I Why Won't Observation in a 2 Slit Experiment Cause 1 Slit Diffraction?

[Flamel]

TL;DR Summary

In double-slit experiments it's said that observing a particle in a slit will result in two distinct bands. Why wouldn't there be single-slit diffraction?

From what I understand, if the two-slit experiment is performed while observing a slit for particles, two distinct bands appear rather than interfering. This is a little confusing, as, from what I understand, diffraction is caused by measuring a particle's position (i.e. using a slit to narrow down its position), which results in uncertainty in momentum and contributes to the interference pattern. If that is the case, why wouldn't even the slit without the detector show single-slit diffraction? Also, if diffraction is caused by momentum uncertainty, why would it result in bands where particles are absent; wouldn't that reduce momentum uncertainty?

 

[Paul Colby]

How does one "observe a slit for particles?" How measurements are performed matters especially in these types of discussions.

 

[Flamel]

How does one "observe a slit for particles?" How measurements are performed matters especially in these types of discussions.

In terms of detectors, some examples might be a device that detects the electrical charge from an electron or perhaps a filter that polarizes photons going through each slit in different directions, but the type of detector isn't necessarily specified when I've heard about this experiment.

 

[PeroK]

Summary:: In double-slit experiments it's said that observing a particle in a slit will result in two distinct bands. Why wouldn't there be single-slit diffraction?

From what I understand, if the two-slit experiment is performed while observing a slit for particles, two distinct bands appear rather than interfering. This is a little confusing, as, from what I understand, diffraction is caused by measuring a particle's position (i.e. using a slit to narrow down its position), which results in uncertainty in momentum and contributes to the interference pattern. If that is the case, why wouldn't even the slit without the detector show single-slit diffraction? Also, if diffraction is caused by momentum uncertainty, why would it result in bands where particles are absent; wouldn't that reduce momentum uncertainty?

There is single-slit diffraction.

 

[Dr_Nate]

There is single slit diffraction. One for each slit. The observed two band pattern, which is accumulated over many particles, is just the addition of the individual single slit diffraction patterns.

Also, if diffraction is caused by momentum uncertainty, why would it result in bands where particles are absent; wouldn't that reduce momentum uncertainty?

I don't understand this question.

 

[vanhees71]

Well, indeed to get a double-slit refraction pattern you need coherence, i.e., pretty well-defined momenta of the particles. The wave function must have large enough spatial spread to cover the two slits and the screen must be far enough away such that the partial waves from each slit (think about the diffraction qualitatively as the superposition of two waves originating from each slit in the sense of Huygen's principle) overlap. If you put the screen close enough to the double slit you can still see from which slit each particle orinates, and since there is no overlap of the partial waves you don't see double-slit interference. This is the most simple example, why you either see double-slit interference or have which-way information or any degree of something in between, i.e., an interference pattern with less than maximal possible contrast (for an ideally coherent wave and infinitely far observation screen (Fraunhofer conditions) between 0 and maximum intensity in the forward direction) and more or less uncertain which-way information. The less certain the which-way information is the sharper the interference pattern becomes and vice versa.

It's in some sense the paradigmatic example for, what Bohr called in his enigmatic way "complementarity". You can either have a very well determined momentum (if you have a sharp contrast interference pattern you can very well determine the de Broglie wave length and thus momentum through $p=2 \pi \hbar/\lambda=h/\lambda$) or you decide to get precise which-way (position) information about the electron but then the interference pattern is less sharp and thus the de Broglie wavelength can be only determined with pretty large uncertainties and thus the momentum is not very accurately determined. As (I think) Bohr put it: You can either look with the momentum eye at the particle or with the position eye but never with both eyes at the same time ;-).

 

[Paul Colby]

In terms of detectors, some examples might be a device that detects the electrical charge from an electron or perhaps a filter that polarizes photons going through each slit in different directions, but the type of detector isn't necessarily specified when I've heard about this experiment.

Well, placing a detector blocking one slit changes the experiment. Detecting the passing charge transfers some energy or momentum to said charge. Once more the two slits experiment is altered with each measurement strategy. Bohr and Einstein went around and around on this back in the 30's. Bohr would always come back with how the measurement interferes changing the result. Einstein lost this debate definitively.

