Uncertainty Principle and the size of an atom

[nobahar]

Hey,
Sorry, but I have a qustion on the uncertainty principle to join the many others.
Just reading a book on physics, and its says that, as a result of the Heisenberg uncertainty principle, if the proton and electron were confined to the same volume of space, the electron would be traveling about 2,000 times faster, as the proton is 2,000 times bigger. How is this a consequence of the uncertainty principle? It must entail the momentum and the position, but I don't see how.
Thanks in advance for your responses.

 

[Naty1]

if the proton and electron were confined to the same volume of space, the electron would be traveling about 2,000 times faster, as the proton is 2,000 times bigger.

m (delta(v))(delta(x)) greater than h, says it all...

 

[gabbagabbahey]

First, it is incorrect to say that the electron will be traveling 2000 times faster; the correct statement would be that the uncertainty in the electron's speed is about 2000 times greater than the the uncertainty in the proton's speed.

By confining the proton and the electron to the same volume, you are essentially saying that the uncertainty in position is the same for both particles (i.e. $\Delta x_{\text{electron}}=\Delta x_{\text{proton}}$ )

So, if you assume that $\Delta x_e\Delta p_e=\frac{\hbar}{2}$ and $\Delta x_p\Delta p_p=\frac{\hbar}{2}$ then you have:

$$\Delta x_e\Delta p_e=\Delta x_p\Delta p_p$$

$$\implies \Delta p_{\text{electron}}=\Delta p_{\text{proton}}$$

Then you simply use the fact that $\Delta p_e=m_e\Delta v_e$ and that $\Delta p_p=m_p\Delta v_p$ (since you presumably know the masses of the electron and proton exactly, the uncertainty in their momenta is due entirely to the uncertainty in their speeds) and you get

$$\Delta v_e =\frac{m_p}{m_e} \Delta v_p \approx 2000\Delta v_p$$

 

[dave_baksh]

gabba> you mean v subscript p in your last line of working

 

[gabbagabbahey]

gabba> you mean v subscript p in your last line of working

Yes, thank you. I've edited my post.

 

[nobahar]

Thanks Gabba,
That's extremely clear and awesome.
Thanks!