Could Planck-Einstein relation be applied to matter waves?

[Haorong Wu]

TL;DR Summary

Could Planck-Einstein relation be applied to matter waves?

My friend gave me some statements which are wrong, but I could not tell why they are wrong.

He wrote,

Since $\omega = \frac E \hbar = \frac {\hbar k^2} {2m} = k v$, then$p=\hbar k =2mv$.

I guess that $E =\hbar \omega$ may only appied to photons, not matter waves. Is that correct?

 

[Demystifier]

$E=\hbar\omega$ is true for both photons and matter waves. However, $E=\hbar^2k^2/2m$ is correct only for non-relativistic matter waves. On the other hand, $E=\hbar k v$ is correct only for photons, for which $v=c$.

 

[Haorong Wu]

$E=\hbar\omega$ is true for both photons and matter waves. However, $E=\hbar^2k^2/2m$ is correct only for non-relativistic matter waves. On the other hand, $E=\hbar k v$ is correct only for photons, for which $v=c$.

Thanks, Demystifier. But I still can not understand it clearly.

Meanwhile, could the problem be the velocity? The velocity for $p= mv$ should be the group velocity of the matter wave which is $v=\frac {dw} {dk}$. If so, I can get a proper answer then.

 

[Demystifier]

The velocity for $p= mv$ should be the group velocity of the matter wave which is $v=\frac {dw} {dk}$. If so, I can get a proper answer then.

That's OK, but you should be careful about what do you take for the function $\omega(k)$. For photons it is $\omega=ck$, while for non-relativistic matter it is $\omega=\hbar k^2/2m$.

 

[Haorong Wu]

That's OK, but you should be careful about what do you take for the function $\omega(k)$. For photons it is $\omega=ck$, while for non-relativistic matter it is $\omega=\hbar k^2/2m$.

Thanks, Demystifier. I will be careful about that.