Transcript Learning paths of high school students in Quantum Mechanics

High school students face QM
basic concepts
Alberto Stefanel
Physics Education Research Unit, Physics Department,
University of Udine, Italy
[email protected]
2. The first study
2004-2005
22 high school pupils of 18-19 years old (21 females and 1 male)
The sample: the 18 students
always present to the several activities.
-Any previous experience in lab
-Traditional approach to physics
-Any knowledge on optics
-9 hours - laboratory and group activities conducted by a researcher
-using worksheets for students stimulating personal involvement and challenge
by open inquiry (Martongelli et al. 2001; Michelini et al. 2004, Stefanel 2006b)
2 hours – tests (pre test – post test)
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04/05
Day
Hours
Aims (Activities)
05/06
Day
Hours
I/1
Pre Test(I)
I/1
II / 2-3
Exploration of the polarization of light transmitted through several polaroids II / 2-3
differently crossed: with the naked eye (on the OHP) (T-C); by means of sensors
(Malus and transmittivity of polaroids) (G)
III / 4
Data elaboration and discussion in computer science laboratory (T and C)
IV / 5
Summary of: the experimental results; the phenomenological laws. Discussion about III /4
the meaning of the different factors involved (T). Single photon simulated
experiments and probabilistic interpretation (C).
V/6
Operative association of a property to photons and its iconographic representation. IV
Mutually exclusive properties. Hypothesis exploration: superposition of states – 5-6
statistical mixture (T – C)
V 7-8
VI / 7
Phenomenology exploration of birefringent crystals (T-C)
VI /9
VII / 8
Impossibility to associate a trajectory to photons (T-C)
VII-10
IX
9-10
Superposition principles for the polarization and generalization for any observable VIII
(T-C); Linear operators and their physical meaning (T-C)
11-1213
X / 11
Post Test,(I)
VIII /
14
Table
1. Synthetic
intervention.
Activity:
A Stefanel,
Highprospect
school of the in-classSTE
- MODENA
2009 (I) individual work; (C) work in groups
3
of 2-3
students;
students
face (G)
QM work in groups of 5-6 students; (T) the whole class.
3. Research methodology
3.1 Monitoring tools
Monitoring tools (Aiello 1997) for the in class intervention:
a) Pre/Post qualitative evaluation
b) Pre/Post test (the same)
15 items
- 2 open answer questions,
- 13 multiple choice questions and motivation
(only one exhaustive answer according to QM - Stefanel 2006a);
c) worksheets for students (URDF 2002; Stefanel 2006b);
d) notes written by the researcher and the class
teacher during and after activity.
e) Adiorecording of class discussion
f) Only for the 2005/06 intervention (after the posttes): discussion on the web; final examination
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3.2 Data analysis methodology
Qualitative analysis
Classification of answers and comments to
- the test questions,
- the worksheets questions,
- the stimula offered
by in-class activity and by the peer interaction
Quantitative
Frequency, distribution, chi-test
phenomenographycal approach (Marton 1986)
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Profiles
Clas – Classic profile. Microscopic systems have an analogous nature to
classic macroscopic systems. All their observables always own well defined
values. In order to describe their evolution, the concept of trajectory can be used,
even if it is necessary to use a statistical approach for lack of information about
the initial state of the system under observation.
Hid – Hidden variables profile (local). Microscopic systems preserve some
properties of the classic macroscopic systems, in particular the trajectory even if it
is not knowable/detectable. Their non-classical behaviour is due to
uncontrollable disturbances, or rather to some their properties that are not
directly accessible/measurable.
Quant – Quantum profile. Classic and quantum systems have different nature.
It is possible to associate dynamic properties to quantum systems only by means
of measurements. These properties are in general imcompatible with those that
characterize the state before the measurement itself. Position and trajectory
loose their meaning. The process of measure can be described only as a
transition between an initial and a final state.
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Q1 - Indicate three topics characterizing QM w.r.t. CM (open answers)
Q2 - Measuring a physical observable, which aspect among the following ones
characterizes in a peculiar way QM w.r.t. CM?
