Quantum Interference: How To Measure the Wavelength of a Particle

Article pubs.acs.org/jchemeduc

Quantum Interference: How To Measure the Wavelength of a Particle Joseph M. Brom* Department of Chemistry, University of St. Thomas, 2115 Summit Avenue, St. Paul, Minnesota 55105, United States S Supporting Information *

ABSTRACT: The concept of wave−particle duality in quantum theory is difficult to grasp because it attributes particle-like properties to classical waves and wave-like properties to classical particles. There seems to be an inconsistency involved with the notion that particle-like or wave-like attributes depend on how you look at an entity. The concept comes into more clear focus with the precise language of mathematics and with an experiment or demonstration of quantum interference that involves the scattering of photons by a single slit. After describing a photon as a quon, a quantum-sized entity, a precise description of the interference of quantum probability amplitudes allows determination of the photon wavelength by classical measurements of laboratory observations. KEYWORDS: Upper Division Undergraduate, Physical Chemistry, Demonstrations, Laboratory Instruction, Hands-On Learning/Manipulatives, Computer-Based Learning, Mathematics/Symbolic Mathematics, Fourier Transform Techniques, Quantum Chemistry

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INTRODUCTION In the physical sciences, the concept of “wave−particle duality” is difficult to grasp because there is a lack of any sense of consistency in the notion. How can it be that electrons or photons, for example, sometimes behave as if they were waves while at other times behave as if they were particles? Perhaps in an attempt to refine the idea of a dual nature of quantum-sized objects one may come across the declaration that when such objects are measured they behave as particles but in between the acts of measurement they behave as waves. Especially for beginning students, the apparent duality of matter remains enigmatic. For advanced students, however, the author believes that one can come to grips with “wave−particle duality” in the context of a quantum experiment to be described in the more precise language of mathematics. The classroom exercise or laboratory experiment here involves quantum interference and enables students to determine the “wavelength” of a quantumsized “particle”.

The 1929 Nobel Prize in Physics was awarded to Louis de Broglie for his radical idea, in 1924, that classical particles like electrons could also display an associated wave-like behavior. Like Einstein, de Broglie offered a simple equation relating the momentum of the particle, e.g., the electron, with the wavelength of the wave somehow associated with the particle:

p = mv = h/λ

As uncomfortable as the idea of “matter-waves” was, again due to a seemingly logical inconsistency in the nature of matter, scientists were forced to reckon with de Broglie’s equation following its vindication by the electron diffraction experiments of Clinton Davisson and Lester Germer, and George P. Thomson. For their 1927 discovery of electron diffraction by crystalline materials, Davisson and Thomson shared the 1937 Nobel Prize in Physics. Quons

From the point of view of classical physics, electrons are thought of as being discrete particles while photons are thought of as being electromagnetic waves. After all, electrons are known to possess discrete mass and charge, but photons are said to possess frequency and wavelength. Electrons, however, are not classical particles. They also possess a discrete intrinsic angular momentum that is expressly nonclassical. It is the quantum spin angular momentum of the electron. Neither are photons classical waves. They also possess a discrete intrinsic angular momentum that is expressly nonclassical. It is the

Einstein and de Broglie

Albert Einstein received the 1921 Nobel Prize in Physics for his 1905 explanation of the experimental photoelectric effect, wherein he proposed the radical idea that classical waves of electromagnetic radiation consisted of particle-like bundles of energy, later termed “photons” by G. N. Lewis in 1926. The energy of each photon is determined by the now ubiquitous equation in all of spectroscopy: Ephoton = hν = hc/λ

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Thus, the energy of the particle, i.e., the photon, is determined by the frequency ν or the wavelength λ of its associated wave. © XXXX American Chemical Society and Division of Chemical Education, Inc.

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Received: October 28, 2016 Revised: January 24, 2017

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DOI: 10.1021/acs.jchemed.6b00830 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

Article

Experiment Background

quantum polarization of the photon. From the point of view of quantum theory, both electrons and photons are examples of pure quantum-stuff, and, as we have seen, are usually termed “particle−waves” even though they are neither particle nor wave in the classical sense. Thus, we prefer here to use the term quon, not only to denote electrons and photons but also to designate atoms, molecules, and any other species of interest in quantum chemistry. Herbert1 coined this evocative term “quon: any entity, no matter how immense, that exhibits both wave and particle aspects in the peculiar quantum manner”. We allude to the peculiar quantum behavior of quons, and describe a quantum experiment for measuring the seemingly classical wavelength associated with that most fleeting of quons, the photon.

