group velocity equals the particle velocity, we also get:

$$v_p = \frac{v}{2}, \quad v_g = v$$

where $v$ is the speed of the particle.

Main Content

1. De Broglie Matter Waves and Their Velocities

• According to de Broglie, every moving particle has an associated wavelength: $$\lambda = \frac{h}{p}$$ where $h$ is Planck’s constant and $p$ is momentum.

• The frequency of the associated wave is related to the particle’s energy: $$E = h\nu$$ These waves are not ordinary physical water waves but probability waves describing the likely location of the particle.

The two velocities arise naturally from wave theory:

Phase velocity

• describes the speed of any one phase point on the wave, such as a crest.

Group velocity

• describes the speed of the entire wave packet formed by superposing many waves.

For a particle to be localized in space, many waves of slightly different wavelengths are combined. This packet moves as a whole, and its speed is the group velocity.

2. Mathematical Relation Between $v_g$ and $v_p$

• By definition: $$v_p = \frac{\omega}{k} \quad \text{and} \quad v_g = \frac{d\omega}{dk}$$

• For a free particle, using quantum relations: $$E = \hbar\omega,\quad p = \hbar k$$ where $\hbar = \frac{h}{2\pi}$.

For a non-relativistic free particle: $$E = \frac{p^2}{2m}$$ Substituting $E = \hbar\omega$ and $p = \hbar k$, $$\hbar\omega = \frac{(\hbar k)^2}{2m}$$ $$\omega = \frac{\hbar k^2}{2m}$$

Now, $$v_p = \frac{\omega}{k} = \frac{\hbar k}{2m}$$

and $$v_g = \frac{d\omega}{dk} = \frac{\hbar k}{m}$$

Thus, $$v_g = 2v_p$$

Since $$v = \frac{p}{m} = \frac{\hbar k}{m}$$ we get $$v_g = v$$ and therefore $$v_p = \frac{v}{2}$$

This is a fundamental result for non-relativistic matter waves.

3. Physical Significance in Quantum Mechanics

• The group velocity is associated with the actual motion of the particle and the propagation of the wave packet.
• The phase velocity can exceed the speed of light in some cases, but this does not violate relativity because phase velocity does not carry matter or information by itself.
• In quantum mechanics, a particle is better represented by a wave packet than by a single sine wave, because a single wave extends everywhere and cannot localize the particle.

Important interpretation:

• The maximum probability of finding the particle lies where the wave packet is concentrated.
• The motion of this concentration is governed by the group velocity.
• The phase velocity is mainly a mathematical wave property and is not directly equal to particle speed.

Example: For an electron moving in free space, the wave packet representing it travels with the electron’s actual speed, while the internal wave crests move at a different speed.

Working / Process

1. Start with the de Broglie and Planck relations

• Use $\lambda = \frac{h}{p}$ and $E = h\nu$, or equivalently $p=\hbar k$ and $E=\hbar\omega$.

2. Write the energy of a free non-relativistic particle

• Use: $$E = \frac{p^2}{2m}$$

• Convert it into a relation between $\omega$ and $k$: $$\omega = \frac{\hbar k^2}{2m}$$

3. Find phase velocity and group velocity

• Phase velocity: $$v_p = \frac{\omega}{k}$$

• Group velocity: $$v_g = \frac{d\omega}{dk}$$

• After substitution: $$v_p = \frac{\hbar k}{2m}, \quad v_g = \frac{\hbar k}{m}$$

• Hence: $$v_g = 2v_p$$

• Since $v = \frac{p}{m} = \frac{\hbar k}{m}$, the group velocity equals the particle speed.

Advantages / Applications

• Helps explain the wave-particle duality of matter in a clear mathematical way.
• Forms the basis for understanding wave packets and the probabilistic interpretation of quantum mechanics.
• Useful in deriving and interpreting the Schrödinger equation, where the particle is treated as a wave function.
• Important in analyzing electron motion in free space, semiconductors, and quantum devices.
• Provides a conceptual distinction between particle speed and wave phase motion, preventing confusion in quantum theory.

Summary

• The relation between group velocity and phase velocity is a central result in matter-wave theory.
• For a non-relativistic free particle, the group velocity equals the particle velocity.
• The phase velocity is half the group velocity in this case.
• This relation supports the wave-packet description of particles in quantum mechanics.