(5 options+ explanation)
Q3, Consider the following probabilistic previsions:
K) the heads outcome in tossing coin has probability ½ to be realized;
J) a photon with vertical polarization has probability ½ to pass through a
polaroid at 45º.
(3 options, explanations)
Q4. The physical meaning of Heisenberg uncertainty relations?
(5 options, explanation)
Q5. Quantum indeterminism and state concept
(2 options, explanation)
Q6. In classic mechanics it is always possible to attribute a trajectory to a particle.
Which statement can be made as far as a quantum particle is concerned
(5 options, explanation)
Q7. The physical meaning of the superposition principle?
(5 options, explanation)
Q8. Physical meaning of Ψ(x)
(5 options, explanation)
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Q4. Which of the following statements better outlines the meaning of Heisenberg
uncertainty relations?
Answer Options
In
Out
a) there are physical quantities pairs of the same system, which can not be
simultaneously determined with arbitrary precision
2
6
b) it is not possible to measure with arbitrary precision the value of a physical
observable
2
3
c) it is not possible to make the indetermination of an observable measurement
arbitrary small
2
3
d) It is never possible, even not in principle, to predict precisely the outcome of a
measurement
3
1
e) it is not possible to measure with arbitrary precision the position and the
momentum of a particle.
5
1
NR
4
4
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4. First study profiles summary
Crossing of the data collected with the
three monitoring tools
coherence check between answers
profiles in the pre-post test and of the
other two monitoring tools (Michelini,
Stefanel 2007, Girep Conf.).
Classification of categorized answer
On the basis of the coherent prevailing of one profile among the others,
the total profile for each student was defined making use of an index
(Michelini, Stefanel 2006a) build according to the C-index proposed by Müller,
Wiesner (2002).
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Among all examined questions, 6 cases have shown a prevailing of the Clas
and Quant profiles at the same time.
Those have been classified in the Confl Profile (conflictual profile).
The profiles distribution after the
activities (Post), indicating changes
from pre to post profiles:
in 14 cases among 18.
The conceptions initialy classified as Hid-profile have been mostly confirmed (4
cases) or they have evolved to the Quant profile. Only in two cases they have
evolved to conflictual ideas
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X = Clas answers* pc
Y = Hid answers * ph
Z = Quant aswers * pq
p =1 – 1,2 – 1,5
1 – no coherent answers
1,2 – 2 coherent answers
1,5 – 4 coherent answers or more
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Hid
Quant
Confl
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5. The second study
2005-06
17 High School pupils of 18-19 years old
(12 females and 5 male)
-Good experience in lab
-Good practice in connection of data
and models
-Knowledge of light polarization
phenomenology
The sample : the 16 students always present to the several activities.
-12 hours - laboratory and group activities conducted by a researcher
(using Work-sheets)
-2 hours – tests (pre test – post test)
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students face QM
STE - MODENA 2009
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04/05
Day
Hours
Aims (Activities)
05/06
Day
Hours
I/1
Pre Test(I)
I/1
II / 2-3
Exploration of the polarization of light transmitted through several polaroids II / 2-3
differently crossed: with the naked eye (on the OHP) (T-C); by means of sensors
(Malus and transmittivity of polaroids) (G)
III / 4
Data elaboration and discussion in computer science laboratory (T and C)
IV / 5
Summary of: the experimental results; the phenomenological laws. Discussion about III /4
the meaning of the different factors involved (T). Single photon simulated
experiments and probabilistic interpretation (C).
V/6
Operative association of a property to photons and its iconographic representation. IV
Mutually exclusive properties. Hypothesis exploration: superposition of states – 5-6
statistical mixture (T – C)
V 7-8
VI / 7
Phenomenology exploration of birefringent crystals (T-C)
VI /9
VII / 8
Impossibility to associate a trajectory to photons (T-C)
VII-10
IX
9-10
Superposition principles for the polarization and generalization for any observable VIII
(T-C); Linear operators and their physical meaning (T-C)
11-1213
X / 11
Post Test,(I)
VIII /
14
Table
1. Synthetic
intervention.
