AN INTRODUCTION TO THIS INVESTIGATION
Quantum mechanics is an area of immense importance to modern technologies and industries, covering a diverse range of applications from semiconductors and lasers to advances in nuclear medicine. Quantum mechanics is also a subject that most students have traditionally found both difficult and abstract. Despite these facts, quantum mechanics has not until recently attracted much pedagogical research and introductory courses are still taught in much the same manner as they have been for the past seventy years.
As an undergraduate, I found my studies in quantum mechanics very challenging both conceptually and mathematically. Yet it was not until later, as a secondary science teacher that I recognised quantum mechanics important place as the ‘flag ship’ of today’s modern physics.
In mid-1994 whilst teaching a secondary school physics module on ‘the wave properties of light’, I noticed that students within my class had difficulties concerning the wave/particle nature of matter. I went to the literature to investigate further and was surprised to find that very little education research was present. Despite the impressive advances in understanding how students conceptualise other areas of physics and chemistry, these had not impacted on or addressed the problems associated with quantum mechanics. I discussed issues concerning the teaching of quantum mechanics with several teaching colleagues and university academics, they suggested the difficulties encountered by students appear from a number of quarters: students lack a physical intuition for the subject; the concepts are often counterintuitive; the subject is shrouded in a highly mathematical formalism and these are further complicated by ongoing debates concerning how this very formalism should be interpreted. At a university level they commented on the gulf between the apparent practical applications and the mathematical formalism which provides an even greater challenge to academic teaching staff, who have limited time to cover the vast amount of material currently prescribed by undergraduate curricula.
Later that year I commenced a Master of Science research project to investigate ‘How students learn quantum mechanics’ (Fletcher 1997), in which I developed and administered a survey based instrument to first and third year university students in order to identify important concepts and conceptual difficulties. The conclusions suggested that the mental models students are working with are tenuous constructs, extended far beyond the point where they are buttressed by perceived relationships with other, better understood concepts. Methodologically it was recognised that there were a number of shortcomings associated with the project, mainly concerning the reliance on written responses taken in a short time. What was now required was to undertake an extensive program of student interviews to build upon this preliminary research.
Hence this doctoral research investigation was born. It was conducted at the University of Sydney and examined how quantum mechanics was taught in both the School of Physics and the School of Chemistry. Semi-structured, in-depth interviews of students and academic staff served as the primary research instrument for the study.
The purpose of this investigation is to explore the teaching and learning processes associated with delivering a tertiary level quantum mechanics curriculum. The investigation aimed to isolate key concepts, identify learning difficulties, identify teaching difficulties and so provide both teachers and curriculum developers with a valuable resource.
Quantum mechanics is the study of matter and radiation in the atomic world. For everyday objects, classical physics (Newtonian Mechanics) adequately describes what we observe; but when we have to deal with the very small, the inadequacies of classical mechanics soon become apparent. Scientists of the early 20th century needed to develop a new theory to describe the physics at the atomic level.
The evolution of this subject can be viewed in three stages : (1885-1912)1 a period in which there accumulated a variety of experiments and explanations that lacked unification; (1913-1922) which centred on the creation and development of Niels H.D. Bohr’s quantum theory; and finally (1923-1927) the period of development and formalisation of what is ‘officially’ called quantum mechanics.
During the period 1885 to 1912 a large number of experimental facts which could not be explained on the basis of existing theory were accumulated: the discovery of ordered series in atomic spectra by Johann J. Balmer, Theodore Lyman, Johannes Ryberg and Friedrich Paschen; the studies of blackbody radiation by Wilhelm Wein, John W.S. Lord Rayleigh and Sir James H. Jeans and its theoretical description by Max K.E.L. Planck; Albert Einstein’s contributions in the quantisation of energy in black body radiation, the photoelectric effect, the specific heat of solids; and Sir Ernest Rutherford’s planetary model of the atom.
The next stage began with the 1913 paper “On the Constitution of Atoms and Molecules” by Bohr which described the planetary model of a hydrogen atom based upon the quantisation of energy and angular momentum of the electron. Bohr’s theory provided an explanation to spectral phenomena and permitted the calculation of Rydberg’s constant. Bohr’s “simplistic” theory brought together many ideas and concepts that guided both experimenters and theoreticians.
Experiments by James Franck and Gustav L. Hertz in 1914, concerning the measurement of electron energy spent on exciting mercury atoms, was direct experimental evidence for the fact that an atom may change its energy only discretely. In 1916 Arnold Sommerfeld and Peter Debye came to the conclusion that the angular momentum components in the direction of the magnetic field are quantised, thus introducing the concept of the quantisation in space. This received confirmation in experiments conducted by Otto Stern and Walther Gerlach in 1922 on splitting of atomic beams in non-uniform magnetic fields.
