De Broglie wavelength of a train

[ConorDMK]

Homework Statement

Imagine you are traveling in a train, and see in the distance how the train path splits into two parallel tracks separated by a distance d.1) Obtain an analytical expression of the de Broglie wavelength of the train.

2) What condition would you use to determine at what point quantum effects become important (i.e. when the interference pattern could be observed)? Remember to discuss the assumptions you have made, and the meaning of these quantum effects, i.e. think on what you would have to observe

3. Using this condition, estimate the velocity the train must be going to
reach this quantum regime. Hint: plug reasonable numbers for the
mass and distance between tracks. What can you conclude?

Homework Equations

I'm not entirely sure with these

1) p=mv
p is momentum, m is mass and v is velocity

KE=p²/(2m)

KE is kinetics energy

λ=h/p=h/(2m(KE))^0.5

λ is the de Broglie wavelength, h is Planck's constant

ΔdΔp≥0.5ħ

d is the distance between the two parallel tracks, h bar is the reduced Planck's constant

The Attempt at a Solution

1)
I decided to simplify the situation down to a two-slit experiment, with the train as a particle and the split in the tracks as two slits. I also decided to place a detector wall at distance 'L' from the slits so that the interference pattern could be seen with the separation of the maxima being 'x' so that

λ/d=x/L

As above, I said that the train has a momentum,

p=mv=(2m(KE))^0.5

so that its de Broglie wavelength is

λ=h/p=h/(2m(KE))^0.5

I stated (Possibly incorrectly) that ΔdΔp≥0.5ħ and that Δd≈0.5d
⇒ 0.5dΔp≥0.5ħ
⇒ Δp≥ħ/d
⇒ 2πdΔp≥h

and so that λ≈(2πdΔp)/p
or by order of magnitude λ≈(dΔp)/p

2)
The interference pattern will be observable when the wavelength of light incident of the train is greater than the separation of the slits (Tracks). The train will then be able to behave as a wave of probabilities and cause an interference pattern that will be picked up be the detector wall.

But the interference pattern may not be observable, because the maxima separation may be too small to be observable,

λ/d=x/L≈Δp/p

so the train must be traveling slow enough for the interference pattern to be observable; as the momentum decreases, the maxima separation increases, providing Δp, d and L remain constant.

3)
I have not yet attempted this, as it requires parts 1 and 2 to be completed and correct.

 

[ConorDMK]

No it's ok, I just forgot one little thing from AS Physics, but I have done it now.