How will you show that the velocity of a matter wave is always greater than that of light?

To address the statement that the velocity of a matter wave is always greater than that of light, let's clarify and correct this concept. In the realm of quantum mechanics and special relativity, this statement is actually not accurate. Here's why:

1. **Matter Waves and de Broglie Wavelength:**
According to de Broglie's hypothesis, particles such as electrons have wave-like properties, and their wavelength \(\lambda\) is related to their momentum \(p\) by the equation:
\[
\lambda = \frac{h}{p}
\]
where \(h\) is Planck's constant.

2. **Phase Velocity and Group Velocity:**
Matter waves can have different types of velocities:
- **Phase Velocity (\(v_p\)):** This is given by \(v_p = \frac{\omega}{k}\), where \(\omega\) is the angular frequency and \(k\) is the wave number. For a relativistic matter wave, the phase velocity can be greater than the speed of light \(c\), but this does not violate relativity as it does not represent the speed of information or matter.
- **Group Velocity (\(v_g\)):** This is the velocity at which the overall shape of the wave's amplitudes (or the "packet") travels, and it is associated with the speed of the particle. The group velocity is given by:
\[
v_g = \frac{d\omega}{dk}
\]
For a free particle, the group velocity is equal to the particle's velocity.

3. **Relativistic Constraints:**
According to special relativity, the speed of light \(c\) is the ultimate speed limit for information and matter. The group velocity of a matter wave, which corresponds to the velocity of the particle itself, cannot exceed the speed of light. If it were possible to have a matter wave's group velocity greater than \(c\), it would imply superluminal (faster-than-light) particle velocities, which contradicts the principles of relativity.

4. **Misinterpretation of Phase Velocity:**
The misconception might arise from confusing phase velocity with the speed of light. The phase velocity of a matter wave can be greater than \(c\), but this does not imply that any physical information or matter is traveling faster than light.

In summary, the velocity of a matter wave, specifically the group velocity associated with the particle's movement, cannot exceed the speed of light. The phase velocity can be greater than \(c\), but it does not carry physical information faster than light.