How can a single slit diffraction create an interference pattern?

[lioric]

I explained that Huygens principle states that each point on the wave front act as a point source which produces spherical waves which produce the interference pattern.

Now his question is that where are these points and wouldn't there be infinite number of points on each wave front creating infinite number of spherical waves? If so wouldn't the interference pattern be negligible since there are so many of them.
He said that inorder to make an interference pattern, shouldn't there be a finite number of point source with a given space.

I sort of get what he's saying so I hand this question to you.
Please help me help him.

 

[PeroK]

Summary:: I have an extremely smart student who asks about things that I sometimes cannot explain because he is of much higher intellect.
He asked this to me and was asking a much deeper explanation.

I explained that Huygens principle states that each point on the wave front act as a point source which produces spherical waves which produce the interference pattern.

Now his question is that where are these points and wouldn't there be infinite number of points on each wave front creating infinite number of spherical waves? If so wouldn't the interference pattern be negligible since there are so many of them.
He said that inorder to make an interference pattern, shouldn't there be a finite number of point source with a given space.

I sort of get what he's saying so I hand this question to you.
Please help me help him.

In general, an infinite number of points can be modeled using a density function and an integral, that being the limit of a sequence of finite sums.

That the length of each interval tends to zero doesn't imply that the limit of the sum tends to zero.

 

[jtbell]

To make an analogy, a line consists of an infinite number of points. Even though each point has zero length, the line has nonzero length. More properly, we can consider the line as a very large number of very short segments whose lengths add up to the length of the line. This remains true even as the number of segments increases towards infinity and the length of each segment approaches zero.

Similarly, in single-slit diffraction, we can consider a line running across the width of the slit as a very large number of very short segments, each of which contributes a very small amount to the amplitude of the wave that arrives at some point on a screen, some distance from the slit. As the number of segments increases towards infinity, and the length of each segment approaches zero, the total amplitude at the screen from all the segments approaches a certain value.

In general, this procedure is the basis of integral calculus, which must be used to calculate single-slit diffraction properly.

 

[lioric]

Ye

To make an analogy, a line consists of an infinite number of points. Even though each point has zero length, the line has nonzero length. More properly, we can consider the line as a very large number of very short segments whose lengths add up to the length of the line. This remains true even as the number of segments increases towards infinity and the length of each segment approaches zero.

Similarly, in single-slit diffraction, we can consider a line running across the width of the slit as a very large number of very short segments, each of which contributes a very small amount to the amplitude of the wave that arrives at some point on a screen, some distance from the slit. As the number of segments increases towards infinity, and the length of each segment approaches zero, the total amplitude at the screen from all the segments approaches a certain value.

In general, this procedure is the basis of integral calculus, which must be used to calculate single-slit diffraction properly.

Yes he understands that. But he is saying that if there is infinite points that would create infinite point sources.
I think he's saying that so many of these points exist that the interference cannot occur.
Could you dumb down the explanation a little bit so I could follow
But I sort of get what your saying

 

[Dr_Nate]

Ye

Yes he understands that. But he is saying that if there is infinite points that would create infinite point sources.
I think he's saying that so many of these points exist that the interference cannot occur.
Could you dumb down the explanation a little bit so I could follow
But I sort of get what your saying

We never actually go to infinity. Mathematicians can say a lot about the different types of infinities, but in physics we just say something like it's uncountable or undefined or not a number. Instead, we say we go to the limit of infinity, which means things are still finite. A huge number of things with tiny but different magnitudes still adds up to a finite number.

 

[PeroK]

We never actually go to infinity. Mathematicians can say a lot about the different types of infinities, but in physics we just say something like it's uncountable or undefined or not a number. Instead, we say we go to the limit of infinity, which means things are still finite. A huge number of things with tiny but different magnitudes still adds up to a finite number.

Calculus, both differential and integral, involves limits. And, I wouldn't say that's part of "finite" mathematics. And, space and time are mapped to the set of real numbers, which is uncountably infinite.

