double slit experiment uncertainty

June 2, 2011

New 'Double Slit' Experiment Skirts Uncertainty Principle

Physicists show that in the iconic double-slit experiment, uncertainty can be eased.

By Edwin Cartlidge of Nature magazine

An international group of physicists has found a way of measuring both the position and the momentum of photons passing through the double-slit experiment, upending the idea that it is impossible to measure both properties in the lab at the same time.

In the classic double-slit experiment, first done more than 200 years ago, light waves passing through two parallel slits create a characteristic pattern of light and dark patches on a screen positioned behind the slits. The patches correspond to the points on the screen where the peaks and troughs of the waves diffracting out from the two slits combine with one another either constructively or destructively.

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In the early twentieth century, physicists showed that this interference pattern was evident even when the intensity of the light was so low that photons pass through the apparatus one at a time. In other words, individual photons seem to interfere with themselves, so light exhibits both particle-like and wave-like properties.

However, placing detectors at the slits to determine which one a particle is passing through destroys the interference pattern on the screen behind. This is a manifestation of Werner Heisenberg's uncertainty principle, which states that it is not possible to precisely measure both the position (which of the two slits has been traversed) and the momentum (represented by the interference pattern) of a photon.

What quantum physicist Aephraim Steinberg of the University of Toronto in Canada and his colleagues have now shown, however, is that it is possible to precisely measure photons' position and obtain approximate information about their momentum, in an approach known as 'weak measurement'.

Steinberg's group sent photons one by one through a double slit by using a beam splitter and two lengths of fibre-optic cable. Then they used an electronic detector to measure the positions of photons at some distance away from the slits, and a calcite crystal in front of the detector to change the polarization of the photon, and allow them to make a very rough estimate of each photon's momentum from that change.

Average trajectory

By measuring the momentum of many photons, the researchers were able to work out the average momentum of the photons at each detector. They then moved the crystal progressively further away from the slits, and so by "connecting the dots" were able to trace out the average trajectories of the photons. They did this while still recording an interference pattern at each detector position.

Intriguingly, the trajectories closely match those predicted by an unconventional interpretation of quantum mechanics known as pilot-wave theory, in which each particle has a well-defined trajectory that takes it through one slit while the associated wave passes through both slits. The traditional interpretation of quantum mechanics, known as the Copenhagen interpretation, dismisses the notion of trajectories, and maintains that it is meaningless to ask what value a variable, such as momentum, has if that's not what is being measured.

Steinberg stresses that his group's work does not challenge the uncertainty principle, pointing out that the results could, in principle, be predicted with standard quantum mechanics. But, he says, "it is not necessary to interpret the uncertainty principle as rigidly as we are often taught to do", arguing that other interpretations of quantum mechanics, such as the pilot-wave theory, might "help us to think in new ways".

David Deutsch of the University of Oxford, UK, is not convinced that the experiment has told us anything new about how the universe works. He says that although "it's quite cool to see strange predictions verified", the results could have been obtained simply by "calculating them using a computer and the equations of quantum mechanics".

"Experiments are only relevant in science when they are crucial tests between at least two good explanatory theories," Deutsch says. "Here, there was only one, namely that the equations of quantum mechanics really do describe reality."

But Steinberg thinks his work could have prectical applications. He believes it could help to improve logic gates for quantum computers, by allowing the gates to repeat an operation deemed to have failed previously. "Under the normal interpretation of quantum mechanics we can't pose the question of what happened at an earlier time," he says. "We need something like weak measurement to even pose this question."

This article is reproduced with permission from the magazine Nature. The article was first publishe d on June 3, 2011.

Quantum physics experiment shows Heisenberg was right about uncertainty, in a certain sense

Director, Centre for Quantum Dynamics, Griffith University

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Howard Wiseman does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

Griffith University provides funding as a member of The Conversation AU.

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The word uncertainty is used a lot in quantum mechanics. One school of thought is that this means there’s something out there in the world that we are uncertain about. But most physicists believe nature itself is uncertain.

Intrinsic uncertainty was central to the way German physicist Werner Heisenberg , one of the originators of modern quantum mechanics, presented the theory.

He put forward the Uncertainty Principle that showed we can never know all the properties of a particle at the same time.

Read more: Explainer: Heisenberg’s Uncertainty Principle

For example, measuring the particle’s position would allow us to know its position. But this measurement would necessarily disturb its velocity, by an amount inversely proportional to the accuracy of the position measurement.

Was Heisenberg wrong?

Heisenberg used the Uncertainty Principle to explain how measurement would destroy that classic feature of quantum mechanics, the two-slit interference pattern (more on this below).

But back in the 1990s, some eminent quantum physicists claimed to have proved it is possible to determine which of the two slits a particle goes through, without significantly disturbing its velocity.

Does that mean Heisenberg’s explanation must be wrong? In work just published in Science Advances , my experimental colleagues and I have shown that it would be unwise to jump to that conclusion.

We show a velocity disturbance — of the size expected from the Uncertainty Principle — always exists, in a certain sense.

But before getting into the details we need to explain briefly about the two-slit experiment .

The two-slit experiment

In this type of experiment there is a barrier with two holes or slits. We also have a quantum particle with a position uncertainty large enough to cover both slits if it is fired at the barrier.

Since we can’t know which slit the particle goes through, it acts as if it goes through both slits. The signature of this is the so-called “interference pattern”: ripples in the distribution of where the particle is likely to be found at a screen in the far field beyond the slits, meaning a long way (often several metres) past the slits.

But what if we put a measuring device near the barrier to find out which slit the particle goes through? Will we still see the interference pattern?

We know the answer is no, and Heisenberg’s explanation was that if the position measurement is accurate enough to tell which slit the particle goes through, it will give a random disturbance to its velocity just large enough to affect where it ends up in the far field, and thus wash out the ripples of interference.

What the eminent quantum physicists realised is that finding out which slit the particle goes through doesn’t require a position measurement as such. Any measurement that gives different results depending on which slit the particle goes through will do.

And they came up with a device whose effect on the particle is not that of a random velocity kick as it goes through. Hence, they argued, it is not Heisenberg’s Uncertainty Principle that explains the loss of interference, but some other mechanism.

As Heisenberg predicted

We don’t have to get into what they claimed was the mechanism for destroying interference, because our experiment has shown there is an effect on the velocity of the particle, of just the size Heisenberg predicted.

We saw what others have missed because this velocity disturbance doesn’t happen as the particle goes through the measurement device. Rather it is delayed until the particle is well past the slits, on the way towards the far field.

How is this possible? Well, because quantum particles are not really just particles. They are also waves .

In fact, the theory behind our experiment was one in which both wave and particle nature are manifest — the wave guides the motion of the particle according to the interpretation introduced by theoretical physicist David Bohm , a generation after Heisenberg.

Let’s experiment

In our latest experiment, scientists in China followed a technique suggested by me in 2007 to reconstruct the hypothesised motion of the quantum particles, from many different possible starting points across both slits, and for both results of the measurement.

They compared the velocities over time when there was no measurement device present to those when there was, and so determined the change in the velocities as a result of the measurement.

Read more: We did a breakthrough 'speed test' in quantum tunnelling, and here's why that's exciting

The experiment showed that the effect of the measurement on the velocity of the particles continued long after the particles had cleared the measurement device itself, as far as 5 metres away from it.

By that point, in the far field, the cumulative change in velocity was just large enough, on average, to wash out the ripples in the interference pattern.

So, in the end, Heisenberg’s Uncertainty Principle emerges triumphant.

The take-home message? Don’t make far-reaching claims about what principle can or cannot explain a phenomenon until you have considered all theoretical formulations of the principle.

Yes, that’s a bit of an abstract message, but it’s advice that could apply in fields far from physics.

• Quantum mechanics
• Quantum physics
• Particle physics
• Uncertainty
• Scientific experiments

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Quantum Mechanics and the Famous Double-slit Experiment

Table of Contents

• Quantum mechanics is known for its strangeness, including phenomena like wave-particle duality, which allows particles to behave like waves.
• The double-slit experiment is a key demonstration of this duality, showing that even single particles, like photons, exhibit wave-like behavior.
• When the experiment measures which slit a particle goes through, it behaves like a particle. When this measurement is not made, it exhibits interference patterns typical of waves.
• Heisenberg’s uncertainty principle plays a crucial role, connecting the accuracy of position measurement to the momentum of particles.
• Matter waves, proposed by de Broglie, suggest that all matter, not just light, exhibits wave-like properties.
• Wheeler’s delayed-choice experiment demonstrates that making a decision about what to measure after a particle has passed through the slits can affect its behavior, effectively rewriting the past.
• This temporal editing could theoretically reach back to the beginning of the universe, the big bang.
• The interpretation of quantum phenomena like this varies, and different physicists may have differing views on the implications of these experiments.

Introduction

Quantum mechanics is famously strange. But Schroedinger’s dead and alive cat, Einstein’s spooky action at a distance, and de Broglie’s matter waves are mere trifles compared to the greatest strangeness: quantum mechanics can be used to edit events that should have already happened; in other words, to re-write history. However, before we can understand how this is possible, we first need to know how Heisenberg’s uncertainty principle accounts for wave-particle duality in the double-slit experiment.

