Velocity of neutron using uncertainty principle

In summary, the conversation discusses using Heisenberg's uncertainty principle to calculate the possible velocities of a neutron in the nucleus of an atom. The equation \sigma_p \sigma_x \geq \frac{\hbar}{2} is mentioned, as well as the assumption that the problem is non-relativistic. The participant also mentions using de Broglie's relations and making an assumption about the neutron's momentum to find its velocity.
  • #1
Avatrin
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Homework Statement


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.

Homework Equations


[tex]\sigma_p \sigma_x \geq \frac{\hbar}{2}[/tex]
[tex]p = \hbar k[/tex]
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.

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 [itex](-\sigma_p,\sigma_p)[/itex] is 64%. Since I am not given any equation, I say that [itex]\sigma_p = 1fm[/itex] sounds reasonable. Then I get:
[tex]\sigma_p \geq \frac{\hbar}{2.0 fm} = 5.27 * 10^{-20} Js/m [/tex]
[tex]\frac{\sigma_p}{m_n} \geq \frac{\hbar}{2.0 fm * m_n} = 0.105c [/tex]

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.
 
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  • #2
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
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?
 

FAQ: Velocity of neutron using uncertainty principle

1. What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

2. How is the uncertainty principle related to the velocity of a neutron?

The uncertainty principle tells us that the position and momentum of a particle cannot be known simultaneously. Since the velocity of a neutron is related to its momentum, the uncertainty principle applies to the velocity of a neutron as well.

3. How can the uncertainty principle be used to calculate the velocity of a neutron?

The uncertainty principle can be used to calculate the velocity of a neutron by determining the uncertainty in its position and momentum. By using the equation ΔxΔp ≥ h/4π (where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant), we can determine the minimum uncertainty in the velocity of a neutron.

4. What is the significance of knowing the velocity of a neutron using the uncertainty principle?

Knowing the velocity of a neutron using the uncertainty principle can provide valuable information about the behavior and properties of neutrons. This information can be used in various fields such as nuclear physics, material science, and astrophysics.

5. Are there any limitations to using the uncertainty principle to determine the velocity of a neutron?

Yes, there are limitations to using the uncertainty principle to determine the velocity of a neutron. The uncertainty principle gives us a minimum uncertainty in the velocity, but it does not tell us the exact value. Other factors such as experimental errors and the system's complexity can also affect the accuracy of the calculated velocity.

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