Dirac’s Approach to Quantum Mechanics in Physics Teacher Education: From Linear to Circular Polarisation

The teaching/learning of quantum mechanics via two-state systems is continuously spreading in secondary schools due to its promising results in physics education research. One possibility is the use of polarisation states of photons. This paper reports on a polarisation-based introduction to quantum mechanics in physics teacher education. A widely used school material prepares teacher trainees for their future work and also improves their conceptual knowledge. This part includes statistical calculations using only secondary school mathematics and a new formulation of the uncertainty relation, using only real numbers. The second step is to prepare the formalism of quantum mechanics using real two-dimensional vectors and matrices. Considering that students may not learn complex linear algebra, we offer a new way to introduce the complete formalism of two-state systems via circular polarisation providing a step-by-step exploration of complex quantum states. This points out the advantage of using complex linear algebra via a physical example, providing the opportunity to reach the elements of advanced quantum physics and quantum computing while deepening the physics background of the secondary school material.


Introduction
Quantum physics is taught in most European secondary schools [1].Traditional approaches based on wave mechanics and historical framework are widespread, but in the last decades two-state approaches using vector formalism [2][3][4] have been appearing continuously, motivated also by the emergence of a new discipline, quantum computing.The interest in two-state approaches is reinforced by the choice of the topic of the 2022 Nobel Prize in Physics [5].In response, a new aspect of physics teacher education is emerging.This paper reports on a pilot project on quantum mechanics conducted in the fall semester of 2022 at Eötvös Loránd University (ELTE), Budapest.The audience consisted of university students participating in physics teacher education and the learning path is based on Dirac's polarisation approach [6], which has been developed into a school material in Udine [2,[7][8].
Paper [9] points out that a common difficulty for teacher trainee is the lack of conceptual and didactic knowledge in quantum mechanics.One of the aims of this paper is to present a well-researched secondary school material [2] to prospective physics teachers which will equip them with didactic and conceptual knowledge and also help them to explore some of the fundamental laws of quantum mechanics and its simplified formalism using only real numbers, as in secondary schools.[9,10] Since this is a university course, this paper extends the secondary school material to university level by providing the appropriate physical background knowledge.This paper extends the results of a previous

General overview
At ELTE, teacher trainees are introduced to quantum mechanics in their 4 th year of university studies.Before that, they learn the basics of linear algebra of real vector spaces, differential equations, complex numbers, probability theory and some basic atomic physics via historical events.It is important to note that students do not encounter complex vectors and matrices before their quantum studies.
The seminar augmenting the main lecture on quantum mechanics includes 13 lessons (90 minutes each) at ELTE.In the pilot project reported here, the first 2 lessons cover the secondary school material [2,[11][12][13].Students explore the phenomena of polarisation through experiments with polarisers, the same as in secondary school.They then explore the elementary features of two-state quantum mechanics using only real numbers, just as secondary school students do, but in a much shorter time.As the papers [11,12] show, the term "eigenvalue" is used in addition to "property" (the term "property" is a secondary school analogue of eigenvalue in [2]).The third and fourth lessons go beyond the secondary school level, students explore the linear algebra behind quantum phenomena using two-dimensional vectors and matrices as reported in [11].After understanding "real number two-state quantum mechanics", students are led to the more complete description of quantum mechanics using complex numbers via circular polarisation.Another purpose is to provide a learning path that is also suitable for getting acquainted with some current features of quantum computing [14][15][16][17][18] supporting the international collaboration of Quantum Technology Education [17].
Section 3 of the paper summarises the secondary school level statistics of quantum mechanics.In Section 4, a more complete exploration of the linear algebra behind two-state quantum mechanics using only real numbers is presented than in [11].In Section 5, this continues with the introduction of complex states, which reveal the description of general two-state systems of quantum mechanics.Table 1 summarises the syllabus of the seminar.This paper focuses on the content of lessons #3-5. Renewal of the mathematical formalism for the complex case. The general uncertainty relation.

