Uncertainty - Finding Momentum and KE

[erok81]

Homework Statement

An electron is conned inside a cubic region of size an atomic radius L = 0.5 nm.

Homework Equations

What is the uncertainty in the momentum p of the electron? Remember that $$\Delta p \Delta x \approx h$$.

The Attempt at a Solution

I am confused when to use $$\Delta p \Delta x \approx h$$ or $$\Delta p \Delta x \approx \frac{\hbar}{2}$$.

So first I solved for p using $$\Delta p \Delta x \approx h$$ since that's what was suggested in the problem.

From the momentum I found there, I then solved for speed using $$\Delta p \Delta x \approx h$$ along with the electron's mass. This gave me a non-relativistic speed.

The next part of the problem reads: what is its minimum kinetic energy? Is the electron relativistic? For this I just used the regular KE=1/2mv².

Then I got thinking about the minimum KE as asked in the question and that is where my confusion came in. Should I be using the second equation instead? I read the wiki page for the Uncertainty Principle and that confirmed my worry that I should be using the second one.

My question is...is my logic for this problem correct (the way I have already solved it)? And when is one used over the other?

 

[anathema]

Your logic for solving the problem is correct, but you should indeed be using the second equation, \Delta p \Delta x \approx \frac{\hbar}{2}, when calculating the uncertainty in momentum.

The first equation, \Delta p \Delta x \approx h, is a simplified version of the uncertainty principle that is often used in introductory physics courses. It is an approximation that works well for macroscopic objects, but for microscopic particles like electrons, the second equation is more accurate. This is because the value of Planck's constant, h, is much larger than the reduced Planck's constant, \hbar, which is used in the second equation.

In general, the first equation is used when the particle's momentum is much larger than \hbar, and the second equation is used when the particle's momentum is on the order of \hbar or smaller. Since electrons have very small momenta, the second equation is more appropriate for this problem.

Therefore, to answer the question about the uncertainty in momentum, you should use the second equation, \Delta p \Delta x \approx \frac{\hbar}{2}, to calculate the uncertainty in momentum and then use the same equation to calculate the uncertainty in position. This will give you a more accurate value for the uncertainty in momentum, which you can then use to determine the minimum kinetic energy of the electron.

In summary, your logic for solving the problem is correct, but you should use the second equation, \Delta p \Delta x \approx \frac{\hbar}{2}, when calculating the uncertainty in momentum and position for a microscopic particle like an electron. The first equation, \Delta p \Delta x \approx h, is only an approximation and is more suitable for macroscopic objects.