The Double-Slit Experiment

The double-slit experiment is famous because it provides an unequivocal demonstration that light behaves like a wave. But the importance of the double-slit experiment extends far beyond that demonstration because, as Richard Feynman said in 1966:

In reality, it contains the only mystery…In telling you how it works, we will have to tell you about the basic peculiarities of all quantum mechanics .

Figure 1: The double-slit experiment. Waves travel from the source (top) until they reach the first barrier, which contains a slit. A semi-circular wave emanates from the slit until it reaches the second barrier, which contains two slits. The two semi-circular waves emanating from these slits interfere with each other, producing peaks and troughs along radial lines that form an interference pattern on a screen (bottom).

Based on wor k first described by Young in 1802, the experimental apparatus consists of two vertical slits and a screen, as shown in Figure 1. The light emanating from each slit interferes with light from the other slit to produce an  interference pattern  on the screen. This seems to prove conclusively that light consists of waves, even though the only entities detected at the screen are individual particles, which we now call  photons . The bright regions in the interference pattern correspond to areas where photons land with high probability and the dark regions to areas where photons land with low probability.

Figure 2: Emergence (a→d) of an interference pattern in a double-slit experiment8, where each dot represents a photon. Reproduced with permission from Tanamura (Wikimedia).

Remarkably, the same type of interference pattern is obtained even when the light is made so dim that only one photon at a time reaches the screen, as shown in Figure 2. With such a low photon rate, it can take several weeks for the interference pattern to emerge. However, the very fact that an interference pattern emerges at all implies that even a single photon behaves like a wave. This, in turn, seems to imply that each photon passes through both slits at the same time, which is clearly nonsense. Only a wave can pass through both slits, but only a photon can hit a single point on the screen. However, this single-photon interference behavior is not the most remarkable feature of the double-slit experiment.

If a light detector is used to measure which slit each photon passes through then the interference pattern (shown in Figures 2d and 3b) is replaced by a  diffraction envelope  (shown in Figure 3a), as if the light consists of two streams of particles which spread out and merge over time.

Figure 3: The double-slit experiment can produce either (a) a broad diffraction pattern, or, (b) an interference pattern. Here, the height of the curve indicates light intensity. This also shows how probability density evolves over distance after photons pass through two slits19. (a) If slit identity (photon position) is measured then the waves from the two slits do not interfere with each other. (b) If slit identity is not measured then the waves from the two slits produce an interference pattern as they travel further from the slits. For display purposes, the distribution at each distance has unit height.

Crucially, a diffraction envelope is exactly what is observed if the waves from different slits are prevented from interfering with each other. This can be achieved by using a photographic plate to record photons as the first slit is opened on its own, and then the other slit is opened on its own. In this case, the image captured by the photographic plate is the simple sum of photons from each slit (i.e. a diffraction envelope), with the guarantee that photons from the two slits could not possibly have interfered with each other. Thus, we can choose to use a light detector to find out which slit each photon passed through, but as soon as we do this, we see a diffraction envelope instead of an interference pattern.

Despite many ingenious experiments, any attempt to ascertain which slit each photon passed through forces the light to stop behaving like a wave (which yields an interference pattern) and to start behaving like a stream of billiard balls (which yields a diffraction envelope). The transformation from wave-like behavior to particle-like behavior is intimately related to Heisenberg’s uncertainty principle , as explained below. However, even this wave-particle duality is not the most remarkable feature of the double-slit experiment.

Matter Waves

If the light is replaced by a beam of electrons then an interference pattern is obtained, as if the electrons were waves (which looks like Figure 2d). This is especially surprising because only a few years before this result was obtained in 1927, electrons were thought to behave exactly like miniature billiard balls. So, just as light can be forced to behave like billiard balls, electrons can be forced to behave like waves.

The radical idea that electrons could behave like waves was proposed by de Broglie (pronounced de Broil) in 1923. In fact, de Broglie made the far more radical proposal that all matter has wave-like properties, now referred to as  matter waves . Since that time, the double-slit experiment has been used to demonstrate the existence of matter waves using whole atoms and even whole molecules of  buckminsterfullerene  (each buckminsterfullerene molecule is about 1 nanometre in diameter and contains 60 carbon atoms).

But I digress, so let’s return to the subject at hand.

Did the Photon Pass-Through One Slit or Two?

The problem is that a photon (or a molecule of buckminsterfullerene), unlike a wave, cannot pass through both slits, which raises the question: did the photon really pass through only one slit, and if so, which one?

To attempt to answer this question, consider what happens if we replace the screen with an array of long tubes, each of which points at just one slit, as shown in Figure 4a. At the end of each tube is a photodetector, such that any photon it detects could have come from only one slit. Note that there should be a pair of detectors at every screen position, with each member of the pair pointing at a different slit. Thus, irrespective of where a photon lands on the screen, the slit from which it originated is measured.

If we were to use this imaginary apparatus then we would find that the distribution of photons is a diffraction envelope, as in Figures 3a and 4a. Crucially, this envelope is identical to the pattern that would be obtained if the slits were opened one at a time, as described above.

Figure 4: An imaginary experiment for demonstrating wave-particle duality. a) If slit identity (photon position at the barrier) is measured using an array of oriented detectors at the screen then momentum (direction) precision is reduced, and a diffraction envelope is observed on the screen, as here and in Figure 5.9a. b) If slit identity is not measured then the screen is allowed to measure direction (photon momentum) with high precision, so an interference pattern is observed, as here. In Wheeler’s delayed-choice experiment, we decide to measure either a) slit identity, or b) photon momentum, but the decision is made after the photon has passed through the slit(s).

Thus, using detectors to measure each photon’s slit identity (i.e. which slit the photon passed through) prevents any wave-like behavior, just as if each photon had traveled in complete isolation as a single particle. If both slots are left open (and no photodetectors are used) then the original interference pattern is restored, as if the individual photons behave like waves (as in Figures 3b and 4b). This is the famous wave-particle duality .

Heisenberg’s Uncertainty Principle

As counter-intuitive as the result above seems, it is consistent with  Heisenberg’s uncertainty principle .  This ensures that when the slit identity is  not  measured, uncertainty in the particle’s screen position is roughly equal to the distance between successive maxima in the interference pattern (i.e. it is of the same order of magnitude as the width of each bright region in Figure 1).

More generally, Heisenberg’s uncertainty principle guarantees   that  any  reduction in the uncertainty in slit identity (position at the barrier) must increase uncertainty in the momentum of photons as they exit the slits; because momentum includes direction, uncertainty in momentum translates to uncertainty in screen position. This is the famous  position-momentum trade-off  traditionally associated with Heisenberg’s uncertainty principle, where position indicates slit identity and momentum indicates screen position here (i.e. position-momentum uncertainty translates to slit identity-screen position uncertainty in the double-slit experiment).

Heisenberg’s uncertainty principle means that, in practice, it is possible to adjust the amount of information gained regarding position by varying the accuracy of the measurement devices. As more information on position (slit identity) is gained, less information on the fine structure of the interference pattern is available. Consequently, as more information about the position is gained, the interference pattern is gradually replaced by a broad diffraction envelope.

As an aside, Heisenberg explained his uncertainty principle by showing that shining a light on an electron must alter the position and momentum of that electron, which therefore introduces uncertainty in the electron’s position and momentum. But Heisenberg’s uncertainty principle does not depend on the uncertainty of measurements (in fact, it is the other way round). Because both light and matter behave as if they are waves, they can be analyzed using Fourier analysis . And if position and momentum are waves then they must obey a key result from Fourier analysis:  Heisenberg’s inequality.  Speaking very loosely, this states that if position wavelengths are long then momentum wavelengths are short, and  vice versa . More precisely, Heisenberg’s inequality implies that reducing position uncertainty increases momentum uncertainty (and  vice versa ) so that the values of both cannot be known exactly, independently of whether or not they are measured.

Wheeler’s Delayed-Choice Experiment

So far, we have chosen to measure two different aspects of each photon: first, by measuring each photon’s position on the screen (which translates into momentum at a slit; see the previous section); and second, by using tube detectors to measure which slit each photon came from (which translates into the photon’s position at the barrier).

Clearly, when measuring photon position, the experimental setup does not change over time, so it seems plausible (or at least acceptable) that each photon could have passed through both slits. Similarly, when measuring slit identity, it seems plausible that each photon passes through only one slit.

But there is an alternative experiment, which involves changing the experimental setup while each photon is in transit between the slits and the screen.  If what we choose to measure alters how each photon behaves then it seems reasonable that we must make this decision before each photon reaches the slits.  However, what if the decision on whether to measure screen position or slit identity is made after each photon has passed through the slit(s) but before it has reached the screen or tube detectors? This is  Wheeler’s delayed-choice experiment , depicted in Figure 4.