 

[lioric]

We never actually go to infinity. Mathematicians can say a lot about the different types of infinities, but in physics we just say something like it's uncountable or undefined or not a number. Instead, we say we go to the limit of infinity, which means things are still finite. A huge number of things with tiny but different magnitudes still adds up to a finite number.

I understand
Thank you very much

Could guys sum this up to what I can say to his doubts?
At the end of tonight's lesson he read about photons and he said "that makes sense.
The interference were two photons apart. I can sort of understand it now"

By any chance could you guys see the issue he's having here.
Now this kid is not a normal kid, he has a limited vision, and is extremely gifted. So I find it that he was also having difficulty exact problem he's having. But the fact that he could settle for " two photons apart, could help you get why he's having this confusion

 

[PeroK]

I understand
Thank you very much

Could guys sum this up to what I can say to his doubts?
At the end of tonight's lesson he read about photons and he said "that makes sense.
The interference were two photons apart. I can sort of understand it now"

By any chance could you guys see the issue he's having here.
Now this kid is not a normal kid, he has a limited vision, and is extremely gifted. So I find it that he was also having difficulty exact problem he's having. But the fact that he could settle for " two photons apart, could help you get why he's having this confusion

To be honest, I can't make much sense of this.

If the kid is smart, then you're not going to be able to gloss over concepts like a continuous distribution of points without a solid knowledge of calculus yourself.

What was wrong with what @jtbell said in post #3? That's he best summary so far.

 

[lioric]

To be honest, I can't make much sense of this.

If the kid is smart, then you're not going to be able to gloss over concepts like a continuous distribution of points without a solid knowledge of calculus yourself.

What was wrong with what @jtbell said in psot #3? That's he best summary so far.

Kid is smart but this just a high school kid.
Some of the knowledge that he's getting is new.
So he's trying fit it together with the already existing knowledge

 

[lioric]

Thank you for all your effort and words
I seem to have found an answer now

 

[sophiecentaur]

But he is saying that if there is infinite points that would create infinite point sources.

And he would be perfectly correct. The sum of contributions from an infinite number of wave sources is NOT a continuum (despite some intuitive idea that it must be). Huygens showed this long before the Leibnitz (standard) form of Calculus was introduced - you can do it yourself graphically of the idea appeals, starting with a small number of sources and then adding to the spaces in between. This is really not a matter of 'faith' that Maths produces good models of reality; we're well beyond that. The OP, as has been stated, must be smart and he (she?) is surely smart enough to deal with the ideas of Calculus and accept that, later, it will all fit into place.

We never actually go to infinity.

Actually we validly predict what will happen if we did.
But there are some instances where it is a valid thing to do and others where it is not. Introductions to Calculus only involve examples where it's "allowed' and a lot of Science involves these. If you have 'smooth' graphs then basic Calculus can work. This link is just one example (Radio Waves behave just like light) which shows the way it can work.

 

[Dr_Nate]

We never actually go to infinity.

Actually we validly predict what will happen if we did.

I'm sorry but I've got to disagree. This sounds like you are suggesting that you can know the answer to something that it is logically impossible to know the answer to. You can't get to the the end of something that is endless.

When we say we approach infinity, we don't mean we are approaching the non-number infinity. The "limit of infinity" or "approaching infinity" are misnomers. They are misnamed because we don't actually mean to approach a non-number. We mean that we approach an arbitrarily large finite number.

 

[sophiecentaur]

Limit to infinity is dodgy but limit as delta x approaches zero is the whole basis of Calculus. And, where appropriate, Calculus surely justifies itself.

 

[PeroK]

I'm sorry but I've got to disagree. This sounds like you are suggesting that you can know the answer to something that it is logically impossible to know the answer to. You can't get to the the end of something that is endless.