The Bohr models continued to develop in the period between 1913 and the early 1920s. Work by Wilson and Sommerfeld allowed some of the ad hoc aspects of the theory (the insistence on circular orbits, for example) to be abandoned. Despite this, however, the model was inherently problematic and the internal contradictions associated with the very idea of quantisation and of discrete quantum jumps became progressively more apparent through the early decades of the century. In 1923 Bohr formally introduced the correspondence principle2 in his article “On the Quantum Theory of Line Spectra”. According to this principle, the laws of quantum physics must turn into the laws of classical physics for large values of quantum numbers of a system. Thus, despite the apparent incommensurability of the classical and quantum theories, the former has been of great importance in the discovery of laws in quantum mechanics.
The birth of Quantum Mechanics proper was marked by a series of experiments, the unification of ideas and concepts, and the development of consistent mathematical models. In 1923 Arthur H. Compton’s X-ray scattering experiments clearly indicated the existence of particle-wave properties of radiation. During 1923-1924 Louis de Broglie suggested in his doctoral thesis that wave-particle duality should be extended to all micro-particles and in 1927 the idea of duality was confirmed in several laboratories worldwide by experiments on electron diffraction.
In 1924 Satendra Nath Bose carried out fundamental studies, which were extended by Einstein in the form of a statistical theory for photons which came to be known as Bose-Einstein statistics. In the framework of this theory, Planck’s formula for blackbody radiation at last found a complete explanation. During 1925 de Broglie introduced the idea of matter waves described by the so called wave function, and Wolfgang Pauli formulated his famous exclusion principle for electrons3.
In 1926 Erwin Schrödinger in his paper “On Quantisation as an Eigenvalue Problem” used the wave concepts to introduce his well known differential equation for a wave function. Thus the calculation of finding the energy levels of a bound micro-particle was reduced to the problem of finding the eigenvalues of a particular differential equation. The same year Schrödinger published a paper demonstrating the equivalence of his method and that of Max Born, Werner Heisenberg and Pascal Jordan. While the formalisation of Schrödinger’s theory was readily accepted, the problem of the interpretation of the wave mechanics and the physical description of the concept of wave function remained the subject of heated debate for many years. Born, in 1926, proposed a probability interpretation of the wave function; matter waves were replaced by probability waves. The impossibility of interpreting the mathematical wave function as the amplitude of a certain real material field (as in electromagnetic fields) was recognised. This in turn meant that de Broglie’s matter waves could not be interpreted as classical waves of any sort. Interestingly mainstream textbooks seldom report the fact that quantum mechanics still has several longstanding questions concerning the interpretation of the formalism4.
Finally in 1927 Heisenberg introduced his uncertainty principle and showed how the concepts of energy, momentum and position could be included in the wave description of the micro-particle. The appearance of these relations marked the final break of quantum mechanics from classical determinism and established quantum mechanics as a statistical theory.
Lamb has captured the essence of the practising physicist’s approach to quantum mechanics by providing what is, effectively, a definition of the subject’s utility:
“^ (Lamb 1969)
For the student, the shift between the macro- and the micro-world is much more than merely a matter of terminology. Classical physics is based upon the relatively simple idea of the summation of forces and velocities. Quantum mechanics, however, is grounded in the notion of the probabilities of different events interfering with one another to result in the chance of an event occurring. The student is thus required to make the mental shift between classical mechanics, centred around the concepts of billiard ball collisions and an idealised motion of a projectile, and those of quantum mechanics centred around the probability of events.
Quantum mechanics is a very successful theory and underlies nearly all our current understanding of the physical world. Since its conception 70 years ago quantum theory has expanded to usefully describe the properties of the atomic nucleus, the behaviour of subatomic particles and the physics and chemistry of molecules and solids.
This knowledge underpins and has led to developments in modern technologies including medical techniques such as magnetic resonance imaging (MRI) which uses the spin properties of hydrogen nuclei to map the tissue structure under examination and advances in semiconductor technology that provide the infrastructure to build today's super computers. It becomes clear that a comprehensive training in the physical sciences is impossible without a serious study of quantum mechanics.
At the majority of tertiary institutions quantum mechanics is prescribed, perceived and therefore taught as a formal discipline. This formalism relies heavily on a mathematical framework and the subject can be broached from two perspectives, using Heisenberg's matrix approach which utilises linear algebra techniques or via Schrödinger's wave mechanical approach which employs differential calculus. In recent years a new teaching resource has emerged with the introduction of software packages which present the students with graphical and pictorial representations of quantum related phenomena. Examples include the exploration of potential energy diagrams in physics and 3-dimensional visualisations of bonding structures in chemistry.
