Quantum probability and quantum spin are foundational concepts in quantum physics, not normally taught in pre-university education. In line with recent efforts to modernize quantum science education in schools, this paper presents learning sequences designed to provide an introductory conceptual understanding of these core physical concepts. This is achieved by choosing physical models that provide tangible classical representations that can be contrasted with quantum observations relevant to modern science and technology. This approach allows activity-based learning without the use of algebra. The concept of quantum probability is taught through activities with phasor wheels in which quantum probability is obtained graphically and interpreted using single-photon interference data. Quantum spin and spin vectors are taught by contrasting quantum spin with classical spin, explored through activities with gyroscopes and spinning tops. Quantum concepts are evoked by emphasizing how spin is different and how it manifests in real-world technologies such as magnets and MRI imaging. We describe learning sequences developed following the Model of Educational Reconstruction in which historical discoveries are combined with activities based on classical models and relevant quantum applications. We summarize test results from programs conducted with students aged 11–15. Most students connect the models and activities to quantum concepts and retain knowledge of quantum spin six months after the program. The successful learning outcomes indicate that teachers can effectively introduce quantum probability and quantum spin concepts to middle school using learning sequences such as those presented here.
I. INTRODUCTION
Despite the significant importance and explanatory power of quantum physics, it is a challenging subject to teach at schools.1 At the tertiary level, quantum physics is taught using an abstract mathematical framework with associated terminology and notation. This approach poses challenges for students, even at the undergraduate level, because most students come without prior exposure to the foundational concepts of quantum physics. Authors such as Bouchée et al.2 emphasize the importance of a holistic understanding of quantum physics beyond just mathematical proficiency. Moreover, if quantum physics is reserved for tertiary studies, we ensure that most of the population remains ignorant of the quantum science behind the devices and technologies we rely on.
As part of the Einstein-First project,3,4 an initiative led by researchers at the University of Western Australia (UWA) within the international Einsteinian Physics Education Research (EPER) collaboration, we present a teaching approach for two fundamental quantum physics concepts: quantum spin and quantum probability. This initiative, involving over 50 partner schools across Australia and globally, aims to reformulate school physics by integrating modern and relevant physics concepts and applications. Einstein-First's primary goal is to ensure that all students are introduced to the fundamental concepts of physical reality before the age of 15. The central pedagogical approach that Einstein-First brings to the curriculum involves learning from tangible objects such as toys and physical models. Key learning is achieved through tactile interaction with the models and understanding their limitations. Naturally, most physical models, like those discussed here, are classical, but they are still powerful means of communicating quantum concepts through a combination of analogy and contrast.5,6
There are two significant reasons for introducing quantum concepts at an early age:
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Children at this age are not yet heavily exposed to classical physics ideas; therefore, they find it easier to accept quantum concepts that older students and adults find mystifying.
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The students are still very engaged in play. They easily identify with toy models, understand and accept their limitations, and recall their learning in the context of the activities they enjoyed.
The objective of this paper is to present learning sequences for teaching quantum probability in the context of interference and quantum spin in the context of magnetism and imaging. These topics were chosen because of their fundamental relevance to modern technology and shared use of vector concepts. Quantum probability emphasizes the profound break with classical determinism and is relevant to many topics, including radioactivity, optical interference, and quantum computers. Quantum spin underpins our understanding of the fundamental constituents of physical reality. It is relevant to many modern applications of quantum science, such as magnetism, magnetic resonance imaging (MRI), quantum computers, atomic structure, superconductivity, and atomic clocks.
The quantum probability sequence begins with the discovery of the phasor concept and Feynman's use of phasors for introducing path integrals and quantum probability.7 Crucial to this sequence is a physical tool used to represent the real and complex phasor components. This was first attempted using Lego-based gadgets by Dobson et al. in 2000.8 The sequence presented here uses a simpler approach in which the mathematical phasor representation of wave motion is connected to a tool called a phasor wheel, first introduced by Choudhary and Blair in 2022.9 Phasor wheels are used in the learning sequence to predict and describe experimental outcomes, including laser pointer hair interference and double-slit interference with single photons. Online coin-tossing experiments connect quantum probability and macroscopic observations when the number of quanta is large.
The quantum spin learning sequence begins with the historical story of discovery and experiments that confirm quantum spin is a form of angular momentum. Then the vector description of spin is introduced along with the discovery that all fundamental particles are either fermions or bosons. Quantum spin is contrasted with classical spin through spinning toys, which develop an understanding of spin vectors, spin flips, spin precession, and the vector addition of spins. This allows us to explore the use of quantum spin in different technologies.
The sequences reported here have been developed according to the Model of Educational Reconstruction (MER),10 which guides the development of the Einstein-First curriculum. The MER structures the educational research and the development of learning tools, encapsulated in the model's three strands: (1) research on teaching and learning, (2) clarification of content, and (3) trials and evaluation of learning resources. It includes initial testing of pilot learning resources in real classroom settings and cycles of feedback for optimization.
The two learning sequences are designed to provide conceptual understanding and vector concepts without the need for algebra in a form suitable for mainstream students who may have weak mathematical skills. A key part of the learning is the historical, scientific endeavor that shows how modern technology emerged from fundamental studies. Science learning can be enriched by teaching about the historical process through which modern technology emerged from fundamental studies. For this reason, we place significant emphasis on the history before detailing the learning sequence. Finally, to encourage implementation by others, we report initial results with school students aged 11–15 that indicate successful learning and retention of quantum concepts.
II. ORIGINS OF PHASORS AND QUANTUM SPIN
In this section, we provide the historical context of the quantum concepts that the two learning sequences are designed to uncover. The language and approach to the topics are designed to be accessible to teachers without a quantum physics background and to provide teachers with appropriate historical context to make the subjects interesting for their students.
Section II A explores the history of phasors that were first used to describe classical waves and later applied to quantum probability waves. Within this section, we provide a summary, at the appropriate level, of the concept of superposition, discussed in terms of all possible paths. Section II B explores the discovery of quantum spin and its key role in dividing all particles into fermions and bosons, combined with examples of systems and technologies in which quantum spin is crucial.