 

[Dr_Nate]

As (I think) Bohr put it: You can either look with the momentum eye at the particle or with the position eye but never with both eyes at the same time ;-).

Well you can actually. Just don't squint too hard with both eyes at the same time. ;p

 

[Flamel]

Well, indeed to get a double-slit refraction pattern you need coherence, i.e., pretty well-defined momenta of the particles. The wave function must have large enough spatial spread to cover the two slits and the screen must be far enough away such that the partial waves from each slit (think about the diffraction qualitatively as the superposition of two waves originating from each slit in the sense of Huygen's principle) overlap. If you put the screen close enough to the double slit you can still see from which slit each particle orinates, and since there is no overlap of the partial waves you don't see double-slit interference. This is the most simple example, why you either see double-slit interference or have which-way information or any degree of something in between, i.e., an interference pattern with less than maximal possible contrast (for an ideally coherent wave and infinitely far observation screen (Fraunhofer conditions) between 0 and maximum intensity in the forward direction) and more or less uncertain which-way information. The less certain the which-way information is the sharper the interference pattern becomes and vice versa.

It's in some sense the paradigmatic example for, what Bohr called in his enigmatic way "complementarity". You can either have a very well determined momentum (if you have a sharp contrast interference pattern you can very well determine the de Broglie wave length and thus momentum through $p=2 \pi \hbar/\lambda=h/\lambda$) or you decide to get precise which-way (position) information about the electron but then the interference pattern is less sharp and thus the de Broglie wavelength can be only determined with pretty large uncertainties and thus the momentum is not very accurately determined. As (I think) Bohr put it: You can either look with the momentum eye at the particle or with the position eye but never with both eyes at the same time ;-).

In that case, what would happen if the position of an electron with a wavelength of 500 nm going through a pair of 2 micrometer slits had its position measured with an uncertainty of 1 micrometer before passing through the slits? Would the interference pattern eventually disappear if you measured the position correctly before an electron enters the slits? Also, based on what you said, does that mean that momentum certainty must be high for diffraction, in contrast to what I said earlier?

 

[PeroK]

In that case, what would happen if the position of an electron with a wavelength of 500 nm going through a pair of 2 micrometer slits had its position measured with an uncertainty of 1 micrometer before passing through the slits? Would the interference pattern eventually disappear if you measured the position correctly before an electron enters the slits? Also, based on what you said, does that mean that momentum certainty must be high for diffraction, in contrast to what I said earlier?

You mean by sending the beam of electrons through a preliminary single slit?

 

[Flamel]

You mean by sending the beam of electrons through a preliminary single slit?

That could be one possible method of determining location, but one could also detect position via detecting the electrical field.

 

[Flamel]

There is single slit diffraction. One for each slit. The observed two band pattern, which is accumulated over many particles, is just the addition of the individual single slit diffraction patterns.I don't understand this question.

Do you mean that the single-slit diffraction patterns are there, but are hard to notice or aren't generally talked about when discussing the double-slit experiment? In terms of my question about momentum, from what I was told before, it seems that the uncertainty in momentum from measuring the particle position via the slit plays a role in diffraction. This seems unusual to me since I would imagine more uncertainty in momentum would result in broader regions where the particle can travel, since areas without any particles would rule out certain momenta and make the momentum more certain if I'm not mistaken. Am I misunderstanding something?

 

[Dr_Nate]

Do you mean that the single-slit diffraction patterns are there, but are hard to notice or aren't generally talked about when discussing the double-slit experiment? In terms of my question about momentum, from what I was told before, it seems that the uncertainty in momentum from measuring the particle position via the slit plays a role in diffraction. This seems unusual to me since I would imagine more uncertainty in momentum would result in broader regions where the particle can travel, since areas without any particles would rule out certain momenta and make the momentum more certain if I'm not mistaken. Am I misunderstanding something?

In your original post you consistently mentioned observing the particle as it went through one slit. In my comment, I was working off that premise.

