A new teaching-learning sequence to promote secondary school students’ learning of quantum physics using Dirac notation

This paper describes the design of a new teaching-learning sequence on quantum physics aimed at upper secondary school students. In this teaching-learning sequence, GeoGebra simulations and interactive screen experiments are used to investigate the behaviour of a single photon at beam splitter and single photon interference in a Michelson interferometer. We propose a minimal formalism using Dirac notation, which avoids complex numbers and elaborate vector calculus, to make a quantitative description of the quantum optics experiments accessible to secondary school students. With this new educational pathway, we take into account findings from physics education research, which suggest that the introduction of a mathematical formalism tailored to students’ abilities might help them to overcome naive-realist views of quanta or space-time descriptions of quantum phenomena, while at the same time facilitating a transition to a functional understanding of quantum models.


Introduction
Learning about topics in quantum physics has increasingly received attention in secondary school education in recent years.Several studies have explored difficulties secondary school students encounter when being introduced to quantum concepts [1][2][3][4][5][6][7][8][9][10][11] and many of the difficulties widespread among learners have been found to originate from inadequate ideas of quantum physics models [12].As a consequence, many learners stick to naive-realist views or space-time descriptions of quantum phenomena despite formal instruction: For example, learners often imagine atoms to be small planetary systems [13] or assign well-defined classical trajectories to photon [6].In more general, physics education research has highlighted that (a) many learners have a tendency to overinterpret the underlying models as accurate representations of reality and that (b) learning quantum physics is related to the development of elaborate views of models [14].The authors of [15] support the above claims with empirical evidence from a study with N = 116 upper-level secondary school learners: Students with a more functional understanding of the photon model outperformed students with a more gestalt-oriented view on a conceptual inventory of basic quantum concepts.
Taken together, prior research suggests that helping students in the transition of a gestalt-to a functional understanding of (quantum) models might be promising in supporting students' conceptual development in the quantum realm.However, the research-based development and evaluation of instructional sequences that focus on fostering the development of a functional understanding of quantum models among learners constitutes a research desideratum to date.This paper describes the design of such a teaching-learning sequence taking into account empirical evidence from prior research according to which introducing a mathematical formalism adapted to the mathematical abilities of the target group might indeed help learners in the development of a functional understanding of (quantum) models.
In the next section 2, we provide a summary of the research background of mathematics in quantum education.In the subsequent sections, we describe the structure of our teaching-learning sequence and outline the content aspects covered (section 3), before we provide an in-depth description of the educational path in section 4.

Mathematics in quantum instruction
A mathematical description of (quantum) physical phenomena inherently leads to an abstract view of the underlying models and has been argued to be useful in helping learners to detach from gestalt thinking.Therefore, physics education researchers have argued in favour of mathematical approaches in quantum instruction: For example, Pospiech et al introduce the Dirac approach to represent two-state systems and found that a mathematical approach promotes the development of elaborate views of basic quantum concepts among students [16].Furthermore, with the help of an acceptance survey, the authors showed that the Dirac formalism is perceived as a useful tool by learners [16].
In further study, Lovisetti et al [17] examine the impact of an axiomatic description of quantum physics based on the fundamental principles of linear algebra on student learning.In a 15 h course with high school students, the researchers demonstrated that the selected method is beneficial to students' learning.Additionally, they discovered that their approach enables learners 'to derive all fundamental quantum properties, predicting the results of some experiments and explaining the behaviour of quantum systems' (p 17).
Scholz et al [18] present an educational concept focusing on quantum optics effects (i.e.anticorrelation and single photon interference) which cannot be explained with any classical model of light in order to 'demonstrate the striking differences between classical physics and quantum physics' (p 1).The authors provide an 'appropriately rigorous quantum optical theory' to explain the experimental results in a 'quantumsemantic way' (p 1) suitable for learners at the undergraduate level.Prior research supports the approach presented by Scholz et al [18] since the discussion of quantum optics experiments in the secondary school classroom has shown to be conducive to student learning [19].
With the design of a novel teaching-learning sequence on quantum physics for the secondary school level we build upon the work by Scholz et al [18] in a threefold sense: (1) We focus on quantum optics experiments to introduce students to basic quantum concepts.(2) We introduce a reduced Dirac formalism that avoids complex numbers and elaborate vector calculus to make a quantitative description of quantum optics experiments accessible to secondary school students with a less profound mathematical background.(3) At the same time, we provide secondary school students with a 'representation of a reduced quantum theory based […] on mathematics and formal arguments' ( [18], p 19) in order to support the students' transition to a functional understanding of the photon model and to promote the rejection of the idea of the photon as a hard ball following a well-defined trajectory.