A. Phasors and their use for estimating quantum probability
Inspired by the mathematics of electromagnetic waves developed by Oliver Heaviside,11 Charles Steinmetz12 in 1893 introduced a new concept that could simplify the addition of multiple alternating currents with different phases. The term “phasor” was not coined until 1944.13 Today, phasors are essential in engineering and physics to describe the addition of monochromatic waves. Following Richard Feynman, we apply phasors to probability waves.7
Phasors are conventionally used in a classical context to represent the amplitude and phase of a sinusoidal wave relative to an arbitrary origin. At one point in space, the phasor sweeps out a circle at the wave frequency, but relative to another wave of the same frequency, the phasor is stationary, with the phasor directions indicating the relative phase of the waves. Students often find phasors difficult to comprehend because, unlike force or velocity vectors, the direction of a phasor is disconnected from the direction of wave propagation. Phasor wheels overcome this difficulty by tangibly connecting the phasor to wave-like motion. A phasor wheel consists of a rotating wheel with a radial phasor vector and a cam rod, used to trace the approximate wave displacement as the wheel is rolled. The phasor wheel makes phasors concrete through a physical representation of the connection between wave motion, circular motion, and vector amplitude, enabling students to recognize and distinguish key wave properties. Figure 1 illustrates a phasor wheel in a classroom setting. The phasor arrow describes the amplitude and phase of the wave. In quantum language (not used with students) the phasor vector represents the complex phasor amplitude.
Students use the phasor wheel to visually connect the wave-like movement at the end of the cam rod with the phasors rotation. The cam rod is attached near the disk edge.
Students use the phasor wheel to visually connect the wave-like movement at the end of the cam rod with the phasors rotation. The cam rod is attached near the disk edge.
In his book QED,7 Richard Feynman demonstrated that his path integral technique can be presented in a simple form using phasors. In the quantum context, the phasors do not represent photons nor anything observable but rather are part of a procedure for predicting the probability for photons to arrive. Rather than assuming light paths based on the classical laws of optics, Feynman's approach considers all the possible paths between the initial and final states. He shows that the classical laws of optics can be derived using path integrals and, hence, are derivable outcomes of quantum physics. His method implicitly demonstrates the non-locality of quantum physics because all possible paths are considered simultaneously. Choudhary and Blair9 showed how this method could be successfully used in the classroom to demonstrate the origins of the laws of refraction, reflection, and diffraction for secondary students.
The program described in this work focuses on explaining interference using the Feynman path integral technique. We limit our discussion to situations that can be simplified to two possible paths: specifically a double slit or a human hair. In the double-slit experiment, the slits are assumed to be narrow enough that they define two possible paths. For the human hair experiment, the hair serves as a single obstruction around which infinite paths can exist. However, only the paths near the edges of the hair contribute significantly to the intensity pattern, especially when the laser beam diameter is small. In particular, the central bright spot and the first dark fringe positions are the same for both experiments if the slit spacing is the same as the hair diameter.
The learning progression gradually introduces students to the relevant language and interpretations needed to explain interference using quantum probability. The sequence ends with students using the phasor wheel to trace the alternative paths to determine where bright and dark spots can form. These experiments can be scaled up or down. Large-scale experiments using phasor wheels like those in Fig. 1 can be done on the classroom floor, or desk-sized experiments can be done using an A3 sheet as described in Sec. III.
We motivate the learning sequence by quoting Richard Feynman: “We choose to examine a phenomenon which is impossible, absolutely impossible to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by ‘explaining’ how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.”14
Students learn how it works by using phasor wheels to graphically determine the probability of photons arriving at a particular location. This involves introducing crucial, but potentially confusing, terminology. First, we introduce the idea that the propagation of a phasor wheel represents an abstract quantity called a probability wave, while the phasor displayed by a phasor wheel (see Fig. 1) represents the (complex) probability amplitude of the probability wave. The motion of the cam rod as the phasor wheel rolls merely emphasizes the wave-like behavior of this mathematical wave, which is fully described by the phasor arrow.
Second, we introduce the rule that if there are two alternative paths for a photon to arrive at a particular location, the probability amplitudes of each wave are added. That is, we add two phasor arrows, which will have different phases when the paths have different lengths (where the difference is not a multiple of the wavelength). We emphasize that probability waves represented by rolling phasor wheels are purely mathematical prediction tools and are not observable. Probability amplitude is a vector quantity, but probability itself is a scalar: it does not have direction. To convert the abstract probability amplitude into measurable probability, we must transform it into an experimentally observable quantity called the probability density.
Probability densities represent the intrinsic probabilistic nature of the quantum world. It is a simple number, such as a percentage chance that a photon will arrive at a particular location in a particular time interval. To transform the probability amplitude into a probability density, we simply have to rotate the probability amplitude phasor to inscribe a circle. In Sec. III, we will illustrate a classroom example of this process. The circle represents an area, and the phasor amplitude vector becomes a number representing an area. This is equivalent to squaring the amplitude vector except for a factor of π.
The greater the probability of photons arriving, the greater the intensity, because intensity is simply the total photon power, represented by the number of photons per second. If you double the intensity of a light source, you should double the probability of photons arriving. From this, by working backward, we can deduce that the probability amplitude phasor has a length proportional to the square root of intensity.
As a single phasor wheel rolls along a path, its probability density stays constant because the arrow remains the same length. However, if there are two alternative paths, the two probability amplitude vectors must be added by vector addition. The addition results in a new phasor called the resultant. Depending on the phase of each phasor, the resultant of the two phasors can be double the length of the individual phasors (maximum probability density), zero (minimum probability density), or in between. When squared by turning them into circles, these vector sums tell us the probability density arising from any two wave paths. Thus, probability density connects to constructive and destructive interference or bright and dark interference fringes. Figure 7 shows a typical worked example in which students undertook graphical interference.
The above procedure is summarized as follows:
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Phasor wheels are used to determine probability amplitude at the end of a chosen path.
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When photons can arrive by two alternative paths, the probability amplitude is the vector sum of the probability amplitude phasors for each path considered separately.
