Homework Help: Velocity of neutron using uncertainty principle

1. Apr 27, 2015

Avatrin

1. The problem statement, all variables and given/known data
A neutron in the nucleus of an atom can move in a range which is about five femtometers long. Use Heisenberg's uncertainty principle to calculate what velocities one can expect to measure.

2. Relevant equations
$$\sigma_p \sigma_x \geq \frac{\hbar}{2}$$
$$p = \hbar k$$
Probably others as well. I am quite sure this problem is non-relativistic since other problems in the same problem set specify that the particles in those problems are supposed to be moving in non-relativistic speeds.
3. The attempt at a solution
I define the center of the area where the particle can move to be x = 0, and I assume the problem is one-dimensional (it is not specified, and not obvious from the text alone). So, the likelihood of finding the particle in $(-\sigma_p,\sigma_p)$ is 64%. Since I am not given any equation, I say that $\sigma_p = 1fm$ sounds reasonable. Then I get:
$$\sigma_p \geq \frac{\hbar}{2.0 fm} = 5.27 * 10^{-20} Js/m$$
$$\frac{\sigma_p}{m_n} \geq \frac{\hbar}{2.0 fm * m_n} = 0.105c$$

Now, I am stuck. I do not even know the expectation value of momentum. How am I supposed to get the expectation value of velocity? I probably have to use de Broglie's relations somewhere, but I am not sure where.

Last edited: Apr 27, 2015
2. Apr 27, 2015

Buddhapus17

As far as I remember from doing a similar problem a while back, all you need to do is say that the uncertainty in position of your neutron is the diameter of your nucleus (5 fm as you said in problem description), and from there you can find the uncertainty in momentum p. Then you have to make an assumption that the uncertainty in momentum of a neutron is equal to the amount of momentum a neutron can have, so then you have that value. Finally you can say that momentum is equal to mv (using the non-relativistic formula) so if you know the mass of the neutron you're all set to find its velocity.

3. Apr 28, 2015

Avatrin

So, essentially, my answer multiplied by five is the actual answer. However, why is the assumption that the uncertainty in momentum is the neutrons momentum a sensible assumption?