Structure of the new teaching-learning sequence
The teaching-learning sequence is aimed at upper secondary school students and is intended to serve as a useful resource for teachers who wish to go beyond a mere qualitative description of quantum physical effects in their teaching presented.The teaching concept comprises a total of five lessons of 45 min each-apart from the fifth lesson, which takes 90 min or two times 45 min.
In terms of content, the teaching concept is divided in three parts (see table 1): In the first part (lesson 1-1), the students repeat key aspects of electromagnetic waves using GeoGebra simulations developed specifically for this teaching concept.The parts two and three cover different quantum experiments (two lessons each): While the students investigate the behaviour of a single photon at the beam splitter in the part two, single photon interference in the Michelson interferometer is discussed in the third part.In both cases, a qualitative description of the experimental outcomes using interactive screen experiments developed by Bronner et al [20] is followed by a quantitative description using a minimal Dirac formalism (see section 4).
Worksheets are provided for all the lessons to (a) allow for a smooth implementation of the teaching-learning sequence in classroom practice, and to (b) serve as a summary of the contents covered for the learners.The worksheets and all further teaching materials such as the GeoGebra simulations and the interactive screen experiments can be obtained online 4 .

Description of the new teaching-learning sequence
Prior to quantum instruction, secondary students most often are introduced to a wave model of light.In order to ensure that the necessary learning prerequisites are available, we developed three GeoGebra simulations to repeat key aspects regarding the behaviour of electromagnetic waves incident on a beam splitter or a Michelson interferometer in part one of the teaching-learning sequence to lay a foundation for the discussion of these experiments in the single photon realm in the subsequent parts.

Part one: repetition on electromagnetic waves
Lesson 1-1 constitutes a repetition on electromagnetic waves.At the beginning, the students explore that when light falls on a beam splitter, it can be detected at both, the reflected and transmitted output ports.In our teaching-learning sequence, we focus on (lossless) 50/50 beam splitters, i.e. the light intensity measured at the two output ports is equal.We developed a GeoGebra simulation 5 to make the students grasp that a beam splitter causes a phase shift of 90 • between the electromagnetic waves at the transmitted and reflected output ports [21].A screenshot from the corresponding GeoGebra simulation is shown in figure 1.
In a further step, the Michelson interferometer is introduced as an experimental set-up to produce interference.Therefore, we developed a further GeoGebra simulation6 (see figure 2).Specifically, the simulation is used to relate the phase differences caused by different arm lengths, i.e. the different distances of the beam splitter from the mirrors M R and M T , with constructive or destructive interference observed on the screen.
To facilitate learners' reasoning about the interference pattern, we introduce the visual representation of phases using phasors.A further GeoGebra simulation 7 allows the students to explore the impact of the phase difference on the interference pattern (see figure 3) in the case of waves with equal wavelength and amplitude which aligns with the experimental situations discussed in the teaching-learning sequence presented in this article: specifically, the GeoGebra simulation helps students realising that if the phase difference between the waves equals 180 • , destructive interference can be observed, while constructive interference can be observed in the case of a phase difference of 0 • .By convention, the phase arrows are chosen to be of length one (see figure 3), as they do not express the amplitude of the waves, but only their phases.
The experiments discussed in this first phase of the teaching-learning sequence are of particular relevance for learning about the fundamental aspects of quantum physics in the single photon regime: For example, according to Marshman and Singh [22], quantum optics experiments 'elegantly illustrate the fundamental concepts of quantum mechanics such as wave-particle duality of a single photon, single photon interference, and the probabilistic nature of quantum measurement' (p 1).All of these quantum aspects are touched upon in the further parts of our teaching-learning sequence making use of the foundations laid in this introductory part presented in this section.