Figure 1 shows schematically the apparatus used to measure the scattering angles of photons after passing through a single slit of small, finite width.

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QUANTUM INTERFERENCE Marcella2 has described the quantum interference patterns, or diffraction patterns, produced by photons interacting with single- and double-slit systems. Rioux3 extended Marcella’s analysis of quantum interference patterns to the optical diffraction from even more interesting two-dimensional arrays of slit patterns. Rioux and Johnson4 have also demonstrated how to illustrate several fundamental principles of quantum theory by using the optical transforms created with laser light scattering from a variety of two-dimensional masks. Muiño5 has described a lecture demonstration and analysis of single-slit diffraction of laser light that visually introduces the Heisenberg uncertainty principle. Again, Rioux6 provided an alternative analysis of this single-slit diffraction pattern of photons based on the Fourier transform between the representations of the photon quantum wave function in coordinate space and in momentum space. Following the approach of Marcella2 and Rioux,3,6 the quantum experiment outlined here involves measurement and analysis of the interference pattern of quantum waves that can then be used to determine the “wavelength” associated with photons from a monochromatic source. This classroom or laboratory exercise is most appropriate for an undergraduate course in physical chemistry or quantum chemistry.

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Figure 1. Schematic apparatus to detect x-component momentum of photons after passing through a single narrow slit of finite width. Unique quantum state selection by the slit determines the probability distribution of detecting a photon at some scattering angle θ. Length a is the distance from the slit to the detection screen, and length 2b is the distance between the first two interference minima. See Supporting Information for details on construction of the apparatus.

Position State Selection

Because the single slit is a position-measuring device along the x-coordinate, the slit effectively prepares the initial quantum state of each photon in the position representation: ⟨x|Ψinitial⟩ = Ψinitial(x) . This Dirac bra−ket pair is the probability density amplitude for locating the photon position in the range from x to x + dx. In other words, this bra−ket pair is simply another way of stating the more familiar Ψ(x) wave function of a quon, and the probability of locating the photon in the range from position x to x + dx is given by eq 4. probability = |⟨x|ψinitial⟩|2 dx

It is important that the slit possess finite width in order to observe quantum interference. If the slit were of infinitesimal width, then any photon passing through the slit would be described by an eigenstate of the position operator. In the position representation, an eigenfunction of position is given by the Dirac delta function, and for such a position state the probability amplitude that the photon emerges from the very narrow slit with any momentum component px is constant. There is no interference pattern to detect. This result would demonstrate the Heisenberg uncertainty principle, and in fact the single slit experiment can be discussed with this purpose in mind.5,6 With the slit of finite width, however, any photon passing through the slit is considered to be in a superposition of position eigenstates. To be specific, we take it that the single slit possesses some fixed, measured width w, centrally located at the origin along the x-axis. Since the slit has finite width, and since we can assume that that any photon has an equal probability of passing through the slit at any point along the slit interval, the initial quantum state function of a photon passing through the slit is constant. Normalizing the wave function gives

QUANTUM EXPERIMENT

The quantum experiment described here has three parts. (1) The f irst part involves a state preparation procedure that places the quon, i.e., the photon, in an initial quantum state described by |Ψinitial⟩. (2) Following state preparation, the second part of the quantum experiment is the measurement of a specified observable that determines the f inal quantum state |Ψfinal⟩ of the photon. (3) The third part of the quantum experiment is to calculate the probability that measurement of a system in a given initial quantum state will find the system in any particular final quantum state. The Born postulate of quantum theory gives the probability amplitude ⟨Ψfinal|Ψinitial⟩, and the described probability is given by eq 3. probability = |⟨ψfinal|ψinitial⟩|2

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(3)

⎧ ⎪ ⎪ ⎪ 1 ⟨x|ψinitial⟩ = ⎨ ⎪ w ⎪ ⎪ ⎩

Given an initial quantum state preparation and a well-defined experiment, the probability distribution for the final quantum state is unique. Thus, we can describe a single-slit scattering of photons experiment in the lab that is a quantum experiment in theory, computation, and observation. B