When we say we approach infinity, we don't mean we are approaching the non-number infinity. The "limit of infinity" or "approaching infinity" are misnomers. They are misnamed because we don't actually mean to approach a non-number. We mean that we approach an arbitrarily large finite number.

This is not right. Mathematically a limit is not a process. Although this is a common misconception. The limit of a sequence or infinite series is, in general, a number that cannot be attained by any finite process. For example:
$$\lim_{n \rightarrow \infty} (\frac 1 2)^n = 0$$
For no finite $n$ do we have $(\frac 1 2)^n = 0$. The limit does not "approach $0$" and $n$ does not "approach" infinity. The limit is a real number in this case, and that real number is $0$. The whole expression is rigorously defined by:
$$\forall \epsilon > 0, \ \exists N: \ n > N \ \Rightarrow \ |(\frac 1 2)^n| < \epsilon$$
Note that this does not define a process, and it employs only inequalities involving real numbers.

This is one of the foundations of real analysis and how the concept of an infinite sequence or series can be rigorously defined.

PS In general
$$\lim_{n \rightarrow \infty} a_n = L$$
Means precisely that:
$$\forall \epsilon > 0, \ \exists N: \ n > N \ \Rightarrow \ |a_n - L| < \epsilon$$
All of calculus is build on this foundation.

 

[Dr_Nate]

This is not right. Mathematically a limit is not a process. Although this is a common misconception. The limit of a sequence or infinite series is, in general, a number that cannot be attained by any finite process. For example:
$$\lim_{n \rightarrow \infty} (\frac 1 2)^n = 0$$
For no finite $n$ do we have $(\frac 1 2)^n = 0$. The limit does not "approach $0$" and $n$ does not "approach" infinity. The limit is a real number in this case, and that real number is $0$. The whole expression is rigorously defined by:
$$\forall \epsilon > 0, \ \exists N: \ n > N \ \Rightarrow \ |(\frac 1 2)^n| < \epsilon$$
Note that this does not define a process, and it employs only inequalities involving real numbers.

This is one of the foundations of real analysis and how the concept of an infinite sequence or series can be rigorously defined. The whole of calculus is build on this foundation.

PS In general
$$\lim_{n \rightarrow \infty} a_n = L$$
Means precisely that:
$$\forall \epsilon > 0, \ \exists N: \ n > N \ \Rightarrow \ |a_n - L| < \epsilon$$

I think for the most part we agree. We don't need to approach anything to find these solutions.

What generally goes unstated when we use vocabulary like "approaches infinity" is that we shouldn't use an equality because that is a process that never reaches the RHS. It approaches it.

 

[jbriggs444]

Limit to infinity is dodgy but limit as delta x approaches zero is the whole basis of Calculus. And, where appropriate, Calculus surely justifies itself.

No. Both the limit toward infinity and limit toward zero are on equally sound theoretical footing. As @Dr_Nate points out, it is about approaching, not reaching.

As @PeroK points out, it is technically not even about "approaching".

Personally, I like the intuition suggested by "approaching", but do agree that it carries unwanted baggage that leads many students to an incorrect understanding.

 

[PeroK]

Personally, I like the intuition suggested by "approaching", but do agree that it carries unwanted baggage that leads many students to an incorrect understanding.

Yes, a limit is really a function of an infinite set. You can, in fact, discard any finite subset of a sequence and not change the limit. This is one of the paradoxes of the limit. You never actually get to a value of $n$ for which $a_n$ is relevant to the value of the limit!

Each individual $a_n$ is strangely irrelevant. No matter how large $n$ is.

 

[Nugatory]

At the end of tonight's lesson he read about photons and he said "that makes sense.
The interference were two photons apart. I can sort of understand it now"

Now he's moving away from understanding - photons don't work anything like how people imagine when they first hear about them, and a "two photons apart" model of the patter is terribly misleading.

He may not be able to resolve his concerns about the infinite number of points until he's learned calculus - it was invented to handle problems of this sort.