Part two: single photon states at a beam splitter
In the second and third parts of the teachinglearning sequence, interactive screen experiments developed by Bronner et al [20] allow students to

Lesson 2-1: anticorrelation of single photon states.
In the first lesson of part two, using an interactive screen experiment (see figure 4), the students observe a lack of coincident events between the detectors D R or D T positioned at the reflected and transmitted output ports of the beam splitter (see figure 5) when a single photon is incident on the beam splitter.Hence, this experiment demonstrates the anticorrelation of single photon states which cannot be explained with any The corresponding worksheet guides the students through the experiment and helps the learners to draw the following conclusions from their observations: (1) The photon behave as indivisible and indistinguishable energy portions of light.(2) They are detected at either detector D T or detector D R (apart from random coincident events at both detectors).(3) The single events (i.e.detection at D T or D R ) cannot be predicted, while 'statistical predictions (for many repetitions) are possible' ( [24], p 3).
The first conclusion leads to the introduction of term quantisation to students, i.e. photon can be regarded as elementary energy portions of light.Furthermore, the experimental observations two and three can be supported through aspects that are at the core of quantum science-quantum superposition and quantum measurement: In our experiment, the single photon state is first 'converted into […] a superposition of transmitted and reflected' ( [24], p 4) through the beam splitter, and is then detected at one of the detectors with equal probabilities of 50%.Thus, quantum randomness results from the measurement performed on a quantum superposition state [25].By introducing a reduced quantum formalism in the following lesson, students are guided to reflect on their experimental observations in more detail.

Lesson 2-2: introduction of a minimal Dirac formalism.
In general, the quantum formalism does not allow a space-time description of the behaviour of quanta (i.e.what occurs between the preparation of a quantum state and the measurement).Instead, the system can be described in terms of quantum states which are introduced in lesson 2-1 of our teaching-learning sequence.We provide a worksheet guiding through this lesson which is accessible through the homepage corresponding to this article 9 .
Before introducing quantum states to students, we suggest reminding them of physical states in classical physics: in classical physics, if we know the state of a physical system, it is possible to make predictions about its evolution over time.For example, if the initial position and velocity of a classical particle are known, it is possible to predict its trajectory using Newton's laws of motion.In quantum physics, in contrast, predicting the time evolution of an initial quantum state is only possible up to the measurement when the probabilities of the possible events can be given.
In our teaching-learning sequence, we use the so-called Dirac notation to describe quantum states: An arbitrary state of a single photon is denoted by |ψ ⟩ ('ket-psi').In our experimental setting (see figure 5), the initial state of the single photon |ψ ⟩ 1 is |ψ ⟩ 1 = φ 0 • • |S⟩.The symbol |S⟩ indicates that the single photon is emitted from the source (S).The symbol φ 0 • is the phase coefficient.The phase coefficients express the phases of the single photon states (similarly in the case of electromagnetic waves), φ 0 • and indicates that we choose the initial state as a reference (0 • ).
The state of the single photon is then changed as the beam splitter creates a superposition state |ψ ⟩ 2 with respect to the arms (R) and (T): The students are guided to make use of their experimental observations in lesson 2-1 to read the superposition state |ψ ⟩ 2 as follows: • The single photon is not in states |R⟩ or |T⟩ simultaneously, but they are in a totally different state, namely the superposition of both, which is expressed by the symbol '+'.• The phase coefficients φ 90 • and φ 0 • express the phases of the possible states: while in the case of transmission there is no phase shift, there is a phase shift of 90 • phase shift in case of reflection at the beam splitter.Similar to circle representation given in the case of electromagnetic waves, the phase coefficient φ α can be represented through an arrow of length 1 with angle α to the x-axis on a circle of radius 1 (see figure 6).
In the lesson, we provide the students with an interpretation of the square lengths of the phase arrows using probabilities: for example, in the case of the single photon at the beam splitter, the square lengths of the phase arrows are related to the probabilities p R and p T of detecting a single photon at either detector D R or D T : This interpretation of the square lengths is supported through the students' experimental observations in the previous lesson: Since their lengths are the same (=1), the probabilities to detect a single photon at the reflected or transmitted output port of the beam splitter (in states |R⟩ or |T⟩) are equal.
In terms of the formalism, while it is clear to the students that the absolute value is needed to obtain the arrow lengths, the square may seem superfluous at first sight.However, at the end of the teaching-learning sequence, students will recognise its inevitability.
There is a crucial difference between the arrow representation of the phase coefficients used in our reduced Dirac formalism compared to the wave formalism which we suggest to make transparent to learners: the length of the phase arrows can not only be one but also can have different (square) lengths.
It is noteworthy that we propose to not normalise the quantum state together with the students because this reduces the mathematical formalism: We justify this didactical decision because no scalar product has been performed yet.Furthermore, we emphasise a crucial difference with waves, where the amplitudes are related to the length of the projections of the phases to the direction of oscillation, rather than to the lengths of the phases themselves.