⎫ ⎪ ⎪ w w⎪ ≤x≤+ ⎬ for − 2 2⎪ ⎪ w 0 for x > + ⎪ ⎭ 2 0 for x < −

w 2

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Article

function in position space, then Ψ(kx) = ⟨kx|Ψ⟩ is a normalized wave function in reciprocal space. In k-space, the axiom we require is

In other words, the initial quantum state of a photon passing through the slit is a normalized superposition state function, a superposition of eigenfunctions of the position operator. It is worth noting that the initial wave function of the photon has the dimension length−1/2. Position selection by scattering photons through the slit is the first part of the quantum experiment.

⟨kx|x⟩ = (2π )−1/2 exp(−ikxx)

In k-space, the Fourier transform of the wave function in x-space is then given by

Momentum State Measurement

The detection screen in Figure 1 measures the final quantum state of each photon in the experiment. The observable at the screen is the x-component of the linear momentum of each photon, px. Perhaps this is not obvious. If the slit were not present to prepare the x-position of a photon, then the x-component of photon linear momentum would be zero. There would be no detection of photons along the x-axis of the screen, other than the origin; there would be no interference effect. However, the position and momentum operators do not commute, and in accord with the Heisenberg uncertainty principle, the act of state preparation of x-position that selects ⟨x|Ψinitial⟩ introduces a probability distribution in the px values of the photon. In other words, only as a result of measurement of px can the final quantum state of each photon be known. Since the distribution of photon arrival at the detection screen is a measurement of px, it is most convenient to express the final wave function in the momentum representation, Ψfinal (px) = ⟨px|Ψinitial⟩. Thus, measurement of the scattering angles along the x-direction is the second part of the quantum experiment.

+w /2

⟨kx|ψinitial⟩ = (2π )−1/2

=

kx =

=

|x⟩⟨x|dx into the bra−ket the resolution of the identity, 1 = ∫ −∞ amplitude:

⟨px |x⟩

1 dx w

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px ℏ

=

p sin θ h sin θ 2π = = sin θ λ ℏ λ ℏ

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⎛ wπ sin θ ⎞ λ2 1 ⎟ sin 2⎜ 3 2 ⎝ ⎠ λ 2wπ sin θ

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⟨px |x⟩⟨x|ψinitial⟩dx

+w /2

∫−w/2

( )

2 sin πw kx

Equation 11 now directly constitutes the third step of the quantum experiment. We have an expression for the probabilities of the outcomes of the measured observables made in the second step of the experiment.

+∞

=

kxw 2

probability density = |⟨kx|ψinitial⟩|2

+∞

∫−∞

1 dx w

Making this substitution for kx into eq 9, we then compute the probability density distribution of scattering angles θ:

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Computing the ⟨px|Ψinitial⟩ probability distribution amplitude will involve the third part of the quantum experiment. To calculate the probability distribution of the momentum states, we require computation of the probability density amplitude ⟨px|Ψinitial⟩. In order to compute this amplitude function, we insert

⟨px |ψinitial⟩ =

exp(−ikxx)

It is worth noting that the final wave function of the photon has the dimension length−1/2. Equation 9 provides the path to the third part of the quantum experiment because this probability density amplitude leads directly to the probability of finding the photon with particular component momentum values along the detector screen. We want to express the k-space wavenumber probability density amplitude in eq 9 directly in terms of the observable scattering angles θ, which reflect the measured kx eigenvalues. From the geometry of the experimental apparatus in Figure 1, and from the de Broglie expression, we can write

By the Born postulate, the probability distribution of the observable px momentum values is probability = |⟨px |ψinitial⟩| dpx

∫

−w /2

Born Postulate

2

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EXPERIMENTAL RESULTS (ON PAPER) Figure 2 shows a plot of eq 11, the probability density distribution of scattering angles versus scattering angle θ, over the range −π/2 ≤ θ ≤ +π/2. (In the plot shown, the probability density is expressed in arbitrary units as the collection of constants preceding the trigonometric terms has been set to unity.) This plot shows the finite-width, single-slit quantum interference pattern, i.e., it is the ⟨kx|Ψinitial⟩ probability density function that exhibits quantum interference. Note that in this quantum analysis of single-slit interference the photons themselves have at no point been taken to be classical electromagnetic waves. Photons are quons, and a quon interferes neither with other quons nor with itself.7 As R. J. Glauber, winner of the 2005 Nobel Prize in Physics for his work in quantum optics, has succinctly stated,7 “The things that interfere in quantum mechanics are not particles. They are probability amplitudes for certain events. It is the fact that probability amplitudes add up like complex numbers that is responsible for all quantum mechanical interferences.” The maxima and minima

