[Q]Expression of Group velocity in QM

[good_phy]

Hi.

We already know Group velocity is $\frac{\partial w }{\partial k }$

But, liboff said Group velocity is $\frac{\partial w }{\partial k }|_{k_{max}}$ where $k_{max}$ is wave number which probablity this value can be measured is higher than other wave number.

is it right? How can?

 

[jtbell]

In general, $\omega$ is a function of k, $\omega(k)$, so $d\omega / dk$ is another function of k. To get a particular velocity, you need to specify some value of k to evaluate $d\omega / dk$ at.

 

[Entropia]

Hello,

Thank you for your question. Yes, the expression for Group velocity in quantum mechanics is indeed given by \frac{\partial w}{\partial k}, where w represents the energy of the particle and k represents the wave number.

To address the second part of your question, it is important to understand the concept of group velocity in quantum mechanics. Group velocity is a measure of how quickly a group of particles is moving, rather than the individual particles themselves. In other words, it is the velocity of the wave packet that describes the motion of a group of particles.

The expression \frac{\partial w}{\partial k}|_{k_{max}} means that the group velocity is measured at the maximum value of the wave number, which is k_{max}. This is because the wave packet is most likely to be found at this value of k, as it corresponds to the peak of the wave packet. This is why the probability of measuring the group velocity at this value is higher compared to other values of k.

I hope this helps to clarify the expression for group velocity in quantum mechanics. If you have any further questions, please feel free to ask. Thank you.