Such an experiment is conceptually easy to set up. Once a photon is in transit between the slits and the screen, we can decide whether to measure the photon’s screen position (by leaving the screen in place, Figure 4b) or slit identity (by removing the screen so that the tube detectors can function, Figure 4a).

The results of an experiment that is conceptually no different from this were published by Jacques et al, 2007 (who used interferometers). Translating from the interferometer experiment performed by Jacques, when the screen is left in place, the photons’ screen positions are effectively measured, which yields an interference pattern on the screen. In contrast, when the screen is removed, an array of detectors is revealed that detects photons, and the pattern of detected photons is consistent with a broad diffraction envelope (which is the sum of two such envelopes; one per slit), as if no interference had occurred.

Crucially, the decision about whether to measure screen position (by leaving the screen in place) or slit identity (by removing the screen) was made (at random) after each photon had passed through the slit(s); so the behavior of each photon as it passed through the slit(s) depended on a decision made after that photon had passed through the slit(s). In essence, it is as if a decision made now about whether to measure slit identity or screen position of a photon (that is already in transit from the slits to the screen) retrospectively affects whether that photon passed through just one slit or both slits.

Incidentally, we can be certain that the decision was made after a photon passed through the slit(s), as follows. If the distance between the slits and the screen is S then the time taken for a photon to travel from the slit(s) to the screen is T = cS seconds, where c is the speed of light . Suppose it was decided at time t to leave the screen in place, and a photon arrived at the screen dt seconds later. Clearly, if dt < T then the photon must have been in transit between the slit(s) and the screen when the decision was made. Armed with this knowledge, we can select the subset P of photons for which dt<T , and record where they landed to find the pattern they collectively made on the screen. Using the same logic, if the decision was made to remove the screen then the tube detectors measure which slit each photon (in the subset P ) came from, with the guarantee that all these photons were in transit when the decision was made. Given a sufficiently large number of tube detectors, we could find the pattern collectively made by photons at the array of tube detectors.

In principle, the slit–screen distance can be made so large that it takes billions of years for each photon to travel from the slit(s) to the screen. In this case, a decision made now about whether to measure the slit identity or screen position of a photon seems to retrospectively affect whether that photon passed through just one slit or both slits billions of years ago.

As we should expect, these results are consistent with Heisenberg’s uncertainty principle. Regardless of when the decision is made, if the detectors measure slit identity (position) then this must increase uncertainty regarding the particle’s screen position (momentum), which leads to the disappearance of the interference pattern. Even though it is far from obvious how  any physical mechanism could produce this result, the fact remains that if such a mechanism did not exist then Heisenberg’s uncertainty principle would be violated.

Re-Writing Quantum History

So, suppose we wanted to re-write a little history. First, how could we do so, and second, how far back in time could that edit be?

Well, as we have seen, a decision made now about whether or not to leave the screen in place effectively determines how photons behaved in the past. Specifically, if we wish to ensure that each photon passed through just one slit then we should remove the screen (allowing the detectors to measure which slit each photon exited). Conversely, if we wish to ensure that photons passed through both slits then we should leave the screen in place. In both cases, this decision can be made after the photons have passed through the slit(s).

Second, the temporal range of our edit depends on how long the photons have been in transit from the double slits to the screen/detectors. This looks as if we need to have a double-slit experiment that has been set up in the distant past. However, there are natural phenomena, like gravitational lensing , which can be used to effectively mimic the double-slit experiment; as if slits are so far away that the photons we measure have been in transit for many years. Thus, in principle, temporal editing can reach back to the big bang , some 14 billion years ago.

Finally, we should acknowledge that not everyone (including Wheeler) agrees that Wheeler’s delayed-choice experiment edits the past. Like most quantum mechanical equations, the equations that define the results of the delayed choice experiment do not have a single unambiguous physical interpretation. However, in order to appreciate the nature of this ambiguity, it is necessary to understand the equations that govern quantum mechanics . For this, there is no shortcut, but there are many detailed maps and guide books.

James V Stone is an Honorary Associate Professor at the University of Sheffield, UK.

V Jacques, et al. Experimental realization of Wheeler’s delayed-choice gedanken experiment. Science, 315(5814):966–968, 2007.

Note :  This is an edited extract from  The Quantum Menagerie by James V Stone (published December 2020). Chapter 1, the table of contents, and book reviews can be seen here  The Quantum Menagerie .

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27.3 Young’s Double Slit Experiment

Learning objectives.

By the end of this section, you will be able to:

• Explain the phenomena of interference.
• Define constructive interference for a double slit and destructive interference for a double slit.

Although Christiaan Huygens thought that light was a wave, Isaac Newton did not. Newton felt that there were other explanations for color, and for the interference and diffraction effects that were observable at the time. Owing to Newton’s tremendous stature, his view generally prevailed. The fact that Huygens’s principle worked was not considered evidence that was direct enough to prove that light is a wave. The acceptance of the wave character of light came many years later when, in 1801, the English physicist and physician Thomas Young (1773–1829) did his now-classic double slit experiment (see Figure 27.10 ).

Why do we not ordinarily observe wave behavior for light, such as observed in Young’s double slit experiment? First, light must interact with something small, such as the closely spaced slits used by Young, to show pronounced wave effects. Furthermore, Young first passed light from a single source (the Sun) through a single slit to make the light somewhat coherent. By coherent , we mean waves are in phase or have a definite phase relationship. Incoherent means the waves have random phase relationships. Why did Young then pass the light through a double slit? The answer to this question is that two slits provide two coherent light sources that then interfere constructively or destructively. Young used sunlight, where each wavelength forms its own pattern, making the effect more difficult to see. We illustrate the double slit experiment with monochromatic (single λ λ ) light to clarify the effect. Figure 27.11 shows the pure constructive and destructive interference of two waves having the same wavelength and amplitude.

When light passes through narrow slits, it is diffracted into semicircular waves, as shown in Figure 27.12 (a). Pure constructive interference occurs where the waves are crest to crest or trough to trough. Pure destructive interference occurs where they are crest to trough. The light must fall on a screen and be scattered into our eyes for us to see the pattern. An analogous pattern for water waves is shown in Figure 27.12 (b). Note that regions of constructive and destructive interference move out from the slits at well-defined angles to the original beam. These angles depend on wavelength and the distance between the slits, as we shall see below.

To understand the double slit interference pattern, we consider how two waves travel from the slits to the screen, as illustrated in Figure 27.13 . Each slit is a different distance from a given point on the screen. Thus different numbers of wavelengths fit into each path. Waves start out from the slits in phase (crest to crest), but they may end up out of phase (crest to trough) at the screen if the paths differ in length by half a wavelength, interfering destructively as shown in Figure 27.13 (a). If the paths differ by a whole wavelength, then the waves arrive in phase (crest to crest) at the screen, interfering constructively as shown in Figure 27.13 (b). More generally, if the paths taken by the two waves differ by any half-integral number of wavelengths [ ( 1 / 2 ) λ ( 1 / 2 ) λ , ( 3 / 2 ) λ ( 3 / 2 ) λ , ( 5 / 2 ) λ ( 5 / 2 ) λ , etc.], then destructive interference occurs. Similarly, if the paths taken by the two waves differ by any integral number of wavelengths ( λ λ , 2 λ 2 λ , 3 λ 3 λ , etc.), then constructive interference occurs.

Take-Home Experiment: Using Fingers as Slits

Look at a light, such as a street lamp or incandescent bulb, through the narrow gap between two fingers held close together. What type of pattern do you see? How does it change when you allow the fingers to move a little farther apart? Is it more distinct for a monochromatic source, such as the yellow light from a sodium vapor lamp, than for an incandescent bulb?

Figure 27.14 shows how to determine the path length difference for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle θ θ between the path and a line from the slits to the screen (see the figure) is nearly the same for each path. The difference between the paths is shown in the figure; simple trigonometry shows it to be d sin θ d sin θ , where d d is the distance between the slits. To obtain constructive interference for a double slit , the path length difference must be an integral multiple of the wavelength, or

Similarly, to obtain destructive interference for a double slit , the path length difference must be a half-integral multiple of the wavelength, or

where λ λ is the wavelength of the light, d d is the distance between slits, and θ θ is the angle from the original direction of the beam as discussed above. We call m m the order of the interference. For example, m = 4 m = 4 is fourth-order interference.

The equations for double slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes, illustrated in Figure 27.15 . The intensity of the bright fringes falls off on either side, being brightest at the center. The closer the slits are, the more is the spreading of the bright fringes. We can see this by examining the equation

For fixed λ λ and m m , the smaller d d is, the larger θ θ must be, since sin θ = mλ / d sin θ = mλ / d . This is consistent with our contention that wave effects are most noticeable when the object the wave encounters (here, slits a distance d d apart) is small. Small d d gives large θ θ , hence a large effect.

Example 27.1

Finding a wavelength from an interference pattern.

Suppose you pass light from a He-Ne laser through two slits separated by 0.0100 mm and find that the third bright line on a screen is formed at an angle of 10 . 95º 10 . 95º relative to the incident beam. What is the wavelength of the light?