Part three: single photon states in a Michelson interferometer 4.3.1. Lesson 3-1: quantum interference.
In lesson 3-1, the beam splitter experiment is extended through two mirrors (one of which can be moved), leading to a Michelson interferometer setup.The corresponding worksheet is accessible via the article homepage 10 and guides the students through an investigation of the behaviour of a single photon in the Michelson interferometer using an interactive screen experiment (see figure 7).
In the interactive screen experiment, students observe that the number of detections at the output port of the interferometer depends on the arm length differences in the interferometer.We have created a GeoGebra animation 11 that shows that the arm length differences cause a relative phase difference between the optical paths, whichsimilar to the case of electromagnetic wavesleads to constructive or destructive interference,  evidenced by increasing or decreasing detection rates (see figure 8).
We reflect on these observations together with our students by reading a part of the 1986 paper by Grangier et al [23] who describe the anticorrelation and the Michelson interferometer experiments with a single photon as follows: 'They illustrate the wave-particle duality of light.Indeed, if we want to use classical concepts, or pictures, to interpret these experiments, we must use a particle picture for the first one ('the photons are not split on a beam splitter') […].On the contrary, we are compelled to use a wave picture ('the electromagnetic field is coherently split on a beam splitter') to interpret the second (interference) experiment' ( [23], pp 178-9).The students realise that a classical description of the experiments is not appropriate, because photons are neither classical particles nor waves.To highlight that the 'quantum interference phenomenon shown experimentally is a consequence of the interplay of superposition and nonlocality' ( [18], p 17), we make use of the reduced Dirac formalism in the final lesson of our teaching-learning sequence.In the general case, the position of the mirror M T is arbitrarily changed.The educational path of the teachinglearning sequence presented in this article culminates in a description of the quantum interference experiment for arbitrary phase shifts using the reduced Dirac formalism to help students map experimental observations to a formal description.However, before we proceed with the calculations with the students, it is crucial to recall that in the reduced quantum formalism the phases on a phasor can be represented by arrows.With this representation in mind, the students accept that these phases are not vectors but have some similar properties: they can be assigned a length and they can be summed.Finally, the students can be introduced to a general superposition state which can be written as where the coefficients are the phase coefficients (α and β), and the states are arbitrary.The interpretation of the phase coefficients in relation to probabilities has already been introduced in lesson 2-2 and can be used in this lesson again: Hence, the ratio of the probabilities of the possible events (1) and ( 2) is the ratio of the absolute squares of the corresponding coefficients: Finally, we guide the students through the formal description of the effect of a Michelson interferometer on a single photon state using the reduced Dirac formalism to obtain single photon interference in accordance with experimental observation.In terms of practical implementation, we again provide our students with the Overview of calculation steps required to obtain a correct description of the effect of a Michelson interferometer on single photon states using the reduced Dirac formalism proposed for the teaching-learning sequence presented in this article. Step The single photon is emitted from the single photon source.
the photon is chosen as a reference (see phase coefficient φ 2 The state of the single photon was changed by the beam splitter.
2, the state of the reflection |R⟩ is in a phase shift of 90 • with respect to the source and the transmission.
Since the arm length difference between the interferometer arms (R) and (T) may vary, we focus on three specific cases (A) to (C).Case (A): The distances between the mirrors and the beam splitter are the same.

A
The effect of the mirrors on the quantum state.
Students know that a reflection from a mirror causes a 180 • phase shift.However, as stated in section 4.3.1,we are only considering the phase differences between the optical paths.Hence, as there is no difference in arm lengths, the state remains the same.

A
After the mirrors, the effect of the beam splitter on the single photon state is considered again.

|D⟩
We consider the two terms of the superposition state |T⟩ differently using the effect of the beam splitter:

|D⟩
where the phase of |S⟩ has increased by 90 • due to the reflection.The final state after the beam splitter therefore is:|ψ

|D⟩
From the previous lessons, the students are aware that the 180 • of relative phase shifts leads to destructive interference, hence φ (Continued.)

A
Finally, the output state can be given.

|ψ ⟩ 5A =
|D⟩ A single photon state can always be detected at the detector.(superposition collapse).

Case (B): The position of the mirror M
T is changed by an eighth of a wavelength causing a 90 • relative phase shift.

3B
The effect of the mirrors on the quantum state.
T is a quarter of the wavelength of the light further away, the state |T⟩ gets a 90 • phase shift (see figure 3).

