Where did I go wrong in deriving quantized energy?

[SpaceNerdz]

TL;DR

I've been trying to derive ω = E/hbar, but to no success. I thought it was fairly straight forward derivation, could some one point out my mistake ?

OK, so I just want to show ω = E/h = kv, but I keep running into errors, I don't know why.

So, let's start with momentum:
p^2 / 2m = E
p^2 = 2mE
p = sqrt(2mE)
h/λ = sqrt(2mE)
hk = sqrt(2mE)
k = sqrt(2mE)/h

So far so good. Now let's start with conserved Energy
E= ½ mv^2
2E/m = v^2
v = sqrt(2E/m)

So, the angular velocity is :
ω = kv
ω = sqrt(2mE)/h *. sqrt(2E/m)
ω = 2E/ h
E = ½ hω

This is weird. Where did the ½ come from ?

I thought its E = hω.

Can someone tell me where I went wrong ? I'm pretty sure the algebra is correct, but am I introducing some concept where I shouldn't ? Please let me know !

 

[BvU]

Hi,

Is this for billiard balls or for marbles ?

Can you typeset your equations ? It is difficult to deciper your $\hbar$ invention.

Any context ?

(guidelines point 7)

Check out E and p http://depts.washington.edu/jrphys/ph248S16/PhotoEffEqu-16.pdf

 

[Nugatory]

Can someone tell me where I went wrong ?

https://www.physicsforums.com/threads/de-broglie-relations-confusion.987026/

 

[Demystifier]

Summary:: I've been trying to derive ω = E/hbar, but to no success. I thought it was fairly straight forward derivation, could some one point out my mistake ?

So, the angular velocity is :
ω = kv

That was the mistake. What made you think that $\omega=kv$? It's true for photons (with $v=c$), but in general it's not true.

Your confusion is also closely related to another frequent confusion about the formulas $E=mc^2$ and $E=mv^2/2$. Can you explain why only one of them has the factor $1/2$?