Key topics for quantum mechanics at secondary schools: a Delphi study into expert opinions

Figures & data

Table 1. The categorisation used by PISA 2006 (OECD, 2006).

Goals for scientific literacy
Competencies	Knowledge	Attitude
Identifying scientific issues	Scientific concepts	Interest in science
Explaining phenomena scientifically	The nature of science	Support for scientific inquiry
Using scientific evidence		Responsibility towards resources and environments

Table 2. Overview of aspects of scientific literacy guiding the topic choice for a curriculum, based on OECD (2006) and DeBoer (2000).

	Aspects of scientific literacy	Description
Competencies	Identifying scientific issues	Enabling students to recognise scientific issues and key features of scientific investigation
	Explaining phenomena scientifically	Enabling students to apply scientific knowledge in a given situation, to interpret scientific phenomena and identify appropriate descriptions
	Being able to make informed decisions	Enabling student to identify, interpret and be critical about scientific evidence-related evidence in media and conversations, reflect on the societal implications and make informed decisions
Knowledge	Understanding of scientific concepts	Promoting knowledge and understanding of topic content and relations between topics which are considered fundamental for students to know
	Understanding the nature of science	Promoting the understanding of scientific inquiry, data analysis, scientific explanations and models, and limitations of scientific knowledge
	Knowledge of science as a cultural force	Promoting knowledge of the historical development of scientific ideas, current understandings in science and their effect on science and society
	Knowledge for future careers	Promoting knowledge about and needed for future careers or further studies in science
	Understanding the relationship between science and technology	Promoting understanding of the nature of technology and the interdependence of science and technology (e.g. technological applications based on scientific inquiry).
Attitude	Interest in science	Promoting students’ engagement in science-related social issues, their willingness to acquire scientific knowledge and skills, and their consideration of science-related careers
	Support for scientific inquiry	Promoting students’ appreciation of and support for scientific inquiry
	Responsibility towards resources and environments	Promoting students’ sense of personal responsibility for maintaining a sustainable environment, and willingness to take action
	Seeing the influence of science in everyday life	Prompting students to see the applications of science in their daily lives, and have a more informed and intelligent experience with the natural world
	Appreciating the beauty of science	Promoting students’ appreciation of and fascination for the natural world

Figure 1. The procedure followed in this Delphi study.

Table 3. Overview of the experts’ research fields.

Research field	Expertise	Number of experts round 1	Number of experts round 2	Number of experts round 3	Number of expert interviews
Physics	High energy physics	2	1	1	1
	Quantum physics	3	3	2	2
	Solid-state physics	2	2	2	2
	Particle physics	2	0	1	1
Mathematics	Quantum mathematics	2	2	1	1
Chemistry	Solid-state chemistry	1	1	1	1
	Polymer chemistry	1	0	0	0
Biophysics	Nano photonics	2	2	2	1
	Biophysics	2	1	1	0
Total		17	12	11	9

Table 4. The most frequently proposed quantum mechanics topics in round one (top 11 out of 89 items, N = 17).

Topic	Number of experts
Spectral lines	16
Tunnelling	12
Photoelectric effect	11
Probability	11
Wave–particle duality	11
Double slit experiment	10
Energy levels and quantisation	10
Hydrogen atom	10
Heisenberg’s uncertainty principle	9
Lasers	9
Wave function	9

Table 5. The categories resulting from analysing the experts’ responses in round 1.

Categories	Experts’ descriptions	Experts’ arguments
Wave–particle duality	A particle shows both wave and particle behaviour. Single photon detection, the double slit experiment, the photoelectric effect, and the delayed choice experiment can illustrate this dual behaviour	Understanding of scientific concepts, understanding the relation between science and technology, seeing the influence of science in everyday life
Wave functions	A particle can be described by a wave function Ψ. The wave function can be a superposition of all possible wave functions. Ψ^2 is a measure of the particle’s probability distribution	Understanding of scientific concepts, understanding the nature of science, understanding the relation between science and technology, identifying and explaining scientific issues, seeing the influence of science in everyday life, appreciating the beauty of science
Atoms	Electron’s energy levels are quantised, which determines spectra and colours of atoms. Quantisation can be explained with Bohr’s atomic model, and with the quantum atomic model. The electron configuration also depends on Pauli’s exclusion principle	Understanding of scientific concepts, knowing science as a cultural force, understanding the relation between science and technology, seeing the influence of science in everyday life
Subatomic particles	Subatomic particles have properties, which can be described by quantum numbers. An important property is spin, which is important in magnetism and electron configuration, and can be illustrated by the Stern–Gerlach experiment	Understanding of scientific concepts, understanding the relation between science and technology
Materials	Molecules and metals have energy bands and band gaps, which determine material properties such as strength, structure, colour and resistance	Understanding of scientific concepts, knowing science as a cultural force, understanding the relation between science and technology, seeing the influence of science in everyday life
Nonlocality	QM violates local realism, which can be illustrated by the fact that entangled particles, when separated, cannot be described independently. This phenomenon shows the counterintuitive character of QM and is important in information technologies	Understanding of scientific concepts, understanding the relation between science and technology
History	QM plays an important role in the history of science and was one of the most important scientific revolutions of the twentieth century. It shows the nature of science	Knowing science as a cultural force, understanding the relation between science and technology, seeing the influence of science in everyday life

Table 6. Overview of the topics selected by the experts in round two (N = 12), together with the number of experts who wanted the topics to be taught at secondary schools.

