Solving for Energy & Momentum in Physics Questions

[anto]

Homework Statement

1)for an object of size 0.5 Angstrom, what is the longest-wavelength photon with which it can be observed?
2)for the object of problem 1, what is the smallest-energy electron which can be used to make the measurement?

Relevant Equations

∆E∆t = h/4(phi)
∆p∆x = h/4(phi)
∆λ = (∆E/E)λ

for the first question, i thougth that 0,5 A is the answer?

for the second question:
i used the E =hc/λ to found the E. but i got a little confused which equations to find ∆E, since there's no ∆t. or should i search the momentum, then use the λ= h/p ?

 

[phinds]

Your title "Uncertainty principle -- Using photons vs. electrons to observe a very small object" is confusing since the HUP has absolutely nothing to do with how things are observed. That is, the HUP is in no way a measurement problem, it is a feature of reality.

 

[Steve4Physics]

for the second question:
i used the E =hc/λ to found the E. but i got a little confused which equations to find ∆E, since there's no ∆t. or should i search the momentum, then use the λ= h/p ?

Hi @anto. Welcome to PF.

As already noted by @phinds, the question has nothing to do with Heisenberg’s uncertainty principle. It’s about applying the diffraction limit of ‘classical optics’.

Your answer to the first part is reasonable. You might want to add ‘the order of‘ or ‘approximately’ to your value. If you wanted a more precise value, you could use what is called the Abbe limit (e.g. see https://en.wikipedia.org/wiki/Diffraction-limited_system#The_Abbe_diffraction_limit_for_a_microscope).

But the questions seems badly worded, I think the real questions are these:

Q1. What is the longest wavelength [of any type of wave] which can be used to resolve points separated by 0.5 Angstrom?

Q2 What is the smallest kinetic energy of an electron which can be used to achieve this resolution?
________________
You have answered Q1. For Q2, note that E = hc/λ is for photon energy, not the kinetic energy of an electron.

Q2 asks for the electron’s kinetic energy which makes the electron’s deBroglie wavelength equal to 0.5 Angstrom (or whatever your answer to Q1 is).

Can you work out a formula for the electron’s kinetic energy (E) in terms of it mass (m), its deBroglie wavelength (λ) and Planck’s constant (h)?

Edit - typo' corrected.