4B
After the mirrors, the effect of the beam splitter on the single photon state is considered again.
)|D⟩ The effect of the beam splitter on the single photon state is considered as before:

|D⟩
Therefore, the final state after the beam splitter is:  3).

C
After the mirrors, the effect of the beam splitter on the single photon state is considered again.
Then the effect of the beam splitter: and The final state after the beam splitter is therefore: However, the 270 different calculation steps in an incorrect order and give them the task of putting the steps in the correct order and justifying their decisions.We suggest that students work in pairs or groups.
Obviously, the first two calculation steps required are similar to the ones given in table 2, leading to the quantum state |ψ ⟩ 2 = φ 90 • |R⟩ + φ 0 • | T⟩ created by the beam splitter.In the next step, we need to take care of the phase of |T⟩ which is increased by an arbitrary angle x, since the length of the interferometer arm (T) is increased by ∆L : x = (360 • • 2∆L) /λ , with λ being the wavelength of the photon.Hence, the state of a single photon is changed by the different arm lengths of the interferometer (R) and (T) leading to the state: Then, the beam splitter has an effect on the state again.The output state can be written by considering the two components of the superposition, as has been done in table 2: Therefore, the final quantum state is: At this point, a very new situation occurs because the sum of phase coefficients has not been considered so far.The phase addition creates a new phase with-in general-a length different from one which removes the previous comfortable situations where the probabilities of the possible events were the same.We can also see that the change in the position of the mirror M T changes the coefficients which results in different probabilities of detecting the photon at the detector.This addition of these phase coefficients can be done just like in the case of vectors, even though they are no vectors.Since the probabilistic interpretation require absolute squares, we just have to calculate the square of the length of these new phases (see figure 10).
First, we calculate the absolute-square of (φ 180 • + φ x ), or in other words, the square length of the corresponding arrow using elementary trigonometry and the Pythagorean theorem: where we used the theorem of addition cos 2 (x/2) − sin 2 (x/2) = cos (x) , in the last step.The same method holds for the other coefficient: The ratio of the absolute squares is equal to the ratio of the probabilities p S of detecting a single photon at the source (state |S⟩) and p D of detecting a single photon at the detector (state |D⟩): ) .
Since p S + p D = 1 must hold and using the well-known trigonometric identity sin 2 (x/2) + cos 2 (x/2) = 1, the probabilities are p S = sin 2 (x/2) and p D = cos 2 (x/2).Moreover, since x = (360 • • 2∆L) /λ, the result can be also written as p S = sin 2 [(360 Our result is fully in harmony with the experimental observation that the students made in lesson 3-1 (see section 4.3.1).This is the moment when we can talk about why the absolute squares meant the probabilities and not just the absolute values.If the squares were left, then the probabilities would just be a sine and cosine function, which is inconsistent with the experimental results.
At this point the students reinterpret the meaning of the superposition state.Previously, we said that the superposition state, e.g.φ 90 • |R⟩ + φ 0 • | T⟩ expressed the probabilistic nature of quantum physics.However, single photon interference leads to another key concept of quantum physics.A single photon cannot be localised, i.e. no classical trajectory can be assigned to a single photon: it is delocalised over the whole interferometer.The state φ 90 • |R⟩ + φ 0 • | T⟩ expresses that a single photon do not have a certain trajectory of reflection (R) or transmission (T).
Through discussing the anticorrelation of single photon states and single photon interference, both on the basis of experimental data and a formalistic description in our teaching-learning sequence, it becomes clear that a photon can be neither a particle nor a wave in the classical sense, and we discuss this with the students.By bringing together the views of experiment and theory, the teaching-learning sequence is designed to help students move away from naive-realistic views of quanta, while at the same time encouraging the development of a functional understanding of the photon model.

Limitations of the reduced Dirac notation
We believe that for the sake of minimal formalism it was not necessary to introduce the normalisation condition of states, since no scalar products are performed.Our introduction is also based on the fact that we only consider 50/50 beam splitters, i.e.only waves with equal amplitudes interfere.However, according to the learning path presented, the cases of different beam splitters can be considered by simply changing the length of the phase arrows to the square root of the probability.The teaching-learning sequence can be continued as suggested by Scholz [18], where a more general interpretation of the phases can be seen, also avoiding complex numbers.
We have replaced the complex phases φ x = exp (ix) by a visualisation of phases as arrows, which have some similarities to vectors (addition and length), although phases are no vectors.It is important to note that in quantum physics an absolute square is required to determine the probabilities, as shown in section 4.2.2.In classical wave formalism, however, the real parts of the complex phases are relevant.