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The probability of photon arrival is the absolute square of the probability amplitude.
The central mystery of quantum physics is that item 3 above is only true if the alternative paths are truly alternative. If the path traveled by a photon is determined by any means at all, the interference pattern vanishes. This has been tested in innumerable experiments. It is as if photons travel on both paths or do not exist until measured. This is the mystery. If photon paths are known, the probability of a photon arriving is just the sum of the probabilities for each path, and there are no dark fringes.
The learning sequence for quantum probability consists of three components: (a) real interference using a laser pointer with a double-slit or a human hair and (b) single-photon interference online videos15 that show interference in the form of sparsely arriving photons with clear statistical randomness, but which accumulate to create a clear interference pattern, and (c) a macro-scale graphical interference experiment using a phasor wheel.
To perform graphical interference, we are constrained by the size of the phasor wheel, which defines the wavelength. To ensure at least two minima are observable on a landscape A3-sized sheet, the slit spacing must be about 12 cm. The wavelength produced by our phasor wheel is about 105 times larger than the wavelength of light, which means that large angles are needed to observe significant phasor direction changes. While the geometry is foreshortened, the physics of phasor addition is unchanged.
While single-photon interference is probabilistic, we use the activities to discuss the transition from randomness to reproducibility when photon numbers become large. Quantum probability is relevant when photon numbers are small, but intensity is appropriate when the number of photons is large. Thus, probability and intensity are low and high-number descriptors of the same thing. In everyday life, photon numbers are so large that we do not notice the statistical granularity of light, but it is always present.
The transition from randomness to reproducibility can be reinforced using online coin-tossing simulators.19 Students recognize how fractional uncertainty reduces when number counts become large. The whole learning sequence, from laser experiments to coin-tossing, provides students with an appreciation of the connection between the quantum and the classical world and the intrinsic statistical nature of quantum reality.
B. Understanding quantum spin as quantized angular momentum
1. Discovery of quantum spin
In this section, we review the evidence that quantum spin is angular momentum. Then, we summarize the counter-intuitive aspects of quantum spin and review the universality of fermions and bosons, phenomena governed by quantum spin.
Soon after the discovery of electrons, protons, and photons (more than 100 years ago), it was discovered that every one of these fundamental components of the universe has angular momentum in the form of quantum spin. We use the history of these discoveries, combined with an intuitive understanding of spin, in the context of spinning toys, to prepare students for a series of spin activities that uncover the relevance of quantum spin to our daily lives. The discovery of quantum spin is generally credited to the 1922 experiment by Stern and Gerlach that measured the deflection of a beam of silver atoms when they passed through an inhomogeneous magnetic field.16 They observed the atom beam was deflecting into two discrete spots on a screen, rather than forming a continuous distribution. The discrete spots implied that the angular momentum and the associated magnetic moment of silver atoms had just two values. Today we describe these values as quantum spins of and , which are defined by Planck's constant as described below.
The Stern–Gerlach experiment explained the results of an experiment conducted by Einstein and de Haas seven years earlier.17 This experiment used a freely rotating ferromagnetic cylinder, in which the atomic spins could be aligned by applying a magnetic field (like the induced magnetization of a nail by a permanent magnet) or randomly oriented when it was not magnetized. When the spins were aligned, the individual atomic spin vectors were constructively added, corresponding to a measurable change in macroscopic angular momentum. This experiment proved that quantum spin is angular momentum. At a classical level, the transfer of angular momentum can be demonstrated in the classroom by spinning a top on a floating saucer. As the top slows down, the saucer starts to spin. Classically, this is simply an example of conservation of angular momentum.
Thirteen years after the Stern–Gerlach experiment, Richard Beth directly proved that light has angular momentum.18 His experiment measured the torque exerted by a beam of polarized light as it passed through a doubly refracting phase plate. The Beth experiment showed that left circularly polarized photons have + 1 quantum unit of spin while right circularly polarized photons have a quantum spin of –1.
The units of quantum spin are embedded in the Planck formula for photon energy E = hf, or , where ω is angular frequency . Planck's constant has the dimensions of angular momentum . Thus, the Planck formula states that the energy of a photon is given by its angular frequency times the quantum unit of angular momentum , which is conventionally written as .
Students may not be familiar with the concept of angular momentum. To build understanding, we first introduce linear momentum, defined as the product of mass and velocity, and relate it to the force required to stop a moving object. Similarly, angular momentum is the rotational counterpart, combining rotational inertia and angular velocity. Rotational inertia relates to the torque required to stop a rotating object.
When we use the term spin, we are specifically referring to angular momentum, not angular velocity. This distinction is crucial for understanding rotational motion in both classical and quantum contexts.
Planck's energy formula states that photons can have any energy value determined by frequency. Yet every photon has exactly the same value of quantum spin. The linear momentum of photons or increases proportional to photon energy, but the angular momentum has identical quantum spin values for low-energy radio photons, such as those coming out of mobile phones, or high-energy gamma-ray photons created when stars collapse to form black holes.
The historical experiments discussed above established a clear connection between classical and quantum spin and motivated the inclusion of spin in school classrooms, but they are not essential for student understanding. The historical experiments validate the use of macroscopic models, such as spinning tops, that, through both similarity and contrast, uncover the counter-intuitive aspects of quantum spin. Table I presents the main points emphasized in the learning sequence. The universality of quantum spin allows the learning sequence to be extended to the classification of fermions and bosons in the Standard Model of particle physics. We present the classification in a reduced form, mentioning only the most relevant fermions and bosons. This optional component contains more advanced content but can be offered as a student self-study investigation.
Quantum spin compared with classical spin.
