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

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## 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?

## Main Question or Discussion Point

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?

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

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
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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.

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
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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
Gold Member
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.

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

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
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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?

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.

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?

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 travelling 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
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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.

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?

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
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Gold Member
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 modelled 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.

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
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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.

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
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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?

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
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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.

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