Summary
In this paper, we have presented a new secondary school teaching-learning sequence that provides a minimal formalism using Dirac notation to help upper-level secondary school students get a grasp of some basic principles (e.g.quantisation, probability interpretation, superposition, measurement effect, absence of trajectory) and phenomena (anticorrelation and quantum interference) of quantum mechanics.Although the formalism is highly simplified, it can be used to introduce students to a new level of representation of quantum physics, with the advantage that they are not limited by narrow models, but provide a faithful description of quantum phenomena.Initial classroom experiences are promising, showing that students can cope with the proposed formalism if carefully guided from experimental observations to formal descriptions-in particular, students have provided first important feedback on the different teaching materials (from GeoGebra simulations to interactive screen experiments and worksheets), which will help us to refine the material in the future.Finally, the teaching-learning sequence seems promising in terms of supporting students' transition to a functional understanding of the photon model and in promoting the rejection of the idea of the photon as a hard ball following a well-defined trajectory.Future research will focus on an empirical evaluation of the teaching-learning sequence proposed in this article to better understand its effect on students' conceptual development.

Figure 1 .
Figure 1.To help students understand the effect of a 50/50 beam splitter on an incident electromagnetic wave, we developed a GeoGebra simulation.

Figure 2 .
Figure 2. Screenshot from the GeoGebra simulation on the Michelson interferometer.The user can manipulate the position of the Mirror M T to investigate the dependence of the interference pattern on the phase differences.The simulation allows to switch light sources (laser and a single photon source) and, hence, is also useful in the single photon regime in a later stage of the teaching-learning sequence.

Figure 3 .
Figure 3. GeoGebra simulation 6 which allows to explore the impact of the phase difference between two waves (red and blue) of equal wavelength and amplitude on the resulting wave (dotted line).

Figure 4 .
Figure 4. Using an interactive screen experiment developed by Bronner et al [20], the students can study the behaviour of a single photon incident on a beam splitter.explore single photon experiments 8 : these interactive screen experiments contain data from a quantum optics laboratory and 'present real experiments with the help of photos from different perspectives and at different times during experimentation' [19, p 3].In our teaching-learning sequence, students are encouraged to investigate the behaviour of a single photon at a beam splitter (part two) and in the Michelson interferometer (part three) using the digital tools, which allow students to 'conduct the experiment […], view it from different perspectives, display and hide tables for documenting the measurement data, or switch devices on and off' [19, p 3].

Figure 5 .
Figure 5.The schematic representation of the experiment investigating the behaviour of a single photon at the beam splitter.

Figure 6 .
Figure 6.The quantum phase coefficients in the state φ 90 • • |R⟩ + φ 0 • • |T⟩ are represented on a circle.As the phase arrows have the same length, the probabilities of finding the photon on the corresponding detectors are the same.

Figure 7 .
Figure 7. Using an interactive screen experiment developed by Bronner et al [20], the students can study the behaviour of a single photon in the Michelson interferometer.

Figure 8 .
Figure 8.Our GeoGebra animation 12 of the single photon interference experiment.

Figure 9 .
Figure 9.The schematic representation of the Michelson interferometer.

4. 3 . 2 .
Lesson 3-2: quantum interference-a description using Dirac notation.The schematic representation of the Michelson interferometer experiment is shown in figure 9.The position of the mirror M T can be changed, but in the beginning of the lesson, we consider specific cases regarding the phase shift caused by the change in position.Students are guided through the formalism, which can be followed in table 2, by being given the task of putting the various calculation steps required to obtain the output quantum state in the correct order on the corresponding worksheet.Specifically, the classes are divided into three groups (A), (B), and (C) and the students of each group deal with one specific case such that the students can reflect on their different results in an ensuing discussion.The cases are as follows: (A) The distances between the mirrors and the beam splitter are the same.(B) The position of the mirror M T is changed by an eight of a wavelength causing a 90 • relative phase shift.(C) The change in position of mirror M T (by a quater of a wavelength) produces a 180 • phase shift.
the phase of |D⟩ is increased by 90 • due to the reflection.

Table 1 .
Overview of the parts, lessons included comprising the teaching-learning sequence presented in this article and the teaching approaches taken.