Secondary school level statistics of photons, conceptual knowledge
After the phenomenology of lesson #1, students investigate single photon thought experiments.If only a single photon falls on a polariser, then the probability of transmission is cos 2 ϑ (the Malus law) where ϑ is the angle between the polarisation direction of photon and the transmission axis of the polariser.[2] These experiments are supplemented by the previous pilot projects [11][12]: measured values can be assigned to the permitted outcomes with a polariser: λ1 = 1 and λ2 = 0 correspond to the event of free passing through and full absorption.These assigned values are called eigenvalues.In a general state, the single photons usually do not possess a unique value with respect to a physical quantity (polarisation) before a measurement is performed.This feature is called quantum uncertainty in [2][3][4].
To characterise the possible outcomes when measuring a given quantity (polarisation) A, students perform statistical calculations as e.g., paper [11][12] suggests.Provided the probability p1 = cos 2 ϑ (p2 = sin 2 ϑ) of measuring eigenvalue λ1 (λ2) is known in a sequence of repeated identical experiments, the expected value ⟨A⟩ of quantity A in a given state is because of the choice of λ1 = 1 and λ2 = 0, this is the classical interpretation of the Malus law; conceptually, the average of a large number of repeated identical measurements.The square of the standard deviation ΔA of quantity A is: (ΔA) 2 = <A 2 > -<A> 2 , providing a conceptual link to the qualitative description of how uncertain a measurement is.This secondary school way of calculating excepted value and standard deviations [19] will also appear in Section 4 in the standard formalism of quantum mechanics providing a connection between secondary school and university knowledge.
An important aspect is to explore the uncertainty relation.If we consider two polarisation measurements (A and B) with different polarisers (neither parallel nor perpendicular) on the same quantum state, students can conclude that these measurements cannot be accurate simultaneously.This fact can also be expressed by an uncertainty relation [12]: Such quantity pairs are often called incompatible [2] or non-commuting observables [20][21][22].