The third bright line is due to third-order constructive interference, which means that m = 3 m = 3 . We are given d = 0 . 0100 mm d = 0 . 0100 mm and θ = 10 . 95º θ = 10 . 95º . The wavelength can thus be found using the equation d sin θ = mλ d sin θ = mλ for constructive interference.

The equation is d sin θ = mλ d sin θ = mλ . Solving for the wavelength λ λ gives

Substituting known values yields

To three digits, this is the wavelength of light emitted by the common He-Ne laser. Not by coincidence, this red color is similar to that emitted by neon lights. More important, however, is the fact that interference patterns can be used to measure wavelength. Young did this for visible wavelengths. This analytical technique is still widely used to measure electromagnetic spectra. For a given order, the angle for constructive interference increases with λ λ , so that spectra (measurements of intensity versus wavelength) can be obtained.

Example 27.2

Calculating highest order possible.

Interference patterns do not have an infinite number of lines, since there is a limit to how big m m can be. What is the highest-order constructive interference possible with the system described in the preceding example?

Strategy and Concept

The equation d sin θ = mλ (for m = 0, 1, − 1, 2, − 2, … ) d sin θ = mλ (for m = 0, 1, − 1, 2, − 2, … ) describes constructive interference. For fixed values of d d and λ λ , the larger m m is, the larger sin θ sin θ is. However, the maximum value that sin θ sin θ can have is 1, for an angle of 90º 90º . (Larger angles imply that light goes backward and does not reach the screen at all.) Let us find which m m corresponds to this maximum diffraction angle.

Solving the equation d sin θ = mλ d sin θ = mλ  for  m m gives

Taking sin θ = 1 sin θ = 1 and substituting the values of d d and λ λ from the preceding example gives

Therefore, the largest integer m m can be is 15, or

The number of fringes depends on the wavelength and slit separation. The number of fringes will be very large for large slit separations. However, if the slit separation becomes much greater than the wavelength, the intensity of the interference pattern changes so that the screen has two bright lines cast by the slits, as expected when light behaves like a ray. We also note that the fringes get fainter further away from the center. Consequently, not all 15 fringes may be observable.

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• 07 August 2018

Two slits and one hell of a quantum conundrum

• Philip Ball 0

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Bands of light in a double-slit experiment. Credit: Timm Weitkamp/CC BY 3.0

Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality Anil Ananthaswamy Dutton (2018)

According to the eminent physicist Richard Feynman, the quantum double-slit experiment puts us “up against the paradoxes and mysteries and peculiarities of nature”. By Feynman’s logic, if we could understand what is going on in this deceptively simple experiment, we would penetrate to the heart of quantum theory — and perhaps all its puzzles would dissolve.

That’s the premise of Through Two Doors at Once . Science writer Anil Ananthaswamy focuses on this single experiment, which has taken many forms since quantum mechanics debuted in the early twentieth century with the work of Max Planck, Albert Einstein, Niels Bohr and others. In some versions, nature seems magically to discern our intentions before we enact them — or perhaps retroactively to alter the past. In others, the outcome seems dependent on what we know, not what we do. In yet others, we can deduce something about a system without looking at it. All in all, the double-slit experiment seems, to borrow from Feynman again, “screwy”.

The original experiment, as Ananthaswamy notes, was classical, conducted by British polymath Thomas Young in the early 1800s to show that light is a wave. He passed light through two closely spaced parallel slits in a screen, and on the far side saw several bright bands. This, he realized, was an ‘interference’ pattern. Caused by the interaction of waves emanating from the openings, it’s not unlike the pattern that appears when two pebbles are dropped into water and the ripples they create add to or dampen each other’s peaks and troughs. With ordinary particles, the slits would act more like stencils for sprayed paint, creating two defined bands.

We now know that quantum particles create such an interference pattern, too — evidence that they have a wave-like nature. Postulated in 1924 by French physicist Louis de Broglie, this idea was verified for electrons a few years later by US physicists Clinton Davisson and Lester Germer. Even large molecules such as buckminsterfullerene — made of 60 carbon atoms — will behave in this way.

You can get used to that. What’s odd is that the interference pattern remains — accumulating over many particle impacts — even if particles go through the slits one at a time. The particles seem to interfere with themselves. Odder, the pattern vanishes if we use a detector to measure which slit the particle goes through: it’s truly particle-like, with no more waviness. Oddest of all, that remains true if we delay the measurement until after the particle has traversed the slits (but before it hits the screen). And if we make the measurement but then delete the result without looking at it, interference returns.

It’s not the physical act of measurement that seems to make the difference, but the “act of noticing”, as physicist Carl von Weizsäcker (who worked closely with quantum pioneer Werner Heisenberg) put it in 1941. Ananthaswamy explains that this is what is so strange about quantum mechanics: it can seem impossible to eliminate a decisive role for our conscious intervention in the outcome of experiments. That fact drove physicist Eugene Wigner to suppose at one point that the mind itself causes the ‘collapse’ that turns a wave into a particle.

Ananthaswamy offers some of the most lucid explanations I’ve seen of other interpretations. Bohr’s answer was that quantum mechanics doesn’t let us say anything about the particle’s ‘path’ — one slit or two — before it is measured. The role of the theory, said Bohr, is to furnish predictions of measurement outcomes; in that regard, it has never been found to fail. (However, he did not, as is often implied, deny that there is any physical reality beyond measurement.) Yet this does feel rather unsatisfactory. Ananthaswamy seems tempted by the alternative idea offered by David Bohm in the 1950s. Here, quantum objects are both particle and wave, the wave somehow ‘piloting’ the particle through space while being sensitive to influences beyond the particle’s location. But Ananthaswamy concludes that “physics has yet to complete its passage through the double-slit experiment. The case remains unsolved.”

With apologies to researchers convinced that they have the answer, this is true: there is no consensus. At any rate, Bohr was right to advise caution in how we use language. There is nothing in quantum mechanics as it stands, shorn of interpretation, that lets us speak of particles becoming waves or taking two paths at once. And there is no reason to regard the wavefunction as more or less than an abstraction. This mathematical function, which embodies all we can know about a quantum object (and features in the iconic equation devised by Erwin Schrödinger to describe the object’s wave-like behaviour) was characterized rather nicely by physicist Roland Omnès. He called it “the fuel of a machine that manufactures probabilities” — that is, probabilities of measurement outcomes.

Refracting all of quantum mechanics through the double slits is both a strength and a weakness of Through Two Doors at Once . It brings unity to a knotty subject, but downplays some important strands. Those include John Bell’s 1964 thought experiment on the nature of quantum entanglement (conducted for real many times since the 1970s); the role of decoherence in the emergence of classical physics from quantum phenomena (adduced in the 1970s and 1980s); and the emphasis on information and causality in the past two decades. Still, given that popularization of quantum mechanics seems to be the flavour of the month — summoning Adam Becker’s 2018 book What is Real? , Jean Bricmont’s 2017 Quantum Sense and Nonsense , a forthcoming book by physicist Sean Carroll, and my own 2018 Beyond Weird — there’s no lack of a wider perspective.

And we need that. Ananthaswamy’s conclusion — that perhaps all the major interpretations are “touching the truth in their own way” — is not a shrugging capitulation. It’s a well-advised commitment to pluralism, shared with Becker’s book and mine. For now, uncertainty seems the wisest position in the quantum world.

Nature 560 , 165 (2018)

doi: https://doi.org/10.1038/d41586-018-05892-6

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Chapter 27 Wave Optics

27.3 Young’s Double Slit Experiment

• Explain the phenomena of interference.
• Define constructive interference for a double slit and destructive interference for a double slit.

Although Christiaan Huygens thought that light was a wave, Isaac Newton did not. Newton felt that there were other explanations for color, and for the interference and diffraction effects that were observable at the time. Owing to Newton’s tremendous stature, his view generally prevailed. The fact that Huygens’s principle worked was not considered evidence that was direct enough to prove that light is a wave. The acceptance of the wave character of light came many years later when, in 1801, the English physicist and physician Thomas Young (1773–1829) did his now-classic double slit experiment (see Figure 1 ).

Why do we not ordinarily observe wave behavior for light, such as observed in Young’s double slit experiment? First, light must interact with something small, such as the closely spaced slits used by Young, to show pronounced wave effects. Furthermore, Young first passed light from a single source (the Sun) through a single slit to make the light somewhat coherent. By coherent , we mean waves are in phase or have a definite phase relationship. Incoherent means the waves have random phase relationships. Why did Young then pass the light through a double slit? The answer to this question is that two slits provide two coherent light sources that then interfere constructively or destructively. Young used sunlight, where each wavelength forms its own pattern, making the effect more difficult to see. We illustrate the double slit experiment with monochromatic (single [latex]{\lambda}[/latex]) light to clarify the effect. Figure 2 shows the pure constructive and destructive interference of two waves having the same wavelength and amplitude.