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Equation 7 is a Fourier integral; it is the Fourier transform of the initial state function in position space to the final state function in momentum space. To proceed and evaluate the Fourier integral, we require an axiom of quantum theory, namely, the ⟨px|x⟩ probability amplitude. Before we proceed into momentum space, however, it is simply convenient to transform from the px variable to the wavenumber variable, kx  px/ℏ. This is a transformation into “reciprocal space” in that while position x has the dimension of length, wavenumber k has the dimension of length−1. (This wave vector definition of k  2π/λ is named the angular wavenumber; spectroscopists commonly prefer to use the definition ν̃  1/λ for the wavenumber of a photon, with the unit of cm−1.) In k-space, we seek the ⟨kx|Ψ⟩ probability density amplitude, where |⟨kx|Ψ⟩|2dkx is equal to the probability of finding the onedimensional momentum of the quon within the range kxℏ to (kx + dkx)ℏ. Note that if Ψ(x) = ⟨x|Ψ⟩ is a normalized wave C



Article

Figure 2. Angular distribution of photons scattered from a single slit of finite width. In this example plot, the slit width w has been set to be w = 40λ where λ is the de Broglie wavelength of the photon. The probability density is expressed in arbitrary units (see text).

of the quantum interference pattern do allow one, however, to calculate the “wavelength” of the proxy-wave associated with each monochromatic photon scattered by the single slit. Interference minima will occur when w·π ·sin θ = n·π λ

n = ±1, ±2, ...

(12) Figure 3. Photographs of the quantum interference pattern produced by red photons passing through an 86 μm slit and directed toward a detection screen located 400 cm from the slit. (Top) The exposure time has been chosen to display several interference minima. (Bottom) The exposure time has been chosen to observe clearly the measurable sharp first interference minima. [Photographs recorded by Gary Mabbott and used with permission.]

By measuring the scattering angle to the first interference minimum, i.e., n = ±1, one measures the so-called wavelength of the quon, i.e., photon wavelength: λ = w·sin θ1

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Ultimately, the wavelength of the photon is the wavelength of the proxy-wave Ψ, not of a classical electromagnetic wave, and derives meaning from the de Broglie hypothesis.

but students are expected to evaluate the precision and accuracy of the wavelength they measure.

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EXPERIMENTAL RESULTS (IN THE LAB) What the math does on paper the quons do in the lab. Such is the mantra of a quantum experiment. In our lab, coherent, monochromatic photons emitting from a He/Ne laser stream toward a narrow slit positioned at length a = 400 cm from the detector screen. This length is determined by students using a tape measure. The width of the slit has been measured in the lab using a calibrated ocular micrometer. The slit width is measured to be 86 μm. Figure 3 displays photographs of the quantum interference pattern of the red photons observed at the detector screen. In Figure 3 (top), the exposure time has been chosen to illustrate the several interference minima observed the darkened lab. The slit width determines the actual number of interference fringes. Figure 3 (bottom) presents more clearly the sharp first minima in the interference pattern. The distance between the first minima on the detector screen is measured with a ruler to be 2b = 58 mm. Trigonometry, as deduced from Figure 1, allows determination of the scattering angle to the first minima to be ±0.0073 radians. By eq 13 the wavelength λ of the red photon is 0.63 μm. The above values are typical results obtained by students in the lab. The goal of this quantum experiment is not to reproduce the literature value (0.6328 μm) for the He/Ne laser red photon wavelength,