. | Property . | Comparison . |
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1. | Units | Quantum spin and classical spin both have the units and properties of angular momentum, which classically is a product of rotational inertia and rotation speed. |
2. | Vector properties | Quantum spin and classical spin both add like vectors and follow the rules of vector addition. Spins can be canceled to attain a system with zero net spin. |
3. | Precession | Both quantum and classical spins precess when a torque is applied. |
4. | Magnetism | Magnetism is directly connected to the quantum spin vectors of charged particles and is only indirectly linked to rotation in classical spinning bodies like the Earth. |
5. | Spin value | Classical spin can take any value including zero, while quantum spin has a fixed value for every fundamental particle. It can only vary according to its vector direction. |
. | Property . | Comparison . |
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1. | Units | Quantum spin and classical spin both have the units and properties of angular momentum, which classically is a product of rotational inertia and rotation speed. |
2. | Vector properties | Quantum spin and classical spin both add like vectors and follow the rules of vector addition. Spins can be canceled to attain a system with zero net spin. |
3. | Precession | Both quantum and classical spins precess when a torque is applied. |
4. | Magnetism | Magnetism is directly connected to the quantum spin vectors of charged particles and is only indirectly linked to rotation in classical spinning bodies like the Earth. |
5. | Spin value | Classical spin can take any value including zero, while quantum spin has a fixed value for every fundamental particle. It can only vary according to its vector direction. |
2. Bosons and fermions
Following the discovery of the half-integer spins of electrons, protons, and neutrons, and the integer spin of photons, it became clear that all fundamental particles can be divided into just two groups: (a) fermions with half-integer spin and (b) bosons with integer spin. The radically different behavior of these two types of particles underpins the understanding of almost everything from atoms to metals, semiconductors, superconductors, lasers, and medical technologies.
In 1939, Markus Fierz, a student of Wolfgang Pauli, proposed an explanation for why spin- particles obey the Pauli exclusion principle,20 in which only one electron can occupy each quantum state, while particles like photons with spin-1 can have an unlimited number in the same state. Fierz's result, which Pauli, Feynman, and others successively improved later, is called the spin-statistics theorem.
The spin-statistics theorem uses the term wavefunction. A wavefunction can be considered a mathematical function describing particle probability amplitude. The spin-statistics theorem states that the wavefunction of a system of identical particles can be either anti-symmetric or symmetric according to whether the wavefunction changes sign when two particles are swapped. Fierz showed that these two cases correspond to a spin of (anti-symmetric) or a spin of 1 (symmetric). Symmetric particles follow the Bose–Einstein energy distribution function that allows an unlimited occupation of a quantum state, while anti-symmetric particles follow the Fermi–Dirac energy distribution function that allows only one particle in each state.
We do not discuss the complexities of wave or distribution functions with students. Instead, we introduce the simplified rule: all fundamental particles are categorized into two groups—bosons and fermions. Integer-spin particles like photons, with spin-1 are allowed to have an unlimited occupation of an energy state, while spin- particles like electrons are allowed only one particle in each energy state. There appear to be no other possibilities.
Lasers are examples of boson systems, where photons are all in the same energy state. The structure of an atom is dependent on electrons being all in different states because they are fermions.
The photon spin of 1 is related to the 90° difference between the two linear polarization angles of light. Teachers interested in discussions about polarization can use simple demonstrations. A wave on a rope can illustrate vertically and horizontally polarized waves, while polarized sunglasses can demonstrate the two polarizations of light.
Gravitons have a theoretical spin of 2 related to the observed 45° angle between the two polarizations of gravitational waves. The Higgs particle has a spin of zero. The Standard Model of particle physics says that the universe is made with five fundamental bosons and twelve fundamental fermions. Just two of the bosons and two varieties of fermions are relevant here, illustrated in Fig. 2.
Simplified diagram of the fundamental constituents of matter: five types of bosons and twelve types of fermions. The middle line lists the two boson and fermion types that are most relevant here. The third line represents the relevant observational consequences. Note: Protons and neutrons are composite particles of three quarks. A proton consists of two up quarks and one down quark, while a neutron comprises of two down quarks and one up quark. Quarks, however, cannot be detected in isolation.
Simplified diagram of the fundamental constituents of matter: five types of bosons and twelve types of fermions. The middle line lists the two boson and fermion types that are most relevant here. The third line represents the relevant observational consequences. Note: Protons and neutrons are composite particles of three quarks. A proton consists of two up quarks and one down quark, while a neutron comprises of two down quarks and one up quark. Quarks, however, cannot be detected in isolation.
The most important boson for this paper is the photon, comprising a spectrum of 28 decades of frequency that includes radio, visible light, and x-rays. Gravitons have not yet been proven to exist, but they are represented by a similar spectrum of gravitational waves that are a major focus of physics discovery today.
The fermions come in two varieties: quarks and leptons. They all have half a unit of quantum spin. Two types of quark triplets create protons and neutrons with a net spin- . They are composite fermions. Electrons and neutrinos are both fundamental fermions, the first crucial to our understanding of everything, and the other one almost invisible even though 1014 pass through our bodies every second. They are crucial for understanding nuclear physics. A neutron with spin- can decay three particles: a proton, an electron, and an anti-neutrino. To conserve the total angular momentum of the system, the combined spins of the proton, electron, and neutrino must match the initial spin of the neutron.
The above description shows how quantum spin is a central concept for understanding matter. We present activities that enable students to grasp these concepts.
3. Spinning toys and quantum spin technologies
To emphasize the relevance of quantum spin, our learning sequence includes examples of macroscopic phenomena in which quantum spin plays a central role. These phenomena can be explained by observing spinning toys combined with the concept of vector addition of spins.
First, students use simple spinning tops to learn a right-hand rule to define the spin vector. They learn that the length of the spin vector is proportional to the total spin and that spin is a combination of rotational inertia and spin speed. Examining the different spinning toys, students can arrange them in order of rotational inertia by testing the difficulty of accelerating the toy's rotation.
4. Adding quantum spin: Stacking tops and permanent magnets
Next, students recognize that spins can be added by graphical vector addition in which spin arrows are placed head-to-tail. Stacking tops allow students to explore spin addition while recognizing the key aspect of classical spin: it can continuously fall to zero.
The combination of spin and electric charge gives every electron a magnetic dipole moment like a tiny bar magnet. Interestingly, neutrons which are electrically neutral also have a magnetic dipole moment that arises because of the charge and spins of the three quarks from which they are made.