When light passes through narrow slits, it is diffracted into semicircular waves, as shown in Figure 3 (a). Pure constructive interference occurs where the waves are crest to crest or trough to trough. Pure destructive interference occurs where they are crest to trough. The light must fall on a screen and be scattered into our eyes for us to see the pattern. An analogous pattern for water waves is shown in Figure 3 (b). Note that regions of constructive and destructive interference move out from the slits at well-defined angles to the original beam. These angles depend on wavelength and the distance between the slits, as we shall see below.

To understand the double slit interference pattern, we consider how two waves travel from the slits to the screen, as illustrated in Figure 4 . Each slit is a different distance from a given point on the screen. Thus different numbers of wavelengths fit into each path. Waves start out from the slits in phase (crest to crest), but they may end up out of phase (crest to trough) at the screen if the paths differ in length by half a wavelength, interfering destructively as shown in Figure 4 (a). If the paths differ by a whole wavelength, then the waves arrive in phase (crest to crest) at the screen, interfering constructively as shown in Figure 4 (b). More generally, if the paths taken by the two waves differ by any half-integral number of wavelengths [[latex]{(1/2) \;\lambda}[/latex], [latex]{(3/2) \;\lambda}[/latex], [latex]{(5/2) \;\lambda}[/latex], etc.], then destructive interference occurs. Similarly, if the paths taken by the two waves differ by any integral number of wavelengths ([latex]{\lambda}[/latex], [latex]{2 \lambda}[/latex], [latex]{3 \lambda}[/latex], etc.), then constructive interference occurs.

Take-Home Experiment: Using Fingers as Slits

Look at a light, such as a street lamp or incandescent bulb, through the narrow gap between two fingers held close together. What type of pattern do you see? How does it change when you allow the fingers to move a little farther apart? Is it more distinct for a monochromatic source, such as the yellow light from a sodium vapor lamp, than for an incandescent bulb?

Figure 5 shows how to determine the path length difference for waves traveling from two slits to a common point on a screen. If the screen is a large distance away compared with the distance between the slits, then the angle [latex]{\theta}[/latex] between the path and a line from the slits to the screen (see the figure) is nearly the same for each path. The difference between the paths is shown in the figure; simple trigonometry shows it to be [latex]{d \;\text{sin} \;\theta}[/latex], where [latex]{d}[/latex] is the distance between the slits. To obtain constructive interference for a double slit , the path length difference must be an integral multiple of the wavelength, or

Similarly, to obtain destructive interference for a double slit , the path length difference must be a half-integral multiple of the wavelength, or

where [latex]{\lambda}[/latex] is the wavelength of the light, [latex]{d}[/latex] is the distance between slits, and [latex]{\theta}[/latex] is the angle from the original direction of the beam as discussed above. We call [latex]{m}[/latex] the order of the interference. For example, [latex]{m = 4}[/latex] is fourth-order interference.

The equations for double slit interference imply that a series of bright and dark lines are formed. For vertical slits, the light spreads out horizontally on either side of the incident beam into a pattern called interference fringes, illustrated in Figure 6 . The intensity of the bright fringes falls off on either side, being brightest at the center. The closer the slits are, the more is the spreading of the bright fringes. We can see this by examining the equation

For fixed [latex]{\lambda}[/latex] and [latex]{m}[/latex], the smaller [latex]{d}[/latex] is, the larger [latex]{\theta}[/latex] must be, since [latex]{\text{sin} \;\theta = m \lambda / d}[/latex].

This is consistent with our contention that wave effects are most noticeable when the object the wave encounters (here, slits a distance [latex]{d}[/latex] apart) is small. Small [latex]{d}[/latex] gives large [latex]{\theta}[/latex], hence a large effect.

Example 1: Finding a Wavelength from an Interference Pattern

Suppose you pass light from a He-Ne laser through two slits separated by 0.0100 mm and find that the third bright line on a screen is formed at an angle of [latex]{10.95 ^{\circ}}[/latex] relative to the incident beam. What is the wavelength of the light?

The third bright line is due to third-order constructive interference, which means that [latex]{m = 3}[/latex]. We are given [latex]{d = 0.0100 \;\text{mm}}[/latex] and [latex]{\theta = 10.95^{\circ}}[/latex]. The wavelength can thus be found using the equation [latex]{d \;\text{sin} \;\theta = m \lambda}[/latex] for constructive interference.

The equation is [latex]{d \;\text{sin} \;\theta = m \lambda}[/latex]. Solving for the wavelength [latex]{\lambda}[/latex] gives

Substituting known values yields

To three digits, this is the wavelength of light emitted by the common He-Ne laser. Not by coincidence, this red color is similar to that emitted by neon lights. More important, however, is the fact that interference patterns can be used to measure wavelength. Young did this for visible wavelengths. This analytical technique is still widely used to measure electromagnetic spectra. For a given order, the angle for constructive interference increases with [latex]{\lambda}[/latex], so that spectra (measurements of intensity versus wavelength) can be obtained.

Example 2: Calculating Highest Order Possible

Interference patterns do not have an infinite number of lines, since there is a limit to how big [latex]{m}[/latex] can be. What is the highest-order constructive interference possible with the system described in the preceding example?

Strategy and Concept

The equation [latex]{d \;\text{sin} \;\theta = m \lambda \; (\text{for} \; m = 0, \; 1, \; -1, \; 2, \; -2, \; \dots)}[/latex] describes constructive interference. For fixed values of [latex]{d}[/latex] and [latex]{\lambda}[/latex], the larger [latex]{m}[/latex] is, the larger [latex]{\text{sin} \;\theta}[/latex] is. However, the maximum value that [latex]{\text{sin} \;\theta}[/latex] can have is 1, for an angle of [latex]{90 ^{\circ}}[/latex]. (Larger angles imply that light goes backward and does not reach the screen at all.) Let us find which [latex]{m}[/latex] corresponds to this maximum diffraction angle.

Solving the equation [latex]{d \;\text{sin} \;\theta = m \lambda}[/latex] for [latex]{m}[/latex] gives

Taking [latex]{\text{sin} \;\theta = 1}[/latex] and substituting the values of [latex]{d}[/latex] and [latex]{\lambda}[/latex] from the preceding example gives

Therefore, the largest integer [latex]{m}[/latex] can be is 15, or

The number of fringes depends on the wavelength and slit separation. The number of fringes will be very large for large slit separations. However, if the slit separation becomes much greater than the wavelength, the intensity of the interference pattern changes so that the screen has two bright lines cast by the slits, as expected when light behaves like a ray. We also note that the fringes get fainter further away from the center. Consequently, not all 15 fringes may be observable.

Section Summary

• Young’s double slit experiment gave definitive proof of the wave character of light.
• An interference pattern is obtained by the superposition of light from two slits.
• There is constructive interference when [latex]{d \;\text{sin} \;\theta = m \lambda \;(\text{for} \; m = 0, \; 1, \; -1, \;2, \; -2, \dots)}[/latex], where [latex]{d}[/latex] is the distance between the slits, [latex]{\theta}[/latex] is the angle relative to the incident direction, and [latex]{m}[/latex] is the order of the interference.
• There is destructive interference when [latex]{d \;\text{sin} \;\theta = (m+ \frac{1}{2}) \lambda}[/latex] (for [latex]{m = 0, \; 1, \; -1, \; 2, \; -2, \; \dots}[/latex]).

Conceptual Questions

1: Young’s double slit experiment breaks a single light beam into two sources. Would the same pattern be obtained for two independent sources of light, such as the headlights of a distant car? Explain.

2: Suppose you use the same double slit to perform Young’s double slit experiment in air and then repeat the experiment in water. Do the angles to the same parts of the interference pattern get larger or smaller? Does the color of the light change? Explain.

3: Is it possible to create a situation in which there is only destructive interference? Explain.

4:   Figure 7 shows the central part of the interference pattern for a pure wavelength of red light projected onto a double slit. The pattern is actually a combination of single slit and double slit interference. Note that the bright spots are evenly spaced. Is this a double slit or single slit characteristic? Note that some of the bright spots are dim on either side of the center. Is this a single slit or double slit characteristic? Which is smaller, the slit width or the separation between slits? Explain your responses.

Problems & Exercises

2: Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm.

3: What is the separation between two slits for which 610-nm orange light has its first maximum at an angle of [latex]{30.0 ^{\circ}}[/latex]?

4: Find the distance between two slits that produces the first minimum for 410-nm violet light at an angle of [latex]{45.0 ^{\circ}}[/latex].

5: Calculate the wavelength of light that has its third minimum at an angle of [latex]{30.0 ^{\circ}}[/latex] when falling on double slits separated by [latex]{3.00 \;\mu \text{m}}[/latex]. Explicitly, show how you follow the steps in Chapter 27.7 Problem-Solving Strategies for Wave Optics .

6: What is the wavelength of light falling on double slits separated by [latex]{2.00 \;\mu \text{m}}[/latex] if the third-order maximum is at an angle of [latex]{60.0 ^{\circ}}[/latex]?

7: At what angle is the fourth-order maximum for the situation in Problems & Exercises 1 ?

8: What is the highest-order maximum for 400-nm light falling on double slits separated by [latex]{25.0 \;\mu \text{m}}[/latex]?