CONCLUSIONS Students learn that an optical diffraction pattern can be explained and understood using the principles of quantum theory. Students are expected to learn that photons are neither particles nor waves, in the classical sense. Photons are examples of quons in the quantum theoretical sense. Any sense of logical inconsistency in the wave−particle duality of a photon is perhaps replaced by the mind-boggling idea that we obtain understanding of photon behavior from a mathematical wave function that serves to represent the photon. Laboratory measurements of quantum interference allow one to determine the wavelength of a photon in the de Broglie sense, but the measured wavelength is due to the interference of quantum probability amplitudes7 resulting from the superposition wave function in position space. Admittedly, the interference pattern observed here by passing monochromatic photons through a single slit could be explained by classical optics. In other words, the experimental observation can be explained by the diffraction of classical electromagnetic waves of light. However, one must always remember that Einstein explained how the classical theory of light cannot account for the photoelectric effect. Nor can classical optics explain the results of single-photon interference experiments8−10 wherein, although the overall shape of the interference pattern D



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(6) Rioux, F. Single-slit diffraction and the uncertainty principle. J. Chem. Educ. 2005, 82, 1210. (7) Glauber, R. J. Dirac’s famous dictum on interference: one photon or two? Am. J. Phys. 1995, 63, 12. (8) Parker, S. A single-photon double slit interference experiment. Am. J. Phys. 1971, 39, 420−424. (9) Rueckner, W.; Titcomb, P. A lecture demonstration of single photon interference. Am. J. Phys. 1996, 64, 184−188. (10) Dimitrova, T. L.; Weis, A. The wave-particle duality of light: A demonstration experiment. Am. J. Phys. 2008, 76, 137−142. (11) Nairz, O.; Arndt, M.; Zeilinger, A. Quantum interference experiments with large molecules. Am. J. Phys. 2003, 71, 319−325. (12) Arndt, M.; Nairz, O.; Vos-Andreae, J.; Keller, C.; van der Zouw, G.; Zeilinger, A. Wave-particle duality of C60 molecules. Nature 1999, 401, 680−682. (13) Eibenberger, S.; Gerlich, S.; Arndt, M.; Mayor, M.; Tüxen, J. Matter-wave interference of particles selected from a molecular library with masses exceeding 10,000 amu. Phys. Chem. Chem. Phys. 2013, 15, 14696−14700.

can be predicted with certainty, one absolutely cannot predict where along the detector screen one will observe the arrival of the localized photon because that is determined completely by chance. Nor would one even be tempted to use a theory of classical waves to explain the observation of interference patterns developed after the passage of “molecular matter-waves” through multiple slits. These sophisticated, beautiful experiments11 have shown that every quon is associated with a wave function that allows the localized quon to “delocalize” by far more than the quon’s extension in space. Buckyball molecules, C60, show a quantum interference pattern12 after passing an effusive beam of C60 through multiple slits. A C60 molecule is another example of a quon. Even large molecules, such as a functionalized tetraphenylporphyrin consisting of 810 atoms and with molecular weight exceeding 10,000 amu, and thus at least approaching some classical limit, have been shown to exhibit quantum interference patterns.13 It would be most effective to show students the results of these “matter-wave” interference experiments postdemonstration of the simpler photon quantum interference experiment described here. These laboratory experiments verify that “wave−particle duality” is real and can be explained quantum theoretically. The quantum experiment described here could be designated either as a classroom lecture demonstration or as a student laboratory exercise. Details of the apparatus used in the classroom or laboratory setting and a copy of the student lab handout are provided in Supporting Information.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.6b00830. Instructor’s notes document (PDF, DOCX) Student lab handout (PDF, DOCX)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Joseph M. Brom: 0000-0003-3818-9040 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author is most grateful to Frank Rioux for providing helpful comments and insight, and to Gary Mabbott for providing the photographs of the quantum interference pattern.

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REFERENCES

(1) Herbert, N. Quantum Reality: Beyond the New Physics; Anchor Press/Doubleday: Garden City, NY, 1985; p 64. (2) Marcella, T. V. Quantum interference with slits. Eur. J. Phys. 2002, 23, 615−621. (3) Rioux, F. Calculating diffraction patterns. Eur. J. Phys. 2003, 24, N1−N3. (4) Rioux, F.; Johnson, B. J. Using optical transforms to teach quantum mechanics. Chem. Educ. 2004, 9, 12−16. (5) Muiño, P. L. Introducing the uncertainty principle using diffraction of light waves. J. Chem. Educ. 2000, 77, 1025−1027. E


Quantum Interference: How To Measure the Wavelength of a Particle

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