Permanent magnetism arises when the quantum spins of unpaired electrons in iron are aligned and added together. Each electron has an associated magnetic dipole moment. The total spin and associated magnetic moment of the electrons then becomes macroscopic. Magnetism is the result of quantum spin, and in classrooms, experiments with magnets can be used to reinforce the explanation of the quantum spin origin of ferromagnetism. Students stack magnets so that they repel each other, and they can feel the repulsive magnetic forces providing a tangible experience with magnetism as in Fig. 3.
Students stack toroidal magnets on a wooden rod to create a magnetic spring from appropriately oriented magnets to explore the power of electron spin addition.
Students stack toroidal magnets on a wooden rod to create a magnetic spring from appropriately oriented magnets to explore the power of electron spin addition.
5. Turning fermions into bosons by pairing
Two tops spun in opposite directions roughly cancel their total spin. By the right-hand rule, the vector arrows cancel when placed head-to-tail. For quantum spins, two spin-half particles (fermions), when combined, either exactly add to create a total spin of one or cancel to create a spin of zero. Atoms are comprised of fermions—protons, neutrons, and electrons. If the total number of fermions is even, the total spin will be an integer number of quantum units. For the atom Helium-4 (4He), with two protons, two neutrons, and two electrons, the spins are normally in canceling pairs, and the total spin is zero. Thus, 4He is a boson. For 3He, which has one fewer neutron, the total spin is 1/2 so it is a fermion. This difference means that 4He can become a superfluid, able to flow with zero friction, at a temperature of when a large fraction of atoms enter the ground state. This is only allowed for bosons.
Interestingly, at low temperatures, electrons in certain metals, as well as 3He atoms, can form into pairs called Cooper pairs, which turn them into bosons. When this happens in metals, they transform into superconductors with zero electrical resistance. Something similar happens to 3He, transforming it into a superfluid at an ultra-low temperature. These phenomena illustrate the powerful effects of quantum spin.
6. Spin precession in tops and proton spin precession for body imaging
Spin precession occurs when an external force, such as gravity, acts on a classical spinning object such as a spinning top or a gyroscope. Spin precession of protons is harnessed in magnetic resonance imaging (MRI) and is used for medical imaging. The quantum spins of protons are made to precess in a magnetic field (Fig. 4). The precession produces a radio signal at a frequency determined by the magnetic field strength. It is sufficient here to simply note that this means of imaging via quantum spin precession provides very detailed images of the body's internal structures.
The MRI technique: (a) The precession of a spinning top due to a gravitational torque and a proton due to a magnetic torque. (b) MRI image, obtained by examining the radio-frequency precession signals when the protons in water molecules precess in a magnetic field; Image Courtesy: Johns Hopkins University; Licence: Wikicommons (https://commons.wikimedia.org/wiki/File:T1-weighted-MRI.png).
The MRI technique: (a) The precession of a spinning top due to a gravitational torque and a proton due to a magnetic torque. (b) MRI image, obtained by examining the radio-frequency precession signals when the protons in water molecules precess in a magnetic field; Image Courtesy: Johns Hopkins University; Licence: Wikicommons (https://commons.wikimedia.org/wiki/File:T1-weighted-MRI.png).
7. Flipping tops and flipping quantum spins
The process of spin precession in a top or gyroscope demonstrates that spinning objects resist changes to their axis of rotation. It takes physical work or energy to reverse the spin axis of the spinning top. At the quantum level, photon energy is required to flip the spin of an electron. The energy of flipping the electron spin in hydrogen atoms produces photons of 21 cm wavelength that allow radio telescopes to see hydrogen throughout the universe, as shown in Fig. 5. For a spin flip, the photon supplies both spin and energy. The spin-1 photon combines with the electron spin- electron, resulting in a total spin of . The quantum probability of the photon–electron interaction is very low but is easily observed in enormous thin clouds of hydrogen gas in galaxies, allowing galaxies to be imaged as shown in Fig. 5. Table II summarizes the complete learning sequence. We start with phasor wheel activities and continue to spinning top activities. Each macroscopic observation is linked to the relevant quantum concept. Details on some activities are given in Sec. III.
(a) Two hydrogen atoms are shown with the quantum spins either aligned (top) or anti-aligned (bottom). The energy of the spin flip needed to change from one state to the other can be provided by a photon of wavelength 21 cm. The photon provides both the energy and the integer spin needed to flip the quantum spin. Image courtesy of https://en.wikipedia.org/wiki/Hydrogen_line. (b) Neutral hydrogen can be imaged in galaxies using 21 cm radio astronomy as shown in this map of galaxy NGC 5457, captured by the NRAO Very Large Array radio telescope. This image of local galaxy M81 captured by NASA Spitzer Space Telescope (Image courtesy of https://aasnova.org/2019/10/14/tracking-gas-in-star-forming-galaxies/#prettyPhoto).
(a) Two hydrogen atoms are shown with the quantum spins either aligned (top) or anti-aligned (bottom). The energy of the spin flip needed to change from one state to the other can be provided by a photon of wavelength 21 cm. The photon provides both the energy and the integer spin needed to flip the quantum spin. Image courtesy of https://en.wikipedia.org/wiki/Hydrogen_line. (b) Neutral hydrogen can be imaged in galaxies using 21 cm radio astronomy as shown in this map of galaxy NGC 5457, captured by the NRAO Very Large Array radio telescope. This image of local galaxy M81 captured by NASA Spitzer Space Telescope (Image courtesy of https://aasnova.org/2019/10/14/tracking-gas-in-star-forming-galaxies/#prettyPhoto).
The learning progression for quantum probability (1–6) and quantum spin (7–12).