9: Find the largest wavelength of light falling on double slits separated by [latex]{1.20 \;\mu \text{m}}[/latex] for which there is a first-order maximum. Is this in the visible part of the spectrum?

10: What is the smallest separation between two slits that will produce a second-order maximum for 720-nm red light?

11: (a) What is the smallest separation between two slits that will produce a second-order maximum for any visible light? (b) For all visible light?

12: (a) If the first-order maximum for pure-wavelength light falling on a double slit is at an angle of [latex]{10.0^{\circ}}[/latex], at what angle is the second-order maximum? (b) What is the angle of the first minimum? (c) What is the highest-order maximum possible here?

13:   Figure 8 shows a double slit located a distance [latex]{x}[/latex] from a screen, with the distance from the center of the screen given by [latex]{y}[/latex]. When the distance [latex]{d}[/latex] between the slits is relatively large, there will be numerous bright spots, called fringes. Show that, for small angles (where [latex]{\text{sin} \;\theta \approx \theta}[/latex], with [latex]{\theta}[/latex] in radians), the distance between fringes is given by [latex]{\Delta y = x \lambda /d}[/latex].

14: Using the result of the problem above, calculate the distance between fringes for 633-nm light falling on double slits separated by 0.0800 mm, located 3.00 m from a screen as in Figure 8 .

15: Using the result of the problem two problems prior, find the wavelength of light that produces fringes 7.50 mm apart on a screen 2.00 m from double slits separated by 0.120 mm (see Figure 8 ).

1:  [latex]{0.516 ^{\circ}}[/latex]

3:  [latex]{1.22 \times 10^{-6} \;\text{m}}[/latex]

7: [latex]{2.06 ^{\circ}}[/latex]

9: 1200 nm (not visible)

11: (a) 760 nm

(b) 1520 nm

13: For small angles [latex]{\text{sin} \;\theta - \;\text{tan} \;\theta \approx \theta}[/latex] (in radians).

For two adjacent fringes we have,

Subtracting these equations gives

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5.2: The Double Slit with Matter

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Exercise \(\PageIndex{1}\): Neutron beams

A beam of very cold neutrons with kinetic energy \(5.0 \times 10^{-6}\, eV\) is directed toward a double slit foil with slit separation 1 m m. What is the angular separation between adjacent interference maxima?

In addition to Bragg diffraction, the wave-like nature of matter can be demonstrated in the same experimental manner as the wave-like nature of light was first demonstrated, by passing the matter wave through a pair of adjacent slits. You should remember the result for the location of interference maxima in a double slit experiment, but nonetheless I’ll remind you:

\[d \sin \theta = n\lambda\]

• \(d\) is the distance between adjacent slits,
• \(\theta\) is the angle at which constructive interference occurs,
• and \(\lambda\) is the wavelength of the disturbance.

The kinetic energy of the neutrons is so small we can use classical physics to determine the momentum. Remembering the classical relationship between kinetic energy and momentum

\[ KE = \dfrac{1}{2} mv^2 = \dfrac{(mv)^2}{2m} = \dfrac{p^2}{2m} =\dfrac{(pc)^2}{2mc^2}\]

\[ \begin{array}{l} pc &= \sqrt{2(KE)mc^2} \\ pc &= \sqrt{2 (5 \times 10^{-6} (939.6 \times 10^6)} \\ pc &= 96.9 \text{ eV} \end{array}\]

and, by DeBroglie’s relation, a wavelength of

\[\begin{array}{l} \lambda = \dfrac{hc}{pc} \\ \lambda = \dfrac{1240 \text{ eVnm}}{96.9 \text{ eV}} \\ \lambda = 12.8 \text{ nm} \end{array}\]

Inserting this result into the double slit relation results in

\[ \begin{array}{l} d\sin\theta = n\lambda \\ (1000 \text{ nm})\sin\theta = (1)(12.8 \text{ nm}) \\ \theta =0.73^0 \end{array}\]

Thus, adjacent maxima are separated by 0.73 degrees.

Thermal Wavelength

How “cold” is a beam of very cold neutrons with kinetic energy 5.0 x 10 -6 eV?

You may have been confused when I referred to the neutron beam in the previous example as being “very cold”. However, physicists routinely talk about temperature, mass and energy using the same language. An ideal (non-interacting) gas of particles at an equilibrium temperature will have a range of kinetic energies. You may recall from your study of the ideal gas [1] that:

\[ KE_{mean} = \dfrac{3}{2}kT\]

• KE mean is the mean kinetic energy of a particle in the sample,
• k is Boltzmann's constant ,
• and T is the temperature of the sample, in Kelvin.

Technically, we shouldn’t talk about the temperature of a mono-energetic beam, since by definition a temperature implies a range of energies. However, let’s be sloppy and assume the energy of the beam corresponds to the mean kinetic energy of a (hypothetical) sample. Then:

\[ \begin{array}{l} KE_{mean} = \dfrac{3}{2}kT \\ T = \dfrac{2KE_{mean}}{3k} \\ T = \dfrac{2(5 \times 10^{-6} \text{ eV}}{3(8.617\times 10^{-5}\text{ eV / K}} \\ T = 0.039 \text{ K}\end{array}\]

So the neutron beam really is pretty cold!

Note that if we wanted to find the DeBroglie wavelength corresponding to this mean kinetic energy, we would find (assuming non-relativistic speeds)

\[\begin{array}{l} KE = \dfrac{(pc)^2}{2mc^2} \\ pc = \sqrt{2(KE)mc^2} \\ pc = \sqrt{2\bigg(\dfrac{3}{2}KT\bigg)mc^2} \\ pc = \sqrt{3mc^2kT} \end{array}\]

\[ \lambda = \dfrac{hc}{pc}\]

\[\lambda = \dfrac{hc}{\sqrt{3mc^2kT}}\]

This is the DeBroglie wavelength corresponding to the mean kinetic energy of a gas at temperature, T. However, a more useful value would be the mean wavelength of all of the particles in the gas. The mean wavelength is not equal to the wavelength of the mean energy. Calculating this mean wavelength, termed the thermal DeBroglie wavelength is a bit beyond our skills at this point, but it is the same as the result above but with a different numerical factor in the denominator:

\[ \lambda_{thermal} = \dfrac{hc}{\sqrt{2\pi n c^2kT}}\]

For an ideal gas sample at a known temperature, we can quickly determine the average wavelength of the particles comprising the sample.

One important use for this relationship is to determine when the gas sample is no longer ideal. If the mean wavelength becomes comparable to the separation between the particles in the gas, this means that the waves begin to overlap and the particles begin to interact. When these waves begin to overlap, it becomes impossible, even in principle, to think of each of the particles as a separate entity. When this occurs, some really cool stuff starts to happen (like superfluidity, superconductivity, Bose-Einstein condensation, etc.).

A Plausibility Argument for the Heisenberg Uncertainty Principle

Imagine a wave passing through a small slit in an opaque barrier. As the wave passes through the slit, it will form the diffraction pattern shown below.

Remember that the location of the first minima of the pattern is given by

From the geometry of the situation,

\[ \tan\theta = \dfrac{y}{D}\]

If the detecting screen is far from the opening,

\[a\sin\theta = \lambda\]

\[a\tan\theta = \lambda\]

\[a\bigg(\dfrac{y}{D}\bigg)=\lambda\]

\[y = \dfrac{\lambda D}{a}\]

Now, consider the “wave” to be a “particle”. The time to traverse the distance from slit to screen is given by

\[t = \dfrac{D}{v_x}\]

while during this time interval the particle also travels a distance in the y-direction given by:

\[ y= v_yt\]

Combining these relations yields

\[ y = v_y \bigg(\dfrac{D}{v_x}\bigg)\]

Combining this “particle” expression with the “wave” expression above gives:

\[ \dfrac{\lambda D}{a} = v_y \bigg(frac{D}{v_x}\bigg)\]

\[\lambda = \dfrac{v_ya}{v_x}\]

Substituting the DeBroglie relation results in,

\[\dfrac{h}{mv_x} = \dfrac{v_ya}{v_x}\]

\[h = mv_ya\]

Notice that the term mv y is the uncertainty in the y-momentum (\(\sigma_{p_y}\)) of the particle, since the particle is just as likely to move in the +y or the –y-direction with this momentum. Also, a is twice the uncertainty in the y-position (\(\sigma_y\)) of the particle, since the particle has a range of possible positions of +a/2 to –a/2.

Therefore, our expression can be written as

\[(2\sigma_y)(\sigma_{p_y}) = h \]

\[(\sigma_y)(\sigma_{p_y}) = \dfrac{h}{2}\]

Thus, the uncertainty in the y-position of the particle is inversely proportional to the uncertainty in the y-momentum. Neither of these quantities can be determined precisely, because the act of restricting one of these parameters automatically has a compensating effect on the other parameter, i.e., making the hole smaller spreads out the pattern, and the only way to make the pattern smaller is to increase the size of the hole!

