How Can Quantum Mechanics Help Solve These Complex Homework Problems?

[DrMon]

Homework Statement

I've actually got a few questions here, some I think I've been reasonably successful in - others I don't know where to begin. I really am confused by some of this stuff, so a nudge in the right direction would be great. I know my lecturer will probably find and read this - (It's not your fault! I've been preoccupied with other subjects :( Probably too much so.)
On the other hand - this means that I will certainly be re-writing/calculating these solutions to I remain anonymous. In other words - If you take the time to help me with this I'll certainly be learning from you :)

Question 1 [20 marks] - Attempted solution below.
In one of the lectures, a recent experiment showing the interference of buckminsterfullerene (C60)
molecules using a double slit apparatus was discussed.
(a) If the average velocity of the C60 molecules is v = 200 m/s, calculate the de Broglie
wavelength for a C60 molecule.
(b) If the C60 particles are incident on two 50 nm slits with a center-to-center separation of 100
nm, what is the distance between the first two maxima deposited on a screen located 2 m
behind the slits. Given that the resolution limit of optical microscopy is comparable to the
wavelength, would you be able to resolve the maxima optically using this set-up?
(c) Suppose one wanted to attempt the Davisson-Germer experiment with C60, and planned to ionise the C60 to C60 2− and then use a potential difference to accelerate the C60 molecule to a velocity sufficient to achieve a wavelength λ = 1.65 Å. How large a potential difference V would be needed to do this?

Question 2 [10 marks]
You wish to study a crystal’s structure by diffracting thermal, non-relativistic neutrons from the
crystal, so you go to a nuclear reactor and set up your diffraction experiment on one of the neutron
beam-lines. The neutrons emerge from the reactor with a range of wavelengths (i.e., a range of
energies/momenta), so you use a ‘chopper’ to select neutrons of a certain wavelength.
The simplest chopper consists of two disks made from a highly neutron absorbing material,
mounted on an axle. The disks are 2 m apart. In each disk is a small slot (to let the neutrons
through). There is an offset of 10° between the two slots and the chopper rotates rapidly.
The distance between the planes of atoms in the diffracting crystal is 1.73Å and you observe strong
diffraction when the angle between the incident and diffracted neutron beams is 163°. Calculate the
minimum rotational speed of the chopper in revolutions per minute.
Question 3 [15 marks]
(a) Rayleigh’s criterion is used to determine when two objects are resolvable by a lens of
diameter d. The angular separation of the objects must be at least θ R where:
θ(subscript r) = 1.22*(λ/d)
In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5
cm, what energy (i) photons and (ii) electrons should be used?
Is this consistent with the uncertainty principle? Explain briefly and support with numerical
estimates if required.
(b) Assume that the uncertainty in the position of a particle is equal to its de Broglie wavelength.
Show that the uncertainty in its velocity is equal to or greater than 1/(4π) times its velocity.

Homework Equations

(included in the questions above)

The Attempt at a Solution

Question 1:

A) λ=h/p=h/mv=(6.626 068 96 *(10)^(-34))/(200*(720.66*1.660538782*(10)^(-27) ) )=2.768512669*(10)^(-12) Meters
B) x=(n*λ*L)/d=(n*2.768512669*(10)^(-12)*2)/(50*(10)^(-9) )
Sub in n=1 and n=2, and find the difference:
(2*2.768512669*(10)^(-12)*2)/(100*(10)^(-9) )-(1*2.768512669*(10)^(-12)*2)/(100*(10)^(-9) )=5.537025342*(10)^(-5) Meters

Given the limit of optical microscopy is comparable to the wavelength of this particular particle then it is reasonable to assume that you would be able to resolve the maxima optically using this setup as the distance is several orders of magnitude larger.

C) 1/λ=√((2*m*e*V) )/h
∴ V = (1/2)*h^2/(λ^2*m*e)=0.00004217267763volts


As for 2 and 3, I am genuinely stumped for the moment. I'll continue doing some more work on them as I wait for a reply, But I've been stuck on these for quite some time now. Please find it in your soul to help out an undergrad who screwed up his time management :)
These are due on thursday, so I've got a little time yet.

 

[AvroArrow]

Thanks a ton, Question 2: The angular difference between the two beams is 163°. Therefore, the time taken for one neutron to travel from one slit to the other is given by:t = (2mL)/(h*sin(163°)) = (2*2)/(6.626*10^(-34)*sin(163°)) = 2.16*10^(-13) sTherefore, the rotational speed of the chopper in revolutions per minute should be:ω = 2π/t = 1.45*10^13 rev/minQuestion 3: A) The angular separation of the objects must be at least θR where:θR = (1.22*λ)/dTherefore, for photons:λ = (1.22*4000*10^(-9))/5*10^(-2) = 9.744*10^(-7) mFor electrons:λ = (1.22*4000*10^(-9))/5*10^(-2) = 8.3*10^(-14) mThis is consistent with the uncertainty principle as the product of the uncertainties in position and momentum is greater than or equal to Planck's constant. Numerical estimates:For photons: ΔxΔp > h (9.744*10^(-7))*(6.626*10^(-34)) > 6.626*10^(-34)For electrons: ΔxΔp > h (8.3*10^(-14))*(6.626*10^(-34)) > 6.626*10^(-34) B) Δx = λ Δp = h/λ Δv = Δp/m = (h/λ)/m = (h/(h/λ))/m = 1/(4π)*v

 

[nlightner]

Hi there! It's great that you are reaching out for help with your quantum mechanics problems. I understand that these concepts can be confusing and it's important to seek clarification when needed. I'll do my best to provide some guidance and resources for you to continue working on these questions.

Question 1:

A) Your solution for calculating the de Broglie wavelength of a C60 molecule is correct. Just a small correction, the mass of a C60 molecule is 720.66 amu (atomic mass units), not 720.66 kg. So the de Broglie wavelength would be 2.768512669*(10)^(-12) meters.

B) Your solution for finding the distance between the first two maxima deposited on a screen located 2 m behind the slits is correct. And yes, it is reasonable to assume that the maxima can be resolved optically since the distance is several orders of magnitude larger than the wavelength.

C) For this part, you need to use the formula for kinetic energy, KE = (1/2)mv^2, and the relationship between kinetic energy and potential difference, KE = eV, where e is the elementary charge. So you can solve for V using the values given in the question.

Question 2:

For this question, you need to use the Bragg's Law, nλ = 2dsinθ, where n is the order of diffraction, λ is the wavelength of the neutron, d is the distance between the planes of atoms in the diffracting crystal, and θ is the angle between the incident and diffracted neutron beams. You can solve for the minimum rotational speed of the chopper using this formula and the given values.

Question 3:

A) For this part, you need to convert the given distance of 4000 nm to meters and use the given formula for Rayleigh's criterion. For photons, you can use the energy-wavelength relationship, E = hc/λ, where h is Planck's constant and c is the speed of light. For electrons, you can use the de Broglie wavelength formula, λ = h/mv, where m is the mass of the electron and v is its velocity. You can compare the values obtained to the uncertainty principle, which states that the product of the uncertainties in position and momentum must be greater than or equal to h/4π.

B) For this part, you