1 | Phasor discovery and concept: waves can be described by arrows |
2 | Phasor wheel activity: tracing waves, adding waves |
3 | Two path interference: observing interference with lasers |
4 | Single-photon interference: recognizing probabilistic nature of interference |
5 | Phasor wheel interference |
6 | Intensity and probability: phasor inscribed area connects intensity and probability |
7 | What is spin? Discovering classical spin with tops |
8 | Proving quantum spin is spin: classic historical experiments |
9 | Quantum spin for photons and electrons: historical context, E = hf and magnetic moments |
10 | How quantum spin differs: quantization, two values, unstoppable |
11 | Discovering fermions and bosons: quantum spin and the nature of matter |
12 | Quantum spin in technology: magnets, imaging brains and imaging hydrogen |
1 | Phasor discovery and concept: waves can be described by arrows |
2 | Phasor wheel activity: tracing waves, adding waves |
3 | Two path interference: observing interference with lasers |
4 | Single-photon interference: recognizing probabilistic nature of interference |
5 | Phasor wheel interference |
6 | Intensity and probability: phasor inscribed area connects intensity and probability |
7 | What is spin? Discovering classical spin with tops |
8 | Proving quantum spin is spin: classic historical experiments |
9 | Quantum spin for photons and electrons: historical context, E = hf and magnetic moments |
10 | How quantum spin differs: quantization, two values, unstoppable |
11 | Discovering fermions and bosons: quantum spin and the nature of matter |
12 | Quantum spin in technology: magnets, imaging brains and imaging hydrogen |
III. IMPLEMENTATION OF ACTIVITIES
A. Phasor wheels for quantum probability
We introduce phasors using the demonstration-size phasor wheel shown in Fig. 1. The 25 cm wheel radius allows one student to roll it by the handle in front of a whiteboard while another student uses the cam rod to trace the wave displacement, which is roughly sinusoidal. Rolling the phasor wheel is a convenient way to create rotation and measure the phase for various paths. Wavelength is the circumference of the wheel and the wave amplitude is set by the cam pick-off diameter equal to the arrow length in this model. Once students are familiar with the phasor wheel, they use the 1.5 cm or 2 cm radius phasor wheels as shown in Figs. 6(a) and 6(b) with an A3 paper template to analyze a double-slit interference experiment, following the procedure outlined in Sec. II.
(a) The tabletop phasor wheel. (b) Students using the tabletop phasor wheel for probability wave interference.
(a) The tabletop phasor wheel. (b) Students using the tabletop phasor wheel for probability wave interference.
Figure 7 shows a completed example. Starting from an agreed phasor angle representing the incident phase of the impinging photons, students roll the wheel along the shortest path to record the phasor angle at the end. The template contains a blank phasor where students record the incoming probability wave phasor for each path. Below is an area for students to add the phasors to find the probability density. The phasors from the two paths are added head-to-tail to find the resultant phasor, which is used to inscribe a circle representing the probability density as described in Sec. II. The spacing of the double slits is designed to give two dark fringes and a bright spot. Numerical values proportional to intensity can also be obtained by squaring the resultant vector lengths (when the resultant is found using a ruler), allowing students to plot graphs of intensity vs position.
A worked example of a graphical double-slit interference tidied up for publication purposes. Students trace the phasor wheels along two paths to the selected points on the wall. The top two rows allow students to record the final phasor of each path at the correct scale. The third row allows students to add the phasors from each path. Students will find the central maximum and two dark fringes on either side. Errors are seen in the lack of symmetry in the final intensity circles, which arise from the qualitative measures of the phasor wheel.
A worked example of a graphical double-slit interference tidied up for publication purposes. Students trace the phasor wheels along two paths to the selected points on the wall. The top two rows allow students to record the final phasor of each path at the correct scale. The third row allows students to add the phasors from each path. Students will find the central maximum and two dark fringes on either side. Errors are seen in the lack of symmetry in the final intensity circles, which arise from the qualitative measures of the phasor wheel.
Students analyze the results of the graphical phasor wheel interference, comparing them with a physical interference pattern using either a double-slit apparatus or a human hair to provide alternative paths. Figure 8 shows a typical interference pattern that results from photons diffracting around the hair. These observations can be compared to the probability density prediction obtained graphically. Students with mathematical interest can measure fringe separations and estimate their hair diameter.
(a) Students set up a hair interference experiment, arranging hair strands on a frame and directing a laser beam toward them to observe the interference pattern projected onto a screen. (b) Magnified image of the interference pattern.
(a) Students set up a hair interference experiment, arranging hair strands on a frame and directing a laser beam toward them to observe the interference pattern projected onto a screen. (b) Magnified image of the interference pattern.
The student feels the resistance as she tries to flip the spin of the bicycle wheel.
The student feels the resistance as she tries to flip the spin of the bicycle wheel.
The phasor wheel activities are supplemented with coin-tossing experiments using online simulators such as Coin Toss.19 These experiments are designed to show the transition from probability to statistical certainty when the numbers become large, mimicking the transition from the probability of arrival observed in single-particle interference videos to intensity observed in the laser experiments.
At this point, students are ready for an extended program on Feynman path integrals in the context of the laws of reflection, refraction, and diffraction that are described in detail by Choudhary and Blair,9 but are not part of the sequence presented here.
B. Activities for exploring quantum spin
Having explored the connection between classical spin (i.e., angular momentum) and quantum spin as described in Sec. II, and examined how quantum spin is dramatically different, as described in Table I, students are ready to undertake the activities with the various spinning tops summarized in Table III.
Summary of the toys used for learning activities. Images: public domain courtesy of commercial suppliers.