Exploring the formalism of real number two-state quantum mechanics
An important feature of quantum mechanics described in detail in [2] is that a vector can be associated with any quantum state, a feature that is obviously recognisable from the polarisation approach, the vector corresponds to the polarisation direction of light.Two unit vectors can be assigned to special states, so-called basis states or eigenstates, in which the measurement of the selected quantity leads to a certain value, one of the eigenvalues.The incoming photons are usually in a general state called the superposition state.
Considering a measurement with a polariser of horizontal orientation, the basis states (eigenstates) are the horizontal unit vector h and the vertical unit vector v (v corresponds to full absorption, because vertically polarised photons certainty became absorbed by a polariser of horizontal orientation).The state uϑ of a photon with polarisation direction ϑ to the horizontal can be written as a linear combination of the basis states: where ψ1 and ψ2 are the coefficients of the linear combination.This is the superposition principle: not only the basis states (eigenstates) are possible states, but also their linear combinations.The coefficients have a physical meaning too, |ψ1| 2 and |ψ2| 2 are the probabilities of measuring the eigenvalues λ1 or λ2, and also the probabilities of projecting the state of the photons into the corresponding basis states h and v, respectively.The probability appearing in the Malus law can be also expressed as the square of the scalar product of two unit vectors, the state uϑ and the basis pointing along the transmission axis (h): as this is the probability of transmission through the polariser.The probability of absorption can be calculated in the same way, (uϑ, v) 2 = cos 2 (uϑ, h) = cos 2 (ϑ -π/2) = sin 2 ϑ.And the orthogonality of states (e.g., v and h) can also be expressed via a scalar product [2]: Because of the probabilistic interpretation: and because of the normalisation 1 = |ψ1| 2 + |ψ2| 2 = uϑ 2 also holds.The use of the absolute value seems unnecessary, since only real numbers are used.However, it is suggested in this paper to emphasise that a state and its negative have the same physical meaning.It will be advantageous in Section 5 when complex states are introduced.This is the point when the university level appears.While in secondary school, the linear combination is used only in the horizontal-vertical frame, avoiding the use of the matrix formalism [2], in university the students learn that any two orthogonal states can be the basis states, which are determined by the choice of the quantity to be measured.The vectors of the basis states point along the transmission axis and a direction perpendicular to it in the case of polarisation.[12] The quantum state can also be represented by a column vector, in horizontal-vertical representation, ) and uϑ = cos ϑ h + sin ϑ v = ( cos ϑ sin ϑ ).The basis states of a measurement of a polariser of orientation φ are uφ = ( cos φ sin φ ) and uφ+π/2 = ( -sin φ cos φ ) (uφ for transmission and uφ+π/2 for absorption).Now, the previously mentioned scalar product in ( 4) can now also be calculated using the matrix product rule, where the transpose of a vector is denoted by a superscript T.
After that, students realise that the matrix A ̂ of the polarisation measurement A is since it possesses eigenvalues λ1 = +1 and λ2 = 0 with eigenstates uφ and uφ+π/2.They can also check that the expected value <A> of measurement A with the corresponding matrix A ̂ in a state uϑ is This is the usual formalism of quantum mechanics [20] in full harmony with the secondary school level (1), indeed gives cos 2 (ϑ -φ) in accordance with the Malus law, since the angular difference between the polarisation direction of the photons and the transmission axis of the polariser is (ϑ -φ) and the eigenvalues are λ1 = 1 and λ2 = 0. We also show that the matrix A ̂ of the polarisation measurement A can be expressed as: where  is the unit matrix, σ ̂x = ( 0 1 1 0 ) and σ ̂z = ( 1 0 0 -1 ) are the Pauli-X and Pauli-Z matrices, respectively [22].One obtains that α = 1/2, β = sin φ cos φ = (1/2) sin (2φ) and δ = 1/2 -sin 2 φ = (1/2) cos (2φ): Equations ( 10) and (11) express that any polarisation measurement matrix can be expressed as a linear combination of the unit matrix  and the Pauli-X and Z matrices.This also means that any matrix representing any arbitrary physical quantity have to be symmetric, since the linear combination contains only symmetrical components: This feature ensures that the eigenvalues are real and the eigenstates are orthogonal to each other, which is in full harmony with the physical interpretation of the eigenvalues, and the experimental experience of polarisation.
Let us consider another (polarisation) measurement represented by the matrix B ̂ written as where ϕ is the direction of the transmission axis of polariser B. The following observations can be made about the simultaneous behaviour of quantities A and B, i.e. two measurements on the same quantum state.If the eigenstates of the two matrices are not the same, the two quantities cannot be accurate simultaneously, so that the uncertainty relation (2) holds.
An exciting property of the real number quantum mechanics has emerged, namely that the traditional way of writing the uncertainty relation as a meaningful inequality [20] is not possible.To show this, secondary school arguments can be found in the paper [11], but here the pilot projects supplement this with the general uncertainty relation ΔAΔB ≥ |<[A ̂, B ̂]>| / 2, where [A ̂, B ̂] is the commutator of A ̂ and B ̂ [20].If we write the two matrices as a linear combination as in ( 11) and ( 13), and also use the special feature of the commutators of σ ̂x and σ ̂z, which is proportional to the matrix ( 0 -1 1 0 ), we see that the commutator is antisymmetric.The expected value <[A ̂, B ̂]> = u ϑ T [A ̂, B ̂] u ϑ turns out to be 0 for every real state uϑ.The conclusion is that the general relation yields ΔAΔB ≥ 0 in this case.It makes no sense to write an inequality for the uncertainties in the real number quantum mechanics, because the standard deviations are nonnegative by definition, so the inequality is empty.Only equation (2), found earlier, properly expresses the uncertainty relation at a secondary school level.

Circular polarisation, introducing complex quantum states
The most obvious way to introduce complex quantum states in polarisation approach is via circularly polarised photons.Books [20][21][22] suggest various possible learning methods, but this paper offers a new way.The main question to be discussed is how to mathematically describe circular polarisation states based on previous knowledge of linear polarisation states.What follows is a step-by-step learning path that also takes into account the students' thoughts.
Let us denote the state of a left-circularly polarised photons by L. The experimental observation that half of the circularly polarised light passes through a linear polariser of any orientation and half is absorbed plays an essential role.This means that a circularly polarised photons must be an equally weighted superposition of any linearly polarised state and a state orthogonal to it.The first task: "Show that the state (