If you measure the particles going through one slit, then each particle's wave function will have been localized at one of the two slits. When your detector screen (not the detector at the slit) collects many data points, you will see two single-slit bands. If, however, you don't detect the particle going through the slits, the wave function goes through both and interferes with itself. In this case, you will see the interference pattern.

I agree that diffraction is affected by momentum. Localizing the particle in the plane of the slit is going to cause a spread in momentum of its wave function in that plane. As we allow it to evolve with time after the slit, it will spread out more in that plane than if the slit was wide.

I think you might be confused by the term: bands. These are not taken to mean the particles will fall only on the silhouette, or shadow, of the slits on the detector as if they were traveling like rays. In the case of single-slit diffraction, when we say bands we mean that it is a broad region that has a peak centered on the silhouette of the slit but this broad region where particles land extends out beyond this gradually decreasing in intensity.

 

[PeroK]

Do you mean that the single-slit diffraction patterns are there, but are hard to notice or aren't generally talked about when discussing the double-slit experiment? In terms of my question about momentum, from what I was told before, it seems that the uncertainty in momentum from measuring the particle position via the slit plays a role in diffraction. This seems unusual to me since I would imagine more uncertainty in momentum would result in broader regions where the particle can travel, since areas without any particles would rule out certain momenta and make the momentum more certain if I'm not mistaken. Am I misunderstanding something?

The simple analysis of single-slit diffraction is that the electron gets "measured" by the slit and owing to the uncertainty principle picks up an uncertainty in lateral momentum as a result. For example, if you start with a wide slit and gradually narrow the slit, then initially the beam gets narrower according to the width of the slit. Until the width of the slit is narrower than the original beam nothing changes. Then, the beam gets narrower simply according to the width of the slit. But, once the width of the slit becomes sufficiently narrow - much narrower than the width of the original beam - the beam starts to spread out and he narrower the slit the more the beam spreads out after the slit.

A slightly more detailed analysis is that electron's wavefunction is spread across a range wider then the slit. This wavefunction, if allowed to evolve, will gradually spread out further. Mathematically, you could model the electron as a moving Gaussian wave packet, which is gradually spreading out in all three directions, relative to the moving centre of the wavepacket.

When this wavefunction interacts with a slit, it is constrained by the width of the slit. Note that a proportion of the electrons are lost to the experiment at this point, as there is a significant probability that the electron will impact the barrier and not pass through the slit. As you narrow the slit, therefore, the beam that reaches the screen gets fainter, as well as diffracting more.

Because the wavefunction is constrained by the slit, it quickly evolves (or collapses) into a wavefunction compatible with an infinite square well. It's briefly no longer a free particle. The narrower the slit, the narrower the potential well and the greater the uncertainty in momentum (in the lateral direction across the slit). If you study the infinite square well, you will find that the mininum energy of the particle (due to lateral momentum) is inversely proportional to the square of the width of the well. This conforms to the uncertainty principle, where the uncertainty in lateral momentum is inversely propertional to the width of the well.

In any case, the wavefunction picks up an component of lateral momentum from its time constrained by the slit.

It emerges from the slit in a superposition of potential well eigenstates, with a range of lateral momentum, which now evolve unconstrained in the lateral direction. Again, the narrower the slit the greater the range in lateral momentum. When this wavefunction interacts with the screen, you get the bell-shaped single-slit diffraction pattern reflecting the probabilitistic spread of lateral momentum.

 

[Flamel]

The simple analysis of single-slit diffraction is that the electron gets "measured" by the slit and owing to the uncertainty principle picks up an uncertainty in lateral momentum as a result. For example, if you start with a wide slit and gradually narrow the slit, then initially the beam gets narrower according to the width of the slit. Until the width of the slit is narrower than the original beam nothing changes. Then, the beam gets narrower simply according to the width of the slit. But, once the width of the slit becomes sufficiently narrow - much narrower than the width of the original beam - the beam starts to spread out and he narrower the slit the more the beam spreads out after the slit.

A slightly more detailed analysis is that electron's wavefunction is spread across a range wider then the slit. This wavefunction, if allowed to evolve, will gradually spread out further. Mathematically, you could model the electron as a moving Gaussian wave packet, which is gradually spreading out in all three directions, relative to the moving centre of the wavepacket.