. | Object . | Description of concepts conveyed with each toys . |
---|---|---|
1. | Spinning top ![]() | Right-hand rule: Students spin tops clockwise and counterclockwise, defining spin vectors using the right-hand rule and also observing spin precession. |
2. | Stacking tops ![]() | Adding up spins: Students successively add spinning tops into a pile. Each top has spin, determined by its mass, shape, and rotation speed. The spin of the entire pyramid is the sum of the spins of all the individual spinning tops. The vector sum of each individual spin represents the combined spin. |
3. | Gyroscope ![]() | Spin precession: A gravitational torque caused by the gyroscope weight causes the spinning gyro to precess around a vertical axis. Precession always occurs when a torque not aligned with the axis of rotation is applied to a spinning body. The precession speed depends on the magnitude of the torque. |
4. | Wheel with handles ![]() | Torques and forces: Students feel the forces acting when they try to flip the spin axis of the spinning wheel. They experience the work required to flip the spin and feel the reaction torques that act back (Fig. 9). |
5. | Flip-over top ![]() | Do flip-over tops flip their spin? The asymmetrical shape and mass distribution cause flip-over tops to wobble, tilt, and turn over. Careful inspection shows that the spin vector remains unchanged. The spin axis moves through the body of the top without changing the spin axis. When the top flips over onto its stem, students check the spin vector direction, confirming no spin reversal. |
. | Object . | Description of concepts conveyed with each toys . |
---|---|---|
1. | Spinning top ![]() | Right-hand rule: Students spin tops clockwise and counterclockwise, defining spin vectors using the right-hand rule and also observing spin precession. |
2. | Stacking tops ![]() | Adding up spins: Students successively add spinning tops into a pile. Each top has spin, determined by its mass, shape, and rotation speed. The spin of the entire pyramid is the sum of the spins of all the individual spinning tops. The vector sum of each individual spin represents the combined spin. |
3. | Gyroscope ![]() | Spin precession: A gravitational torque caused by the gyroscope weight causes the spinning gyro to precess around a vertical axis. Precession always occurs when a torque not aligned with the axis of rotation is applied to a spinning body. The precession speed depends on the magnitude of the torque. |
4. | Wheel with handles ![]() | Torques and forces: Students feel the forces acting when they try to flip the spin axis of the spinning wheel. They experience the work required to flip the spin and feel the reaction torques that act back (Fig. 9). |
5. | Flip-over top ![]() | Do flip-over tops flip their spin? The asymmetrical shape and mass distribution cause flip-over tops to wobble, tilt, and turn over. Careful inspection shows that the spin vector remains unchanged. The spin axis moves through the body of the top without changing the spin axis. When the top flips over onto its stem, students check the spin vector direction, confirming no spin reversal. |
Table III summarizes the main concepts linked to each toy, which in turn link to quantum spin concepts. Students work in small groups, undertake tasks, and relate their observations to relevant macroscopic quantum spin phenomena.
The spinning top (Table III) is used to introduce the concept of spin and define spin vectors. The stacking tops illustrate the vector addition of spins and allow students to practice vector additions. They relate vector addition to the alignment of electron spins in permanent magnets. The gyroscope illustrates spin precession, connected to the context of MRI imaging using the quantum spins of protons. The wheel with handles allows students to experience the work required to flip a spin and recognize that energy is required to flip a spin. This is contrasted with the flip-over, a flipping top, that appears to flip its spin without the need for an energy source. However, students observe that the final spin vector is unchanged: this is not a spin flip but merely a change in the top/bottom orientation of the mass. These two activities are linked to hydrogen imaging of distant galaxies.
IV. CLASSROOM IMPLEMENTATION
Following the Model of Educational Reconstruction, we conducted pilot trials of the learning sequences to measure evidence of students' learning and identify improvements.
A. School trials
Three different programs incorporated the learning sequence presented above (Sec. II). The programs included the historical context and activities from the above learning sequence as well as worksheets and tests developed for each age group. Representative samples of initial trial results are given below for three groups of students (groups A, B, and C) who learned either the quantum probability sequence or quantum probability and the quantum spin sequence. We describe the three groups, the teaching context and the participants below.
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Author K.A. supported a physics teacher who presented seven 45-minute lessons that began by introducing photon properties, phasors, laser interference, and quantum probability. The class included 19 students aged 14–15, with no prior experience of quantum physics concepts or vector addition. The students had learned about classical wave properties in their normal classes with their regular teacher.
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Author A.L. presented a series of eight 45-minute lessons to 44 students aged 11–12 who had no prior knowledge of quantum physics.
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Author A.L. presented a series of seven 55-minute lessons in an after-school program for 20 students aged 11–15. The students were divided into two age groups for the activities, covering the full learning sequence of Table II. The students had previously received approximately one hour of preliminary instruction on the topics of what photons are and how to add vectors (arrows).
B. Tests and data evaluation
Post-tests were created to assess students' understanding. The questions related directly to the activities undertaken and took approximately 20 minutes for students to complete. Quantum concepts are not traditionally covered in the school curriculum and we had no benchmarks or prior data to calibrate our expectations of student outcomes. Test items were validated by a panel of eight academics including physicists, educational researchers, and curriculum specialists. The experts examined validity in terms of language and age appropriateness. Group C completed a six-month retention test (N = 19). Student data were anonymized and marked. Full marks were given to responses demonstrating a full Einsteinian understanding of the relevant concept.
V. CLASSROOM RESULTS
In this section, we summarize students' results from these initial trials to provide evidence of students' success in understanding the presented concepts. Table IV summarizes the test questions labeled Q1–Q12 and the class results. Questions are given in summary form, omitting associated pictures and visual content.
Summary of the questions and class results labeled by groups (mc indicates multiple choice and sa indicates short answer).