1/√2)h + (1/√2)v cannot be a left-circularly polarised state, even though this state satisfies the condition that the probability of passing through a polariser of horizontal direction is 1/2."
Most students are able to show that this superposition state is the diagonal state u45 and cannot be a circular polarisation state because photons in state L have 1/2 probability of passing through a polariser with 45° permitted direction, but this probability is 1 in the given state.
The next task is the extension of the concept of linear combination.It is easy to recognize that finding a suitable superposition is hopeless if only real numbers are used."Where on the plane are the vectors that are an equal weight combination of the horizontal and vertical bases?Are there any that are an equal weight combination of the diagonal (u45) and antidiagonal (u135) bases?"The students realise that only the null vector satisfies the second condition.This is the moment when we can incorporate the advantage of using the absolute value in the interpretation of the transition probability in superposition states (6)."Previously, the use of the absolute square |ψ1,2| 2 instead of the square (ψ1,2) 2 for the probabilistic interpretation of coefficients in superposition was taken to emphasise that a quantum state and its negative have the same physical meaning.Do you have any idea utilizing the concept of absolute value which can help to describe the circular polarisation states?"Some students hit upon the idea of allowing the imaginary unit i as a coefficient.To support this idea, the next task is to investigate the state "What is the probability of measuring values certainly in state h and v (14)?(This state is different from the diagonal u45, and is denoted by L since it corresponds to the state of left circular polarisation, as will be shown later. )" The answer is obvious, the probability of measuring state h and v is |1/√2| 2 = 1/2 and |i/√2| 2 = 1/2 respectively.Continuing the exploration, the following question is asked: "What is the probability of measuring state u45 and u135 in state ( 14)? (Note the change of the bases.)"The answer is: So the two probabilities are |(1 + i)/2| 2 = 1/2 and |(i -1)/2| 2 = 1/2.It is clear that ( 14) is an equally weighted superposition of the bases h and v, but also of u45 and u135, simultaneously.Moreover, it can be shown that the state L is an equally weighted superposition of any two orthogonal linear polarisation states.To this end, we use the exponential form of complex numbers and write (15) as e iπ/4 u45 + i √2 e iπ/4 u135.(16) More generally, L can be written as a linear combination of arbitrary orthogonal linear polarisation basis states: e iφ uφ+π/2, (17) where φ corresponds to the transmission axis of a polariser, and φ + π/2 corresponds to the full absorption of a photon of polarisation φ.The feature that |e iφ | = 1 is important because it shows that introducing the phase φ in the coefficients does not make any difference regarding the probability of the measurement.Multiplying a state by a factor e iφ does not change its physical meaning (multiplication by -1 and by i are special cases of this).This property is expressed by rewriting (17) as The formula ( 18) is very similar to (14) showing the fact that the coordinate frame can be arbitrary, and half of the circularly polarised light passes through a linear polariser of any choice of φ.At this point the mathematical rules of the real number two-state quantum mechanics summarised in Section 4 have to be modified.Since all states are represented by unit vectors, the definition of the scalar product (probability amplitude) should be changed from (7) to if u is a complex quantum state, the symbols * and † denote the complex conjugate of a number and the adjoint of a vector, respectively.The expected value of the quantity A in an arbitrary complex state u is (replacing ( 9)): The discussion of complex quantum states followed here provides for those who have no prior knowledge of complex linear algebra, through physical examples, pointing out the advantage of this mathematical structure.
Based on this, the students can write the right-circularly polarised state of photons as a superposition of two linearly polarised basis states, denoted by R, using the rule of orthogonality ((L, R) = 0): For better understanding of the complex phase, it is worth mentioning that a complex state u can be e iζ v, (22) which also corresponds to a superposition state.The state L described by (14)  Another aim is to find the matrix of measuring circularly polarised states.It should be a 2×2 complex matrix, to which a natural way to associate eigenvalues is λ1 = +1 (left polarisation) and λ2 = -1 (right polarisation) with the eigenstates L (14) and R (21), respectively.A direct investigation shows that such a matrix is It can be checked that C ̂L = +1L and C ̂R = -1R, then L † C ̂L = L † • (+1)L = 1.The matrix C ̂ is nothing other than σ ̂y, the Y-component of the Pauli matrices [20][21][22].It is important to note that C ̂ = σ ̂y is not symmetric, as it would be in the real number description.At this point we can see that the matrices describing physical quantities must be Hermitian, i.e. matrices equivalent to their own conjugate transpose, and Hermitian matrices have real eigenvalues and orthogonal eigenstates.
Comparing the matrix of circular polarisation (24) with that of linear polarisation (8) the simplicity of the former is striking.This is because no preferred directions play a role in a circular polarisation problem, in contrast to the linear case, where the orientation of the polariser is essential.In fact, circular polarisation is a rotationally symmetric state mirroring a fundamental property of photons.This property is the conservation of the intrinsic angular momentum component pointing along the direction of propagation, called spin in this respect.In fact, C ̂ is proportional to the spin matrix whose eigenvalues are known to be ±ℏ [20][21][22].
As can be seen from ( 17) and ( 21), the state uφ describing the free transmission of photons from a linear polariser of orientation φ can be expressed with the bases L and R of circular polarisation as This means that a general linearly polarised state uϑ can be described equivalently both in the basis of uφ and uφ+π/2, special cases of which are h and v, and in the bases of L and R. Of course, this statement also holds for the complex quantum state u.
As a generalisation of (10), we find that any Hermitian matrix A ̂ can be written as where α, β, γ and δ are real numbers [22].Using the form of (26) and the special feature of the commutators of the Pauli matrices, it can be easily seen that, the general uncertainty inequality ΔAΔB ≥ |<[A ̂, B ̂]>| / 2 is not empty.The right-hand side of the inequality will not be zero as before, it will be a function which depends on the state and turns out to be zero in special states.We can also conclude that the true mathematics of two-state systems with complex quantum states contains the real number two-state quantum mechanics as a special case with perhaps surprising features, as appeared with respect to e.g., the uncertainty relation.