A more careful analysis (for circular openings rather than slits) shows that the minimum uncertainty in the product of position and momentum can be reduced by a factor of 2 p , resulting in:

\[(\sigma_y)(\sigma_{p_y}) \geq \dfrac{1}{2\pi}\dfrac{h}{2}\]

\[(\sigma_y)(\sigma_{p_y}) \geq \dfrac{\hbar}{2}\]

where the symbol ħ is defined to be Planck’s constant divided by 2\(\pi\).

Contributors and Attributions

Paul D’Alessandris ( Monroe Community College )

[1] The factor 3/2 is true only for particles. If the gas is comprised of diatomic molecules the factor is 5/2 at low temperatures and 7/2 at high temperatures. What counts as a low or high temperature depends on the molecule. More complex molecules have even more complex factors.

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The Uncertainty Principle

Michael Fowler, University of Virginia

Waves are Fuzzy

As we have said, the wave nature of particles implies that we cannot know both position and momentum of a particle to an arbitrary degree of accuracy—if Δ x  represents the fuzziness in our knowledge of position, and Δ p  that of momentum, then

Δ x Δ p ∼ h ,  

where h  is Planck’s constant.  We formulated this by constructing a wave packet moving along a line.  In the real world, particles are three dimensional and we should say

Δ x Δ p x ∼ h ,  

with corresponding equations for the other two spatial directions.  The fuzziness about position is related to that of momentum in the same direction . 

Let’s see how this works by trying to measure y  position and y  momentum very accurately.  Suppose we have a source of electrons, say, an electron gun in a CRT (cathode ray tube, such as an old-fashioned monitor).  The beam spreads out a bit, but if we interpose a sheet of metal with a slit of width w ,  then for particles that make it through the slit, we know y  with an uncertainty Δ y = w .   Now, if the slit is a long way downstream from the electron gun source, we also know p y  very accurately as the electron reaches the slit, because to make it to the slit the electron’s velocity would have to be aimed just right. 

But does the measurement of the electron’s y  position—in other words, having it go through the slit—affect its y  momentum?  The answer is yes.  If it didn’t, then sending a stream of particles through the slit they would all hit very close to the same point on a screen placed further downstream. But we know from experiment that this is not what happens—a single slit diffraction pattern builds up, of angular width θ ∼ λ / w ,  where the electron’s de Broglie wavelength lambda is given by p x ≅ h / λ  (there is a negligible contribution to lambda from the y  -momentum).  The consequent uncertainty in p y  is

Δ p y / p x ∼ θ ∼ λ / w .  

Putting in p x = h / λ ,  we find immediately that

Δ p y ∼ h / w  

so the act of measuring the electron’s y  position has fuzzed out its y  momentum by precisely the amount required by the uncertainty principle.

Trying to Beat the Uncertainty Principle

In order to understand the Uncertainty Principle better, let’s try to see what goes wrong when we actually try to measure position and momentum more accurately than allowed. 

For example, suppose we look at an electron through a microscope.  What could we expect to see?  Of course, you know that if we try to look at something really small through a microscope it gets blurry—a small sharp object gets diffraction patterns around its edges, indicating that we are looking at something of size comparable to the wavelength of the light being used.  If we look at something much smaller than the wavelength of light—like the electron—we would expect a diffraction pattern of concentric rings with a circular blob in the middle.  The size of the pattern is of order the wavelength of the light, in fact from optics it can be shown to be ∼ λ f / d  where d  is the diameter of the object lens of the microscope, f  the focal distance (the distance from the lens to the object). We shall take f / d ∼ 1 ,  as it usually is.  So looking at an object the size of an electron should give a diffraction pattern centered on the location of the object.  That would seem to pin down its position fairly precisely. 

What about the momentum of the electron?  Here a problem arises that doesn’t matter for larger objects—the light we see has, of course, bounced off the electron, and so the electron has some recoil momentum.  That is, by bouncing light off the electron we have given it some momentum.  Can we say how much?  To make it simple, suppose we have good eyes and only need to bounce one photon off the electron to see it.  We know the initial momentum of the photon (because we know the direction of the light beam we’re using to illuminate the electron) and we know that after bouncing off, the photon hits the object lens and goes through the microscope, but we don’t know where the photon hit the object lens.  The whole point of a microscope is that all the light from a point, light that hits the object lens in different places , is all focused back to one spot, forming the image (apart from the blurriness mentioned above).  So if the light has wavelength λ ,  its constituent photons have momentum ∼ h / λ , and from our ignorance of where the photon entered the microscope we are uncertain of its x  -direction momentum by an amount ∼ h / λ .   Necessarily, then, we have the same uncertainty about the electron’s x  -direction momentum, since this was imparted by the photon bouncing off. 

But now we have a problem.  In our attempts to minimize the uncertainty in the electron’s momentum, by only using one photon to detect it, we are not going to see much of the diffraction pattern discussed above—such diffraction patterns are generated by many photons hitting the film, retina or whatever detecting equipment is being used.  A single photon generates a single point (at best!).  This point will most likely be within of order λ  of the center of the pattern, but this leaves us with an uncertainty in position of order λ . 

Therefore, in attempting to observe the position and momentum of a single electron using a single photon, we find an uncertainty in position Δ x ∼ λ ,  and in momentum Δ p x ∼ h / λ .   These results are in accordance with Heisenberg’s Uncertainty Principle Δ x Δ p x ∼ h .  

Of course, we could pin down the position much better if we used N  photons instead of a single one.  From statistical theory, it is known that the remaining uncertainty ∼ λ / N .   But then N  photons have bounced off the electron, so, since each is equally likely to have gone through any part of the object lens, the uncertainly in momentum of the electron as a consequence of these collisions goes up as N .   (The same as the average imbalance between heads and tails in a sequence of N  coin flips.)

Noting that the uncertainty in the momentum of the electron arises because we don’t know where the bounced-off photon passes through the object lens, it is tempting to think we could just use a smaller object lens, that would reduce Δ p x .   Although this is correct, recall from above that we stated the size of the diffraction pattern was ∼ λ f / d ,  where d  is the diameter of the object lens and f  its focal length.  It is easy to see that the diffraction pattern, and consequently Δ x ,  gets bigger by just the amount that Δ p x  gets smaller! 

Watching Electrons in the Double Slit Experiment

Suppose now that in the double slit experiment, we set out to detect which slit each electron goes through by shining a light just behind the screen and watching for reflected light from the electron immediately after it had passed through a slit.  Following the discussion in Feynman’s Lectures in Physics, Volume III , we shall now establish that if we can detect the electrons, we ruin the diffraction pattern!

Taking the distance between the two slits to be d ,  the dark lines in the diffraction pattern are at angles

( n + 1 2 ) λ elec = d sin θ .  

If the light used to see which slit the electron goes through generates an uncertainty in the electron’s y   momentum Δ p y ,  in order not to destroy the diffraction pattern we must have

Δ p y / p < λ elec / d ,  

(the angular uncertainty in the electron’s direction must not be enough to spread it from the diffraction pattern maxima into the minima).  Here p  is the electron’s full momentum, p = h / λ e l e c .  Now, the uncertainty in the electron’s y  momentum, looking for it with a microscope, is Δ p y ∼ h / λ light .

Substituting these values in the inequality above we find the condition for the diffraction pattern to survive is

λ light > d ,  

the wavelength of the light used to detect which slit the electron went through must be greater than the distance between the slits. Unfortunately, the light scattered from the electron then gives one point in a diffraction pattern of size the wavelength of the light used, so even if we see the flash this does not pin down the electron sufficiently to say which slit it went through .  Heisenberg wins again.

How the Uncertainty Principle Determines the Size of Everything

It is interesting to see how the actual physical size of the hydrogen atom is determined by the wave nature of the electron—in effect, by the Uncertainty Principle.  In the ground state of the hydrogen atom, the electron minimizes its total energy. For a classical atom, the energy would be minus infinity, assuming the nucleus is a point (and very large in any case) because the electron would sit right on top of the nucleus.  However, this cannot happen in quantum mechanics.  Such a very localized electron would have a very large uncertainty in momentum—in other words, the kinetic energy would be large. This is most clearly seen by imagining that the electron is going in a circular orbit of radius r with angular momentum   h / 2 π .   Then one wavelength of the electron’s de Broglie wave just fits around the circle, λ = 2 π r .  Clearly, as we shrink the circle’s radius r ,   λ  goes down proportionately, and the electrons momentum p = h / λ  increases. Adding the electron’s electrostatic potential energy we find the total energy for a circular orbit of radius r  is:

K . E . + P . E . = h 2 8 m π 2 r 2 − e 2 4 π ε 0 r .  

Notice that for small enough r ,  the kinetic energy term dominates, and the total energy grows as the atom shrinks. Evidently, there is a value of r  for which the total energy is a minimum. This is easy to find by differentiating the total energy with respect to r  and finding where it is zero. This gives

− h 2 4 m π 2 r + e 2 4 π ε 0 = 0 ,  

from which we find

r min = ε 0 h 2 π m e 2 .  

The total energy for this radius is, incredibly,  the exact right answer for the ground state of  the hydrogen atom.  Very nice, but we don’t deserve it, because we have used a very naïve picture, as will become clear later.