When this wavefunction interacts with a slit, it is constrained by the width of the slit. Note that a proportion of the electrons are lost to the experiment at this point, as there is a significant probability that the electron will impact the barrier and not pass through the slit. As you narrow the slit, therefore, the beam that reaches the screen gets fainter, as well as diffracting more.

Because the wavefunction is constrained by the slit, it quickly evolves (or collapses) into a wavefunction compatible with an infinite square well. It's briefly no longer a free particle. The narrower the slit, the narrower the potential well and the greater the uncertainty in momentum (in the lateral direction across the slit). If you study the infinite square well, you will find that the mininum energy of the particle (due to lateral momentum) is inversely proportional to the square of the width of the well. This conforms to the uncertainty principle, where the uncertainty in lateral momentum is inversely propertional to the width of the well.

In any case, the wavefunction picks up an component of lateral momentum from its time constrained by the slit.

It emerges from the slit in a superposition of potential well eigenstates, with a range of lateral momentum, which now evolve unconstrained in the lateral direction. Again, the narrower the slit the greater the range in lateral momentum. When this wavefunction interacts with the screen, you get the bell-shaped single-slit diffraction pattern reflecting the probabilitistic spread of lateral momentum.

Do regions without particl

The simple analysis of single-slit diffraction is that the electron gets "measured" by the slit and owing to the uncertainty principle picks up an uncertainty in lateral momentum as a result. For example, if you start with a wide slit and gradually narrow the slit, then initially the beam gets narrower according to the width of the slit. Until the width of the slit is narrower than the original beam nothing changes. Then, the beam gets narrower simply according to the width of the slit. But, once the width of the slit becomes sufficiently narrow - much narrower than the width of the original beam - the beam starts to spread out and he narrower the slit the more the beam spreads out after the slit.

A slightly more detailed analysis is that electron's wavefunction is spread across a range wider then the slit. This wavefunction, if allowed to evolve, will gradually spread out further. Mathematically, you could model the electron as a moving Gaussian wave packet, which is gradually spreading out in all three directions, relative to the moving centre of the wavepacket.

When this wavefunction interacts with a slit, it is constrained by the width of the slit. Note that a proportion of the electrons are lost to the experiment at this point, as there is a significant probability that the electron will impact the barrier and not pass through the slit. As you narrow the slit, therefore, the beam that reaches the screen gets fainter, as well as diffracting more.

Because the wavefunction is constrained by the slit, it quickly evolves (or collapses) into a wavefunction compatible with an infinite square well. It's briefly no longer a free particle. The narrower the slit, the narrower the potential well and the greater the uncertainty in momentum (in the lateral direction across the slit). If you study the infinite square well, you will find that the mininum energy of the particle (due to lateral momentum) is inversely proportional to the square of the width of the well. This conforms to the uncertainty principle, where the uncertainty in lateral momentum is inversely propertional to the width of the well.

In any case, the wavefunction picks up an component of lateral momentum from its time constrained by the slit.

It emerges from the slit in a superposition of potential well eigenstates, with a range of lateral momentum, which now evolve unconstrained in the lateral direction. Again, the narrower the slit the greater the range in lateral momentum. When this wavefunction interacts with the screen, you get the bell-shaped single-slit diffraction pattern reflecting the probabilitistic spread of lateral momentum.

How does the momentum uncertainty generate the minima in the interference pattern? Does it have something to do with the nodes in the infinite square well, and if so why are there no obvious minima when you know which slit the particle traveled through?

 

[Dr_Nate]

How does the momentum uncertainty generate the minima in the interference pattern? Does it have something to do with the nodes in the infinite square well, and if so why are there no obvious minima when you know which slit the particle traveled through?

I think you need to realize that the superposition of all types of waves can lead to destructive and constructive interference. This isn't just a result in quantum mechanics.

Check out this simulation: https://phet.colorado.edu/sims/html/wave-interference/latest/wave-interference_en.html. You can choose to explore the wave interference of water, sound, and light.