Concepts	Number of experts	Examples	Number of experts	Applications	Number of experts
‘de Broglie’ wavelength	12	Double slit experiment	12	Solar cells	9
Particle behaviour of light	12	Atomic structure	12	Quantum information	9
Probability	12	Periodic table	12	STM	8
Energy levels and quantisation	12	Spectral lines	12	Lasers	8
Wave–particle duality	11	Photoelectric effect	11	LEDs	8
Wave function	11	Hydrogen atom	10	Quantum computers	8
Heisenberg’s uncertainty principle	11	Bohr’s atomic model	10	Single photon detection	7
Tunnelling	11	Colour	10	Spectral analysis of stars	7
Pauli’s exclusion principle	11	Magnetism	10	Transistors	7
Spin	11	Orbitals	9	Quantum cryptography	7
Momentum	10	Material properties	9	Atomic clock	6
Fermions and bosons	10	1D infinite well	8	Fluorescence	6
Superposition	8	Radioactive decay	8	Neon lamps	6
Time evolution	8	Schrödinger’s cat	8	MRI	6
Quantum numbers	8	Bonds	8	IC’s and chips	6
QM at a macroscopic scale	7	Semi-conduction	8	Quantum teleportation	6
Entanglement	7	Conduction	7	GPS	5
History of QM	7	Heat radiation	7	Microwaves	5
Complementarity	6	Polarisation	6	CCD	5
Zero point energy	6	Energy bands	6	Giant magneto resistance	5
Subatomic particles	6	Super-conduction	6	CT scan	4
Standard model	6	Chemical reactions	5	SEM	3
Bohr versus Einstein	6	Stern–Gerlach experiment	5	Random generators	3
Foundations of QM	6	Delayed choice Experiment	4	Single molecule microscopy	3
Schrödinger equation	5	Crystal structures	4	Flash memory	3
Stationary states	5			Bennet–Brassard protocol	3
Measurement	5			PET scan	2
EPR paradox	5
Development of atomic models	5
Free vs. localised particle	4
Locality and causality	3
Bell’s inequalities	3

Figure 2. The number of experts in round three (N = 11) who considered the listed topics indispensable, desirable, optional or dispensable.

Table 7. Mean expert ranking in the third round (N = 11) on the importance of the selected quantum topics for the secondary school curriculum. Rank 1 is considered most important.

Rank	Concepts	Mean rank	Examples	Mean rank	Applications	Mean rank
1	Wave–particle duality	2.10	Double slit experiment	2.10	Solar cells	2.70
2	Particle behaviour of light	3.50	Spectral lines	4.20	STM	3.10
3	Wave function	4.20	Photoelectric effect	4.30	LEDs	3.60
4	De Broglie wavelength	4.60	Atomic structure	4.60	Lasers	3.70
5	Probability	4.80	1D infinite potential well	6.00	Quantum information	3.90
6	Energy levels and quantisation	6.80	Hydrogen atom	6.30	Quantum computers	4.00
7	Heisenberg’s uncertainty principle	7.40	Periodic table	6.60
8	Superposition	9.10	Bohr’s atomic model	8.20
9	Spin	9.40	Radioactive decay	9.40
10	Tunnelling	9.70	Bonds	11.40
11	Pauli principle	10.60	Orbitals	11.60
12	Momentum	11.20	Magnetism	11.70
13	Quantum numbers	11.50	Schrödinger’s cat	12.20
14	Fermions/bosons	11.70	Colour	12.20
15	Time evolution	13.40	Material properties	12.40
16			Semiconductors	12.80

Table 8. The arguments regarding scientific literacy used by the experts (N = 9) during the interviews.

	Goals for scientific literacy	No. of experts	No. of fragments	Topics mentioned
Competencies	Identifying and explaining scientific issues	5	7	Heisenberg’s uncertainty principle, energy levels and quantisation, Schrödinger’s cat, quantum information, quantum computers
	Being able to make informed decisions	–	–	–
Knowledge	Understanding of scientific concepts	9	67	All
	Understanding the nature of science	3	4	Double slit, wave function
	Knowing science as a cultural force	4	4	Material properties
	Being aware of career opportunities	1	1	Quantum information, quantum computers
	The relationship between science and technology	2	2	Wave–particle duality, probability, semiconductors
Attitude	Interest in science	–	–	–
	Support for scientific inquiry	1	1	Quantum information, STM.
	Responsibility towards resources and environments	–	–	–
	Seeing the influence of science in everyday life	8	11	Wave–particle duality, ‘de Broglie’ wavelength, Heisenberg’s uncertainty principle, quantisation and energy levels, tunnelling, atoms, 1D infinite potential well, radioactive decay, spin, fermions/bosons, material properties, lasers
	Appreciating the beauty of science	2	3	Wave–particle duality, spin, tunnelling, quantum information, quantum computers

OECD. (2006). Assessing scientific, reading and mathematical literacy, A framework for PISA 2006. Paris: OECD Publishing.

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DeBoer, G. E. (2000). Scientific literacy: Another look at its historical and contemporary meanings and its relationship to science education reform. Journal of Research in Science Teaching, 37(6), 582–601.

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