Group . | No. . | Questions . | % . |
---|---|---|---|
A | Q1 | Phasor addition (graphical) | 89 |
A | Q2 | Select phasor pairs to give the highest intensity (mc) | 74 |
A | Q3 | Connection between phasors and intensity spots (sa) | 68 |
A | Q4 | Use of photon terminology to explain question 3 (sa) | 42 |
B | Q5 | Phasor addition for estimation intensity in the interference pattern (mc) | 90 |
C | Q6 | Explaining interference pattern with probability (mc) | 79 |
C | Q7 | Connection between probability and intensity in interference pattern (mc) | 84 |
B | Q8 | Spin vector direction using right-hand rule (graphical) | 63 |
B | Q9 | Spin vector addition and cancellations (graphical) | 73 |
Retention test | |||
C | Q10 | Discreet nature of quantum spin (mc) | 95 |
C | Q11 | Quantum spins and magnetism (false/true) | 95 |
C | Q12 | Recall of two quantum spin concepts (sa) | 100 |
Group . | No. . | Questions . | % . |
---|---|---|---|
A | Q1 | Phasor addition (graphical) | 89 |
A | Q2 | Select phasor pairs to give the highest intensity (mc) | 74 |
A | Q3 | Connection between phasors and intensity spots (sa) | 68 |
A | Q4 | Use of photon terminology to explain question 3 (sa) | 42 |
B | Q5 | Phasor addition for estimation intensity in the interference pattern (mc) | 90 |
C | Q6 | Explaining interference pattern with probability (mc) | 79 |
C | Q7 | Connection between probability and intensity in interference pattern (mc) | 84 |
B | Q8 | Spin vector direction using right-hand rule (graphical) | 63 |
B | Q9 | Spin vector addition and cancellations (graphical) | 73 |
Retention test | |||
C | Q10 | Discreet nature of quantum spin (mc) | 95 |
C | Q11 | Quantum spins and magnetism (false/true) | 95 |
C | Q12 | Recall of two quantum spin concepts (sa) | 100 |
Students in group A completed a post-test that included four items relating to phasor addition in the context of a hair interference experiment, photon paths, and quantum probability (Q1, Q2, Q3, and Q4). In group A, 89% of students correctly added phasors (Q1), while 68% correctly explained the connection between phasor lengths and the intensity observed in an interference pattern (Q3). Question 4 was designed to test students' understanding of the connection between phasors and photon arrival probability. Only 42% of students explicitly made this connection. Often, there was insufficient emphasis on the connection between intensity and arrival probability in responses. In line with the Model of Educational Reconstruction, the learning sequence was modified for groups B and C to emphasize photon arrival probability. This includes online coin-tossing experiments to reinforce the concepts of probability. Group C was explicitly tested on quantum probability, with results indicating that this modification was successful based on outcomes to different questions targeting the same understanding (Q7 and Q8).
Group B students were younger than group A, but had similar results to group A on items relating to phasor addition and the connection between phasor resultant, interference patterns, and intensity (Q2 and Q5) with scores of 74% and 90%, respectively.
Group C students were asked two questions explicitly testing their understanding of quantum probability (Q6 and Q7). Their scores of 79% and 84% indicate that most students grasped the connection between phasor length, quantum probability, and intensity.
Groups B and C participated in both the quantum probability and quantum spin learning sequences. In Group B, 73% of students could grasp the addition of spin (Q9), while 63% could determine the direction of the spin vector of a spinning top (Q8). We noted that most errors regarding the definition of spin vectors were associated with difficulty defining rotation direction from the diagram rather than an inability to use the right-hand rule.
A. Retention test
A retention test was given to group C six months after the program to test students' knowledge of quantum spin. The test revealed 95% of students remembered the discrete nature of quantum spin, correctly choosing: “[spin] can only take specific values” (Q10). Furthermore, 95% of students recalled that quantum spin can be added, causing macroscopic phenomena like magnetism (Q11). In the test, students were asked to write two special things about quantum spin (Q12). All students were able to recall relevant aspects of quantum spin. Responses included: “quantum spin is constant,” “spins can add up,” “it makes magnets,” “spin can flip,” “it can be used for imaging,” and “it can precess” (Q12).
It was clear from the responses that all students remembered the toys presented in Table III, and students connected the toys with the macroscopic spin phenomena connected to them, such as magnetism and imaging.
VI. CONCLUSION
We have demonstrated a route by which school science can be made relevant to the modern, technology-driven era by combining knowledge of historical experiments with modern experiments on single-particle interference, as well as exploring the connections between quantum spin and fundamental science. This approach illuminates the possibility that classical and quantum concepts can be taught as an integrated whole rather than as separate and distinct areas of study.
The observed successful outcomes indicate that learning sequences such as those presented here can allow the concepts of modern physics to be introduced to students aged 11–15. They support the general case that children can appreciate the quantum description of matter at an early age if presented at the appropriate level. Thus, we conclude that the modernization of science education in the context of Einsteinian physics is possible and could prepare students for future learning and future careers. We have been funded to conduct additional trials of the proposed learning sequences over the next five years to collect data for a thorough statistical analysis of the learning outcomes and to evaluate how this learning influences students' attitudes toward modern physics.
The program outlined here is part of an effort to radically restructure science education from primary school to middle school. An 8-year curriculum called Eight Steps to Einstein's Universe is designed to ensure that all students gain a conceptual understanding of the fundamental concepts of the Einsteinian science that powers modern technology and is our best understanding of physical reality. We are currently offering a micro-credential course, Einsteinian Science for School Teachers, to enable educators to effectively present our carefully structured content across the age range 7–15.3,4
SUPPLEMENTARY MATERIAL
See the supplementary material for short instructional videos for delivering activities, a worksheet, and examples of test questions.
ACKNOWLEDGMENTS
The authors express their deepest gratitude to all members of the Einstein-First collaboration for their invaluable contributions. The authors especially thank Dr. Jesse Santoso, Dr. Shon Boublil, Dr. Tejinder Kaur, and Johanna Stalley for their contributions to the development of learning sequences, questionnaires, and activity trials. The authors are also grateful to Professor David Treagust and Professor Marjan Zadnik for their valuable advice, and to David Wood for generously sharing his extensive experience in working with schools and educators. Additionally, the authors extend their gratitude to Professor Ju Li for providing essential guidance and organizational support, along with other team members. The authors sincerely thank Peter Rossdeutscher and Professor Howard Golden for their efforts in securing donation funds to supplement our ARC Linkage funding (LP180100859), which enabled the development of our online training programs. The authors are also grateful to the ARC Centre of Excellence for Gravitational-Wave Discovery (OzGrav) for their unwavering support, particularly in facilitating the creation of school kits for their activities. Additionally, the authors acknowledge the enthusiastic support of their Einsteinian Physics Education Research (EPER) collaborators. K.A. and A.L. acknowledge their respective Australian Government Research Training Program (RTP) Scholarships and a UWA fees scholarship. The authors extend thanks to the West Australian Department of Education for their generous support, the Independent Schools Association of Western Australia for enabling many of our trials, and the Science Teachers Association of Western Australia for their continuous and essential assistance. Finally, the authors are deeply grateful to the principals, teachers, and students of the participating schools for allowing them to conduct the program and for granting permission to use their photographs and data for research purposes. This research was supported by the Australian Research Council (LP180100859). The research was carried out under the University Ethics Approval No. 2019/RA/4/20/5875.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.