Despite the revolution that quantum mechanics has inspired in twentieth century physics and chemistry, introductory quantum mechanics has been taught in the same fashion for the past seventy years and until recently there has been little pedagogical research directed towards the teaching of tertiary level quantum mechanics. We do not have tools to monitor a student’s conceptual development in the subject. It is not clear what problems and difficulties the student actually experience and it is not known how these difficulties may link outside the discipline, say to mathematics.
As educators it is imperative that we are able to convey to students in an efficient, effective, appropriate environment, the key ideas and concepts encompassed in quantum mechanics and to monitor students’ conceptual development.
As the need for students to understand quantum phenomena increases, so does our need to understand the learning processes adopted by students to grapple with these abstract and counterintuitive concepts. As a physics education researcher faced with this quandary a host of questions arose.
As stated earlier the aim of this research was to isolate key concepts, identify learning difficulties, identify teaching difficulties and so provide both teachers and curriculum developers with a valuable resource. To achieve this aim and address the list of questions, research data would best be collected from a range of sources guided by a flexible and responsive research methodology.
Phenomenology5 was adopted as the primary philosophical standpoint for this research, as it provided a number of open and responsive methods to explore the lived experience of the students and lecturers. Three approaches were chosen and adapted within this philosophical view which provided a flexible and responsive research environment. A grounded theory approach (Straus and Corbin 1998) allowed the research initially to cast a wide net over a number of data sources including examination scripts, interviews, texts and focus groups providing the foundations on which to build the study. A phenomenological approach (Cohen and Manion 1994, Holloway 1997) was adopted to conduct interviews and progressively focus the research allowing key themes to be recognised. Lastly aspects of the phenomenographic approach (Marton 1989, Prosser and Trigwell 1999) concerning variation influenced the analysis phases.
The thesis comprises three parts - the first concerns itself with the research setting; the second part reports the results of the grounded and phenomenological research phases; and the third part combines the results and reports the overall research findings.
Reporting the results from research phase
Bransden, B.H. and Joachain, C.J., (1989) Introduction to Quantum Mechanics (Longman Scientific and Technical, New York), p31
Cohen, L. and Manion, L., (1994) Research Methods in Education 4th edition (Routledge, London), pp29-31, 292-296
Fletcher, P.R., (1997) ^ (Unpublished, University of Sydney)
Holloway, I., (1997) Basic Concepts for Qualitative Research (Blackwell Science, Oxford), pp116-120
Marton, F., “Phenomenography – A Research Approach to Investigating Different Understandings of Reality”, (1989) ^ , Vol. 21 (3), pp29-39
Mermin, N.D., “Quantum Mysteries Redefined”, (1994) American Journal of Physics, Vol.62 (10), pp880-887
Lamb, W.E. “An operational interpretation of non-relativistic quantum mechanics”, (1969) ^ , Vol. 22, pp23-28
Prosser, M., and Trigwell, K., (1999) Understanding Learning and Teaching : the experience in higher education (The Society for Research into Higher Education and Open University Press)
Strauss, A.L., and Corbin, J.M., (1998) Basics of Qualitative Research : Techniques and Procedures for Developing Grounded Theory 2nd edition (Sage Publications, London)
AN INTRODUCTION TO THIS INVESTIGATION 1
1.1 INTRODUCTION 1
1.2 WHAT IS QUANTUM MECHANICS? 2
1.3 IMPORTANCE OF TEACHING QUANTUM MECHANICS 6
1.4 RESEARCH QUESTIONS 7
1 It is difficult to determine the commencement date of this period. The date 1885 has been chosen as it was the year the first experiments on atomic line spectra were conducted by Balmer.
2 The correspondence principle as articulated by B.H. Bransden and C.J. Joachain (1989) in^ s is as follows. ... quantum theory results must tend asymptotically to those obtained from classical physics in the limit of large quantum numbers.
3 The Pauli Exclusion Principle states that no two electrons in the same atom can have the same quantum numbers. This Principle underpins a great deal of modern studies in chemistry, explaining, for example, the structure of the Periodic Table.
4 One of the most obvious areas of current discussion concerns Bell’s Theorem. References may be found in the bibliographies of several recent articles, for example : Mermin, N.D., “Quantum Mysteries Redefined”, American Journal of Physics, Vol. 62 (10) pp880-887 (1994). Other areas of critical discussion include EPR Paradox and Hidden Variable theories.
5 Phenomenology is not a research method but is primarily a philosophy and an attitude to human existence, but it has been widely used in educational circles as a method to explore the lived experience of people.