Uncertainty in Double-Slit Diffraction

[mrjeffy321]

I am asked to show that it would be impossible to determine exactly which slit a single photon of light passed through in a double slit diffraction grating without destroying the resulting interference pattern since in order to do so, we would need to know both the momentum and position of the photon to such a degree not allowed by the Heisenberg Uncertainty principle (ΔxΔp would need to be much less than h-bar / 2).

I am not sure how I would go about showing this…I am some-what stuck as to where to start.

I know the momentum of the photon is: p = h / λ.
To get a interference pattern, the size of the slits need to be roughly on the same order as the wavelength of the light. If we know the slits are on the same order as the wavelength of the light and in order for a photon to pass through either one slit or the other we would need to know the photon’s position on this same scale. But what about momentum? We know the photon needs to be traveling in the direction of the screen (not backwards), but this is not a very constraining piece of knowledge.

Once I get a sense of the sizes of Δx and Δp, I need to show that this is << h-bar / 2. And then from the uncertainty principle we know this is impossible, so that means that we cannot know which slit the photon passes through without destroying the interference pattern.

 

[Jhero]

First, let's define the terms in the Heisenberg Uncertainty principle. Δx is the uncertainty in position and Δp is the uncertainty in momentum. h-bar is the reduced Planck's constant, which has a value of approximately 1.0545718 × 10^-34 joule seconds.

In order to show that it is impossible to determine exactly which slit a single photon of light passed through in a double slit diffraction grating without destroying the resulting interference pattern, we need to understand the concept of wave-particle duality. This concept states that particles, such as photons, can exhibit both wave-like and particle-like behavior. This means that a single photon can behave like a wave and interfere with itself, creating an interference pattern, or it can behave like a particle and pass through one of the two slits.

Now, let's consider the uncertainty in position and momentum of a single photon passing through the double slit. As you mentioned, the momentum of a photon is given by p = h / λ, where h is Planck's constant and λ is the wavelength of the light. The wavelength of light is on the order of nanometers, while the size of the slits is also on the order of nanometers. This means that Δx and Δp are both on the same order of magnitude.

According to the Heisenberg Uncertainty principle, the product of Δx and Δp must be greater than or equal to h-bar / 2. This means that in order to determine the position of the photon with a high degree of precision, Δx must be very small, which in turn means that Δp must be large. However, if Δp is large, it means that the uncertainty in momentum is also large, which makes it impossible to determine which slit the photon passed through.

In other words, in order to know the position of the photon with a high degree of precision, we need to know its momentum with a high degree of precision as well. But this violates the Heisenberg Uncertainty principle, as the product of Δx and Δp must be greater than or equal to h-bar / 2.

Therefore, it is impossible to know both the position and momentum of a single photon passing through the double slit diffraction grating with enough precision to determine which slit it passed through without destroying the interference pattern. This is due to the inherent uncertainty in the behavior of particles at the quantum level, as described by the