How Does Quantum Uncertainty Affect Nucleon Kinetic Energy and Electron Orbits?

[Odyssey]

Hello everybody, I got two questions on my assignment that I am stuck on...it'd be great if you guys can give me some hints to get me in the right direction.

1) Using the uncertainty principle, find the minimum value (in MeV) of the kinetic energy of a nucleon confined within a nucleus of radius $$R=5x10^-15 m.$$

2) Show that the electron orbits in the semi-classical Bohr model are not real. Do this by showing that any attempt to measure the orbit radius to an accuracy $$\Delta{x}<<R_{n+1}-R_{n}$$ is the radius of the electron in the hydrogen atom, would cause an uncertainty in the energy $$E_{n}$$ which is larger than the binding energy in that orbit. (Hint: This problem requires that you make suitable approximations).

Of the two questions, I am more desperate for number 2 ...I don't know where to begin!

 

[Gokul43201]

According to the posting guidelines, we can't help you unless you start.

What do you know about the Uncertainty Principle?

 

[Odyssey]

About the uncertainty principle...I just know the basics...that if $$\Delta{x}$$ and $$\Delta{p}$$ are the uncertainties in position and momentum, respectively, then their product must be at least $$\hbar /2$$. Do I set the radius equal to $$\Delta{X}$$??

 

[ccesare]

Correct, you would set the radius to $$\Delta x$$.

If you solved the uncertainty principle for the minimum momentum in this case, you could easily find the minimum kinetic energy.

 

[Odyssey]

Thank you for the help. Here's what I got after setting the radius equal to $$\Delta{x}$$...please check my work. :)

$$\Delta{p}\geq{\hbar /2\Delta{x}}$$
$$\Delta{p}\geq{(1.05*10^-34Js)/(2)(5*10^-15m))}$$
$$\Delta{p}\geq{1.05*10^-20kgm/s}$$

Then I used the minimum value of momentum and the mass of the nucleon to find the minimum kinetic energy...
$$E_{k}\geq{p^2/2m}$$
$$E_{k}\geq(1.05*10^-20kgm/s)^2/(2)(3.34*10^-27kg)$$
$$E_{k}\geq1.65*10^-14J$$

 

[ccesare]

$$\hbar$$ should be $$1.05*10^{-34} Js$$

 

[Odyssey]

yes, that was a typo there. $$\hbar$$ should be $$1.05*10^{-34} Js$$. The final value I get is $$E_{k}\geq1.65*10^-14J$$.