 

[PeroK]

How does the momentum uncertainty generate the minima in the interference pattern? Does it have something to do with the nodes in the infinite square well, and if so why are there no obvious minima when you know which slit the particle traveled through?

It might be an interesting exercise to calculate what happens to the eigenstates of the infinite well when the well is removed. I guess that's why it's simpler just to invoke the uncertainty principle!

If you look at the Gaussian wave packet, then the smaller the initial spread, the quicker it spreads out. Even if you just modeled the wavefunction here as a very narrow Gaussian, then there is a much quicker subsequent spread in the wavefunction laterally after passing through the slit.

But, a spreading Gaussian wouldn't explain anything other than a bell-shaped region on the screen. Whereas, considering the slit as an infinite well would predict a more complex pattern of a superposition of wavefunctions, each spreading out laterally at different rates.

 

[Flamel]

I think you need to realize that the superposition of all types of waves can lead to destructive and constructive interference. This isn't just a result in quantum mechanics.

Check out this simulation: https://phet.colorado.edu/sims/html/wave-interference/latest/wave-interference_en.html. You can choose to explore the wave interference of water, sound, and light.

I understand that other types of waves interfere, but what is unclear to me is how momentum uncertainty plays a role in it. Before I thought it was just due to the position probability wave bending around slits and interfering. Would it be more accurate to view it as a pattern that arises from the different momenta, and if that is the case why would determining the slit a particle passed through, and thus increasing position certainty and momentum uncertainty, prevent particles going through the slits from interfering?

 

[PeroK]

I understand that other types of waves interfere, but what is unclear to me is how momentum uncertainty plays a role in it. Before I thought it was just due to the position probability wave bending around slits and interfering. Would it be more accurate to view it as a pattern that arises from the different momenta, and if that is the case why would determining the slit a particle passed through, and thus increasing position certainty and momentum uncertainty, prevent particles going through the slits from interfering?

In the double-slit experiment, it is not different particles interfering with each other; it's each single particle interfering with itself.

 

[Flamel]

In the double-slit experiment, it is not different particles interfering with each other; it's each single particle interfering with itself.

That is true, but if a particle passed through each of the slits at the same time and their positions were measured by a detector, they wouldn't interfere the same way as they would without a detector, correct?

 

[PeroK]

That is true, but if a particle passed through each of the slits at the same time and their positions were measured by a detector, they wouldn't interfere the same way as they would without a detector, correct?

I don't understand what you mean by that. I guess you understand the basic idea of quantum interference, re complex probability amplitudes canceling each other out?

 

[Flamel]

I don't understand what you mean by that. I guess you understand the basic idea of quantum interference, re complex probability amplitudes canceling each other out?

Yes, I understand that there are probability amplitudes that constructively and destructively interfere. I think what is confusing is how momentum uncertainty apparently also plays a role in the formation of interference patterns.

 

[PeroK]

Yes, I understand that there are probability amplitudes that constructively and destructively interfere. I think what is confusing is how momentum uncertainty apparently also plays a role in the formation of interference patterns.

If there were no momentum uncertainty, then with a single slit you would get a single impact point on the screen; and, with a double slit you would get two single impact points on the screen.

The simple analysis, therefore, of both these experiments is that the electrons pick up a larger lateral momentum uncertainty from the smaller lateral position uncertainty enforced by the slits.

 

[Flamel]

If there were no momentum uncertainty, then with a single slit you would get a single impact point on the screen; and, with a double slit you would get two single impact points on the screen.

The simple analysis, therefore, of both these experiments is that the electrons pick up a larger lateral momentum uncertainty from the smaller lateral position uncertainty enforced by the slits.

Momentum uncertainty isn't the only parameter needed for diffraction is it? If that were the case, why would detecting which slit a particle passed through result in reduced diffraction, since the momentum uncertainty should increase with that measurement?

 

[PeroK]

Momentum uncertainty isn't the only parameter needed for diffraction is it? If that were the case, why would detecting which slit a particle passed through result in reduced diffraction, since the momentum uncertainty should increase with that measurement?

If you detect the particle after the slit(s) then that is another interaction/measurement. What happens then depends on the detection process. One aspect of this is that if you have a reliable detector at one slit and you see nothing, then you know the particle went through the other slit unmolested. It's possible that when you detect a particle in this case, then it disrupts the experiment.

 

[Flamel]

If you detect the particle after the slit(s) then that is another interaction/measurement. What happens then depends on the detection process. One aspect of this is that if you have a reliable detector at one slit and you see nothing, then you know the particle went through the other slit unmolested. It's possible that when you detect a particle in this case, then it disrupts the experiment.

Yes, I understand that it influences the results, but why would increasing momentum uncertainty decrease diffraction if the momentum uncertainty is part of what creates diffraction?

 

[PeroK]

Yes, I understand that it influences the results, but why would increasing momentum uncertainty decrease diffraction if the momentum uncertainty is part of what creates diffraction?

I don't understand that question. In what scenario are we increasing momentum uncertainty and reducing diffraction?

 

[Flamel]

I don't understand that question. In what scenario are we increasing momentum uncertainty and reducing diffraction?

By observing which slit a particle passes through, it would increase momentum uncertainty because we now know more specifically the position of the particle, correct? This also reduces diffraction and results in two relatively narrow bands rather than the typical interference pattern between the two slits. Does my reasoning make sense?

 

[PeroK]

By observing which slit a particle passes through, it would increase momentum uncertainty because we now know more specifically the position of the particle, correct? This also reduces diffraction and results in two relatively narrow bands rather than the typical interference pattern between the two slits. Does my reasoning make sense?

You would have to be specific about how you detect a particle. If you capture the electron, then the experiment ends. If you blast the electron with photons, then you may knock it out of the experiment.

In any case, if you do something more significant to the particle than letting it pass through a narrow slit, then the experiment is no longer simple diffraction. What you will then get are results based on what happened to the electron after it passed through the slit.

I wonder if you have in mind some sort of classical process? I.e. once the electron is through the slit you'll get some sort of classical behaviour? All interactions with the particle must be considered quantum mechanically. Including the post-slit detection process.

 

[Flamel]

You would have to be specific about how you detect a particle. If you capture the electron, then the experiment ends. If you blast the electron with photons, then you may knock it out of the experiment.

In any case, if you do something more significant to the particle than letting it pass through a narrow slit, then the experiment is no longer simple diffraction. What you will then get are results based on what happened to the electron after it passed through the slit.

I wonder if you have in mind some sort of classical process? I.e. once the electron is through the slit you'll get some sort of classical behaviour? All interactions with the particle must be considered quantum mechanically. Including the post-slit detection process.

Why would measuring it some other way way prevent diffraction? Don't quantum eraser experiments show that measurements can be made on a particle and will allow diffraction to occur, so long as it can't be determined which slit the particle came through?

 

[Paul Colby]

Why would measuring it some other way way prevent diffraction?

Measuring an optical wavelength photon basically removes it from the field. Why would this not change the diffraction?

 

[Flamel]

Measuring an optical wavelength photon basically removes it from the field. Why would this not change the diffraction?

The particle doesn't necessarily need to be a photon. I believe you could also add a pair of polarizers in opposite directions over the slits and use polarization to determine which slit it came from if I'm not mistaken, so i think the photons don't necessarily need to be destroyed to measure their general location.

 

[PeroK]

Why would measuring it some other way way prevent diffraction? Don't quantum eraser experiments show that measurements can be made on a particle and will allow diffraction to occur, so long as it can't be determined which slit the particle came through?

I don't know where you are getting these ideas. The thread seems to be just repeating itself now.

You say "prevent" diffraction, but surely we are talking about a potential diffraction pattern being destroyed by a post-slit measurement?

If you detemine which way a particle went, then you get single slit diffraction (*). If you don't you get a double-slit interference pattern. Are you confusing "diffraction" with "interference pattern"?

(*) This is the case if you measure which way non-intrusively. This is not easy to do. In the case where you have a detector after one slit, that detector will interact with 50% of the electrons. As a result those electons may not form a diffraction pattern because after they diffracted they interacted with the measuring apparatus. In the simplest case the elecrons are captured and never reach the screen; or, they may be knocked off course and form a much more random pattern all over the screen. If electrons come through a slit and are hit by a baseball bat, then no they don't form a neat single slit diffraction pattern.

But, the electrons that aren't measured to come through that slit are indirectly measured to come through the other slit and they should form a neat single-slit diffraction pattern.

In a typical experiment, therefore, where you measure electrons after one slit, you get a combination of a neat single-slit diffraction pattern for the 50% of electrons that came through the unobserved slit and something messier for the 50% of electrons that were observed, depending on the post-slit measurement process.

 

[Paul Colby]

The particle doesn't necessarily need to be a photon.

You might think about how not being a photon actually would change your question. I don't think it does.

I believe you could also add...

You might think about how adding these polarizers completely changes the original two slit experiment. You have different geometry, materials, altered field boundary conditions. For normal light sources the diffraction pattern in the experiment you describe is determined by the classical EM boundary value problem. The photon probability of detection is proportional to the classical energy density at the detector. The discussion about which slit a photon went through is just noise IMO.

 

[vanhees71]

How does the momentum uncertainty generate the minima in the interference pattern? Does it have something to do with the nodes in the infinite square well, and if so why are there no obvious minima when you know which slit the particle traveled through?

The double-slit (or also single slit or more slits) experiment is calculated in the very same way as for classical electromagnetic waves, though it's much simpler, because only one scalar field is involved. What comes out in Fraunhofer approximation is that the interference pattern at the "infinitely far" observation screen is just the Fourier transform of the openings.

If you want of course the more detailed analysis of what happens at finite distances from the slits you need the Green's function. Then you can also study what happens with wave packets (sharp in momentum vs. sharp in position and looking close to the slits and far away etc.). I'm not aware that this has been calculated in any textbook (but I'm sure one can find it somewhere when googling long enough).

 

[vanhees71]

I don't understand that question. In what scenario are we increasing momentum uncertainty and reducing diffraction?

If the beam is not "monochromatic" enough the contrast of your interference pattern goes down of course.

 

[kent davidge]

If the beam is not "monochromatic" enough the contrast of your interference pattern goes down of course.

because then the beam is no longer coherent?

 

[vanhees71]

Yes. It's because the position of the maxima and minima of intensity behind the double slit depend on the momenta (de Broglie wavelenths) of the particles. So if you have a spread in momentum the minima and maxima get "washed out" and you get "less contrast" of your interference pattern.

 

[Dr_Nate]

Yes, I understand that there are probability amplitudes that constructively and destructively interfere. I think what is confusing is how momentum uncertainty apparently also plays a role in the formation of interference patterns.

Momentum uncertainty is related to spatial uncertainty. That's really all the Heisenberg Uncertainty principle says. It isn't about interference patterns. You really should go to the math at this point. $\Delta x$ and $\Delta p$ represent equations. They are standard deviations of the wave function in position space or momentum space. Those two spaces are related by Fourier transforms; another set of equations.

 

[vanhees71]

Sure, it is also about interference patterns.

First of all look at the incoming wave: If you want double-slit interference the wave packet has to be spatially broad enough to cover (more or less with equal intensity) both slits. This implies that the momentum uncertainty must be small enough to get a sufficient width in position space. Note that the usual textbook treatment uses the limit of plane waves, i.e., as generalized momentum eigenstates (which are strictly speaking not represent true states, because they are no square-integrable functions though; so you have to take their interpretation with a grain of salt), i.e., your two slits are "illuminated" with strictly equal intensity since the spatial width goes to infinity.

The same holds behind the slits: If you put your observation screen (nowadays, e.g., a CCD cam or some other "pixel detector") too close the "partial waves" originating from each slit (in the sense of Huygen's principle) have no significant overlap, i.e., there's no interference and thus no refraction pattern though here you get some which-way information. Only if the partial waves overlap at a far enough put onservation screen you get interference and a double-slit interference pattern at the cost of loosing which-way information.

In some sense the uncertainty principle particularly manifests in this paradigmatic example of wave-function interference phenomena at slits and gratings!