# Double Slit - any interference?

1. Oct 6, 2012

### ZedCar

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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.

Last edited: Oct 6, 2012
2. Oct 6, 2012

### TSny

That's not the correct answer. Note that 20 m/s is the velocity, not the momentum.
See if you can show that λ will be too big to show double-slit interference.

3. Oct 6, 2012

### ZedCar

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?

Last edited: Oct 6, 2012
4. Oct 6, 2012

### TSny

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

5. Oct 6, 2012

### ZedCar

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

6. Oct 6, 2012

### TSny

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

7. Oct 6, 2012

### ZedCar

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

Do the error messages indicate there is no interference?

8. Oct 6, 2012

### 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!

9. Oct 6, 2012

### ZedCar

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.

10. Oct 6, 2012

### 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.

11. Oct 6, 2012

### ZedCar

Thanks very much TSny !