Problem (with solution) about Heisenberg's microscope

[s3a]

Homework Statement

The problem along with its solution are included as ProblemSolution.jpg.

Homework Equations

(Eq. 1) Δx ≈ λ/(sinθ)
(Eq. 2) Δp_x ≈ 2h(v/c)(sinθ) = 2(h/λ)(sinθ)
(Eq. 3) Δx Δp = ħ/2

The Attempt at a Solution

I am confused about several things. Firstly, geometrically, what is Δx and λ? I ask because I am trying to make mathematical sense of the Δx ≈ λ/(sinθ) equation rather than just memorizing it. I am not looking for any rigorous proof or anything unnecessarily challenging but just an intuitive derivation.

Secondly, why does introducing the Δp_x ≈ 2h(v/c)(sinθ) = 2(h/λ)(sinθ) equation help in answering the question; that is, how does this equation “show that if we minimize Δx by reducing λ, this will result in a loss of information about the x-component of the electron momentum?”

Also, if the electrons and photons are in 100% horizontal (x-axis) motion, then why don't they simply slow down in the same direction or move in the opposite direction (still on the x-axis)? (The diagram implies that photons will move along the arrow-lines that point toward the lens.)

I also don't understand (1.5.3). Is (1.5.3) supposed be something like (Eq. 3) from above?

I would really like to understand this problem inside-out so any help in figuring this problem out will be greatly appreciated!

If more information is needed, just ask.

Thanks in advance!

 

[anathema]

Thank you for your questions. Let me try to address them one by one.

1. Δx and λ are both measures of distance. Δx represents the uncertainty in the position of the electron, while λ represents the wavelength of the photon. In the context of this problem, Δx is related to the diffraction pattern of the electron as it passes through the slit, while λ is related to the wavelength of the light used to illuminate the slit. The equation Δx ≈ λ/(sinθ) is derived from the geometry of the diffraction pattern and the relationship between the slit width and the distance between the slits.

2. The equation Δp_x ≈ 2h(v/c)(sinθ) = 2(h/λ)(sinθ) is important because it relates the uncertainty in the x-component of the electron's momentum (Δp_x) to the uncertainty in the position of the electron (Δx). This equation is derived from the uncertainty principle, which states that the product of the uncertainties in position and momentum must be greater than or equal to a certain value (ħ/2). Therefore, by minimizing Δx (i.e. reducing λ), we are increasing Δp_x, which means we have less information about the x-component of the electron's momentum.

3. When the electrons and photons are in 100% horizontal motion, they are still subject to the laws of diffraction and interference. This means that, even though they are moving in the same direction, they can still interfere with each other and produce a diffraction pattern. The arrow-lines in the diagram represent the paths of the photons after they have passed through the slit and have been diffracted.

4. (1.5.3) is not meant to represent a specific equation, but rather a general statement about the relationship between the uncertainty in position (Δx) and the uncertainty in momentum (Δp). This relationship is described by the uncertainty principle, which is given by (Eq. 3) in your Homework Equations.

I hope this helps to clarify the problem for you. Please let me know if you have any further questions.