Conclusions
This paper presents a method of teaching/learning quantum mechanics in physics teacher education based on a recent pilot project at Eötvös Loránd University, Budapest.The paper describes the structure of the quantum mechanics seminar, which builds on a secondary school level well-researched polarisation-based approach to quantum mechanics, using only real numbers, by favouring a conceptual knowledge.
Teacher trainees perform secondary school experiments and trivial statistical calculations just as students do in school.After that they explore some fundamental laws of quantum mechanics and related linear algebra, restricted to real numbers and two-dimensional vector spaces, since complex numbers are not available in secondary schools.After exploring the real number two-state quantum mechanics, university students are led to complex quantum states using the example of circular polarisation in a new step-by-step way proposed here.
We believe that the proposed teaching/learning path meets the challenges of physics teacher education.The first part of the approach provides professional secondary school teaching material, while at the same time teacher trainees become acquainted with the learning difficulties of secondary school students, didactic proposals for their future work, and get a background knowledge of the higher mathematics behind the school material.They also acquire conceptual knowledge with a continuous transition from real numbers to complex quantum states.The discussion of complex quantum states provides an opportunity to introduce complex linear algebra to those who have no prior knowledge of it, using physical examples, pointing out the advantage of this mathematical structure.University students can also experience that complex linear algebra complements real number linear algebra in a natural way.
The learning path can be continued with another discrete system, the physics of electron spin, and after that with higher dimensional quantum systems, e.g., quantum entanglement of particles.The seminar can be completed with the application of the Schrödinger equation and time dependence in quantum mechanics.In our experience, teacher trainees, who have understood the description of twostate systems via photon polarisation, are better able to understand the whole quantum mechanical apparatus, so we find this type of learning path promising, especially in the age of quantum computing.

Table 1 .
The proposed syllabus of the first seminar lessons on quantum mechanics in physics teacher education at ELTE.
(18)18)) is fully circularly polarised because e iπ/2 = i, but if ζ is different from π/2 or 3π/2, some linear polarisation may be present in the state.This can be demonstrated by changing the bases of (22) e.g., to the pair u45, u135: