De Broglie Relations Confusion

[lekh2003]

I've been attempting to run through some quantum mechanics and I've seen something extremely odd, and I just can't spot my mistake.

I know the relationships: $p = \frac{h}{\lambda}$ and $E = hf$. I also know the relationship $E = \frac{p^2}{2m}$.

I tried to show using the energy-momentum relation and the de Broglie wavelength equation that $E = hf$ is true, but its simply inconsistent.

\begin{align}
E &= \frac{p^2}{2m}\\
&= \frac{(\frac{h}{\lambda})^2}{2m}\\
&= \frac{(\frac{hf}{v})^2}{2m}\\
&= \frac{\frac{E^2}{v^2}}{2m}\\
&= \frac{E^2}{2mv^2}\\
&= \frac{E^2}{2(\frac{h}{\lambda})v}\\
&= \frac{E^2}{2E}\\
&= \frac{E}{2}
\end{align}

Where am I messing this up?

 

[PeroK]

You've used the phase velocity: $v_p = f\lambda = \frac{v}{2}$.

In the de Broglie mechanics it's the group velocity that is the velocity of the particle: $v_g = v$.

 

[lekh2003]

Thank you so much! Everything makes sense now, I read up some more on a StackExchange thread.

 

[vanhees71]

In other words you need to use the correct dispersion relation for free Schrödinger waves, which results from
$$E=\hbar \omega = \frac{p^2}{2m}=\frac{(\hbar k)^2}{2m},$$
i.e.,
$$\omega = \frac{\hbar}{2m} k^2.$$
In your notation it's
$$2 \pi f=\frac{h}{4 \pi m} \frac{(2 \pi)^2}{\lambda^2} \; \Rightarrow \; f=\frac{h}{2m \lambda^2}.$$

 

[iosman001]

The neighborhood of an event includes points that are timelike, null, and spacelike separated from it; the light cone does not bound the neighborhood. The topology of spacetime is still $R^4$ even though the metric on it is not the one induced by that topology. Thanks

 

[Nugatory]

The neighborhood of an event includes points that are timelike, null, and spacelike separated from it; the light cone does not bound the neighborhood. The topology of spacetime is still $R^4$ even though the metric on it is not the one induced by that topology. Thanks

Did you attach this post to the wrong thread? It looks off-topic here.

 

[iosman001]

The neighborhood of an event includes points that are timelike, null and spacelike separated from it; the light cone does not bound the neighborhood. The topology of spacetime is still R4R4R^4 even though the metric on it is not the one induced by that topologyhttps://chatavenue.vipThanks

thanks my issue has been fixed.