The point of this exercise is to see that in quantum mechanics, unlike classical mechanics, a particle cannot position itself at the exact minimum of potential energy, because that would require a very narrow wave packet and thus be expensive in kinetic energy.  The ground state of a quantum particle in an attractive potential is a tradeoff between potential energy minimization and kinetic energy minimization. Thus the physical sizes of atoms, molecules and ultimately ourselves are determined by Planck’s constant.

Open Yale Courses

You are here, phys 201: fundamentals of physics ii,  - quantum mechanics i: key experiments and wave-particle duality.

The double slit experiment, which implies the end of Newtonian Mechanics, is described. The de Broglie relation between wavelength and momentum is deduced from experiment for photons and electrons. The photoelectric effect and Compton scattering, which provided experimental support for Einstein’s photon theory of light, are reviewed. The wave function is introduced along with the probability interpretation. The uncertainty principle is shown to arise from the fact that the particle’s location is determined by a wave and that waves diffract when passing a narrow opening.

Lecture Chapters

• Recap of Young's Double Slit Experiment
• The Particulate Nature of Light
• The Photoelectric Effect
• Compton's Scattering
• Particle-Wave Duality of Matter
• The Uncertainty Principle

Young's Double-Slit Experiment ( AQA A Level Physics )

Revision note.

Double Slit Interference

• The interference of two coherent wave sources 
• A single wave source passing through a double slit
• The laser light source is placed behind the single slit
• So the light is diffracted, producing two light sources at slits A  and B
• The light from the double slits is then diffracted, producing a diffraction pattern made up of bright and dark fringes on a screen

The typical arrangement of Young's double-slit experiment

Diffraction Pattern

• Constructive interference between light rays forms bright strips, also called fringes , interference fringes or maxima , on the screen
• Destructive interference forms dark strips, also called dark fringes or minima , on the screen
• Each bright fringe is identical and has the same width and intensity
• The fringes are all separated by dark narrow bands of destructive interference

The constructive and destructive interference of laser light through a double slit creates bright and dark strips called fringes on a screen placed far away

Interference Pattern

• The Young's double slit interference pattern shows the regions of constructive and destructive interference:
• Each bright fringe is a peak of equal maximum intensity
• Each dark fringe is a a trough or minimum of zero intensity
• The maxima are formed by the constructive interference of light
• The minima are formed by the destructive interference of light

The interference pattern of Young's double-slit diffraction of light

• When two waves interfere, the resultant wave depends on the path difference between the two waves
• This extra distance is the path difference

The path difference between two waves is determined by the number of wavelengths that cover their difference in length

• For constructive interference (or maxima), the difference in wavelengths will be an integer number of whole wavelengths
• For destructive interference (or minima) it will be an integer number of whole wavelengths plus a half wavelength
• There is usually more than one produced
• n is the order of the maxima or minima; which represents the position of the maxima away from the central maximum
• n  = 0 is the central maximum
• n  = 1 represents the first maximum on either side of the central,  n  = 2 the next one along....

Worked example

Determine which orders of maxima are detected at M as the wavelength is increased from 3.5 cm to 12.5 cm.

The path difference is more specifically how much longer, or shorter, one path is than the other. In other words, the difference in the distances. Make sure not to confuse this with the distance between the two paths.

Fringe Spacing Equation

• The spacing between the bright or dark fringes in the diffraction pattern formed on the screen can be calculated using the double slit equation:

Double slit interference equation with w, s and D represented on a diagram

• D  is much bigger than any other dimension, normally several metres long
• s is the separation between the two slits and is often the smallest dimension, normally in mm
• w  is the distance between the fringes on the screen, often in cm. This can be obtained by measuring the distance between consecutive maxima or minima.
• The wavelength ,  λ of the incident light increases
• The distance , s  between the screen and the slits increases
• The separation , w between the slits decreases

Dependence of the interference pattern on the separation between the slits. The further apart the slits, the closer together the bright fringes

Calculate the separation of the two slits.

Since w , s and D are all distances, it's easy to mix up which they refer to. Labelling the double-slit diagram as shown in the notes above will help to remember the order i.e. w and s in the numerator and D underneath in the denominator.

Interference Patterns

• It is different to that produced by a single slit or a diffraction grating

The interference pattern produced when white light is diffracted through a double slit

• Each maximum is of roughly equal width
• There are two dark narrow destructive interference fringes on either side
• All other maxima are composed of a   spectrum
• The shortest wavelength (violet / blue) would appear   nearest   to the central maximum
• The longest wavelength (red) would appear   furthest   from the central maximum
• As the maxima move   further away   from the central maximum, the wavelengths of   blue   observed   decrease   and the wavelengths of   red   observed   increase

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1.37: The Double-Slit Experiment

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Thomas Young used the double-slit experiment to establish the wave nature of light. Richard Feynman used it to demonstrate the superposition principle as the paradigm of all quantum mechanical phenomena, illustrating wave-particle duality as stated above: between release and detection quons behave as waves.

The Experiment

The slit screen on the left produces the diffraction pattern on the right when illuminated with a coherent radiation source.

The Quantum Mechanical Explanation

Illumination of a double-slit screen with a coherent light source leads to a Schrödinger "cat state", in other words a superposition of the photon being localized at both slits simultaneously.

\[| \Psi \rangle=\frac{1}{\sqrt{2}}[ |x_{1}\rangle+| x_{2} \rangle ] \nonumber \]

Here x 1 and x 2 are positions of the two slits. It is assumed initially, for the sake of mathematical simplicity, that the slits are infinitesimally thin in the x-direction and infinitely long in the y-direction.

Because the slits localize the photon in the x-direction the uncertainty principle \(\left(\Delta \mathrm{x} \Delta \mathrm{p}_{\mathrm{x}}>\frac{\mathrm{h}}{4 \pi}\right)\) demands a compensating delocalization in the x-component of the momentum. To see this delocalization in momentum requires a momentum wave function, which is obtained by a Fourier transform of the position wave function given above.

In this case the Fourier transform is simply the projection of the position wave function onto momentum space. See the Appendix for further information on the Fourier transform.

\[\langle p | \Psi\rangle=\frac{1}{\sqrt{2}}[\langle p | x_{1}\rangle+\langle p | x_{2}\rangle]=\frac{1}{2 \sqrt{\pi}}\left[\exp \left(-\frac{i p x_{1}}{\hbar}\right)+\exp \left(-\frac{i p x_{2}}{\hbar}\right)\right] \nonumber \]

The quantum mechanical interpretation of the double-slit experiment, or any diffraction experiment for that matter, is that the diffraction pattern is actually the momentum distribution function, \(|<p| \Psi>\left.\right|^{2}\). This is illustrated below. The following calculations are carried out in atomic units (h = 2 \(\pi\)).

\[\mathrm{p} :=-30,-29.9 \ldots30 \quad \Psi(\mathrm{p}) :=\frac{\frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{x}_{1}\right)+\frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp \left(-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{x}_{2}\right)}{\sqrt{2}} \nonumber \]

The momentum distribution shows interference fringes because the photons were localized in space at two positions: the maxima represent constructive interference and the minima destructive interference. Notice also that the momentum distribution in the x-direction is completely delocalized; it shows no sign of attenuating at large momentum values. This is due to the fact that under the present model the photons are precisely localized at x 1 and x 2 - in other words the slits are infinitesimally thin in the x-direction as specified above.

This of course is not an adequate representation of the actual double-slit diffraction pattern because any real slit has a finite size in both directions. It's really not a problem that the slit is infinite in the y-direction, because that means the photon is simply not localized in that direction. So all we need to do to get a more realistic double-slit diffraction pattern is make the slits finite (not infinitesimal) in the x-direction. This is accomplished by giving the slits a finite width, \(\delta\), in the x-direction, and recalculating the momentum wave function.

Slit width:

\[\delta :=0.2 \quad \Psi(\mathrm{p}) := \frac{\int_{\mathrm{x}_{1}-\frac{\delta}{2}}^{\mathrm{x}_{1}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{x}) \cdot \frac{1}{\sqrt{\delta}} \mathrm{d} \mathrm{x}+\int_{\mathrm{x}_{2}-\frac{\delta}{2}}^{\mathrm{x}_{2}+\frac{\delta}{2}} \frac{1}{\sqrt{2 \cdot \pi}} \cdot \exp (-\mathrm{i} \cdot \mathrm{p} \cdot \mathrm{x}) \cdot \frac{1}{\sqrt{\delta}} \mathrm{d} \mathrm{x}}{\sqrt{2}} \nonumber \]

In summary, the quantum mechanical interpretation of the double-slit experiment is that position is measured at the slit screen and momentum is measured at the detection screen. Position and momentum are conjugate observables which are connected by a Fourier transform and governed by the uncertainty principle. Knowing the slit screen geometry makes it possible to calculate the momentum distribution at the detection screen. Varying the slit width, \(\delta\), gives a clear and simple demonstration of the uncertainty principle in action. Narrow slit widths give broad momentum distributions and wide slit widths give narrow momentum distributions.