CHAPTER ONE - School of Physics

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CHAPTER 1
AN INTRODUCTION TO THIS INVESTIGATION
1.1 INTRODUCTION
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.
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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.
1.2 WHAT IS QUANTUM MECHANICS?
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
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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
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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.
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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 waveparticle 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
2
The correspondence principle as articulated by B.H. Bransden and C.J. Joachain (1989) in
Introduction to Quantum Mechanics 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.
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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:
“The only easy [answer] is that quantum mechanics is a discipline
that provides a wonderful set of rules for calculating physical
properties of matter.” (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.
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.
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1.3 IMPORTANCE OF TEACHING QUANTUM MECHANICS
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
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encompassed in quantum mechanics and to monitor students’ conceptual
development.
1.4 RESEARCH QUESTIONS
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.
Attitudinal

What are the students’ perceptions of the subject?

What are the teachers’ perceptions of the subject?
Content

Is there a set of key concepts associated with subject?

How important is mathematics?
Learning

What types of difficulties are the students facing?

What are the internal and external links being made by the students?

What is the role of visualisations and analogies in the learning process?

How do students approach problem solving?

What learning ‘styles’ do students adopt?

Can students articulate how they learn?

What are the qualitatively different ways students learn quantum
mechanics?

What is the variation in how key quantum mechanical concepts are
perceived by students?
Teaching

What are the difficulties faced by the lecturers?

How are analogies used in the teaching?

What are the key ideas, concepts and skills that the lecturers are trying to
convey to the student?
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.
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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.
1.5 LAYOUT OF THE THESIS
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.
Research Setting

Chapter 2: Review of Related Research - Provides a comprehensive
review of related research covering waves, optics, statistics and specific
quantum mechanics education research in chemistry and physics.

Chapter 3: Research Framework - Describes the research framework and
the theoretical viewpoint from which the research was conducted. The
selected research methodologies; grounded theory, phenomenological
analysis and phenomenographic approach are briefly discussed.
Reporting the results from research phase

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Chapter 4: Development of Research Questions - A grounded theory
approach was employed in order to reveal a selection of appropriate
interview questions. The results of this phase of the study are a set of
interview questions.
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.
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
Chapter 5: Search for Underlying Themes - The interview questions
developed from the grounded research form the starting point for a
qualitative study. The results of the phenomenological analysis are a set
of identified themes.
Report Findings

Chapter 6: The Results - Presents the key findings

Chapter 7: Implications for Teaching and Learning – Mapping of the
themes onto a common framework, recommendations for teachers and
curriculum developers are summarised
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CHAPTER 1 - REFERENCES
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) Master of Science Thesis - How Students Learn Quantum
Mechanics (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) Journal of Thought, 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) Physics Today, 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)
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CHAPTER 1 ..........................................................................................................................................1
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
1.5
LAYOUT OF THE THESIS .....................................................................................................8
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