Double Slit - any interference?

[ZedCar]

Homework Statement

Can wave-like behaviour, eg interference or diffraction, be observed with the following?

Electrons with a velocity of 20 m/s passing through a double slit with a separation of 2nm

Homework Equations

The Attempt at a Solution

The solution is given in the book.

λ = 2.65 x 10^-35 m

Then it states this is too large to show interference or diffraction.

But what I'm wondering is, if the double slit separation distance and the wavelength measurement is known, can one simply know straight away that there is no interference or diffraction if the double slit separation distance is smaller than the wavelength.

As is the case here.

 

[TSny]

The solution is given in the book.

λ = 2.65 x 10^-35 m

That's not the correct answer. Note that 20 m/s is the velocity, not the momentum.

Then it states this is too small to show interference or diffraction.

See if you can show that λ will be too big to show double-slit interference.

 

[ZedCar]

That's not the correct answer.

Sorry. Don't know why I wrote that number.

It should be λ = 3.64 x 10^-5 m

Does that look better?

So the fact that λ is larger than the double slit separation distance, does that automatically mean there is no interference?

 

[TSny]

Yes, that look's good.
Do you know the formula for calculating the maxima or minima of a double slit?

 

[ZedCar]

Do you know the formula for calculating the maxima or minima of a double slit?

MAX
d sin α = k λ
d ... spacing between slits
α ... angle
k ... order of the maximum (0, 1, 2, ...)
λ ... wavelength

MIN
d sin α = (k + ½) λ
d ... spacing between slits
α ... angle
k ... order of the minimum (0, 1, 2, ...)
λ ... wavelength

 

[TSny]

MAX
d sin α = k λ

Good. Divide both sides by d to solve for sin α. See if you can find the angle α to the first-order maximum.

 

[ZedCar]

Good. Divide both sides by d to solve for sin α. See if you can find the angle α to the first-order maximum.

For k, I made k=1

Then attempted to solve for both max and min, but got an error message.

Do the error messages indicate there is no interference?

 

[TSny]

Think about why you got an error. What is the maximum value that sin α can possibly have? (This is just a question about the properties of the sine function.) But then, if λ is greater than d, what can you say about the value of kλ/d for k = 1, 2, 3,...

Note that in this problem, λ is almost 20,000 times larger than d!

 

[ZedCar]

Think about why you got an error. What is the maximum value that sin α can possibly have? (This is just a question about the properties of the sine function.) But then, if λ is greater than d, what can you say about the value of kλ/d for k = 1, 2, 3,...

Note that in this problem, λ is almost 20,000 times larger than d!

Well, sin a can equal, at most, 1.

And if λ is greater than d this would imply sin a is greater than 1, which it cannot be.

So, with sin a equalling kλ/d is greater than 1, this means there can be no interference.

 

[TSny]

Yes, if λ > d then the first-order maximum will not occur (and so neither will the second, third, or higher maxima occur). Likewise, by considering the formula for minima, you can show that if λ > 2d, then no minima will occur. In your case λ is really huge compared to d or 2d. So, there would be no interference fringes appearing.

 

[ZedCar]

Thanks very much TSny !