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.