- #1

raul_l

- 105

- 0

## Homework Statement

A particle in classical mechanics with a velocity v becomes a wave packet with a group velocity v_g in quantum mechanics.

I have to show that [tex] \vec{v} = \vec{v}_g [/tex]

## Homework Equations

[tex] \vec{v}_g = \frac{\partial \omega(\vec{k})}{\partial \vec{k}} [/tex]

[tex] \omega(\vec{k}) = \hbar^{-1} \sqrt{m_{0}{2} c^4+\hbar^2 k^2 c^2} [/tex]

## The Attempt at a Solution

[tex] \vec{v}_g = \frac{\partial}{\partial \vec{k}} \hbar^{-1} \sqrt{m_{0}{2} c^4+\hbar^2 \vec{k} \vec{k} c^2} = \frac{\vec{k}}{\hbar \sqrt{m_{0}{2} c^4+\hbar^2 k^2 c^2}} = \frac{\vec{k}}{\hbar \sqrt{m_{0}{2} c^4+\hbar^2 k^2 c^2}} = \frac{m\vec{v}}{\hbar^2 E} = \frac{m\vec{v}}{\hbar^2 m c^2} = \frac{\vec{v}}{\hbar^2 c^2} [/tex]