Activity:
A Stefanel,
Highprospect
school of the in-classSTE
- MODENA
2009 (I) individual work; (C) work in groups
14
of 2-3
students;
students
face (G)
QM work in groups of 5-6 students; (T) the whole class.
Q2 - Measuring a physical observable, which aspect among the following ones
characterizes in a peculiar way quantum mechanics w.r.t. classic mechanics?
Answer options
Pre
Post
A)Under some conditions, dicrete (not continuous) values of the measured
observable are obtained
4
0
5
6
C) In general, systems initially prepared in the same state evolve in a different
way when subjected to a process of measure
2
5
D) The interaction with the measurement apparatus produces a perturbation on
the system
0
4
3
1
4
0
B) Results of measurements are predictable only in probabilistic terms
E) The result of a measurement is affected by an ineliminable indetermination
NA
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Q3, Consider the following probabilistic previsions:
K) the heads outcome in tossing coin has probability ½ to be realized;
J) a photon with vertical polarization has probability ½ to pass through a polaroid at 45º.
Answer options
Pre
Post
A) In the K case we do not know initial conditions precisely enough, in the J
case initial conditions are known, but the phenomenon itself has a probabilist
nature.
6
10
B) In both cases we do not know initial conditions precisely enough.
1
0
C) In the K case we do not know initial conditions precisely; in the J case, even
knowing the initial conditions, we do not know with enough precision how the
interaction photon-polaroid happens.
8
5
NR
0
0
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Q6. In classic mechanics it is always possible to attribute a trajectory to a particle. Which
statement can be made as far as a quantistic particle is concerned (chooce only one option):
Answer options
Pre
Post
A) it is possible to attribute a trajectory, but it is not possible to experimentally
determine with arbitrary precision all the information needed to determine it with
arbitrary precision
1
1
B) it is possible to attribute a trajectory, but all the information needed to
determine it is not experimentally accessible
5
0
C) it is possible to attribute a trajectory only when a position measurement is
performed
1
1
D) it is impossible to attribute a trajectory to a particle because it is always
subjected to casual perturbations
3
2
E) it is not possible, even not in principle, to associate a trajectory to a particle
3
11
2
0
NR
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Q6. In classic mechanics it is always possible to attribute a trajectory to a
particle. Which statement can be made as far as a quantistic particle is
concerned (chooce only one option):
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Photons-polaroid interaction
Hypotesis A: N 45° polarization photons =N/2 vert. pol. ph.+N/2 horiz. pol. ph ?
Iconographic representation of photons property:
The photon posses either A photon can not posses
one properties or one
simultaneously two
other (just one property)
different properties
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= ** U 
The properties are
incompatible
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The interaction with a polaroid
changes the property of the
photon (active role of the
polaroid)
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According to the comparizon of experimental
results and the hypotesis A results
May we attribute a trajectory to a photon?
No: 13
Yes: 2
NA : 1
Antino: NO - “if I hypotize a trajectory I do
not obtain the experimental results”
Marta: No – “We can not attribute for certain
to each photon the followed trajectory”
Stefania: NO “ We can not establish a priori the trajectory followed, because the
quantum uncertainty”
Dafne: NO “for each photon, I can not establish which direction it followed”
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Discussione sulla traiettoria e i sistemi quantistici in ambiente WEB
Giacomo: In un sistema quantistico (almeno da quello che credo aver capito)
non si può stabilire né a priori né a posteriori la traiettoria di una particella, ma si
può solo sapere da dove parte e dove viene ri-(leva o vela?)-to. questo perché
se si cerca di attribuire una traiettoria alla particella automaticamente si
interferisce con essa e si altera la misura.
Sean (replica a Giacomo): Io neppure credo che si possa stabilire la traiettoria
di una particella in meccanica quantistica. Penso ciò perché abbiamo capito che
lo stato di un corpo è inconoscibile fino a quando lo andiamo a rivelare/rilevare,
e facendolo lo alteriamo, e ne modifichiamo le proprietà... tra una sorgente ed
un rilevatore la nostra particella (fotone) penso si possa considerare essere
ovunque, non potendo sapere quale sia il suo percorso ne come si sposta...
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Marta: ….é lec ito supporre che tra la sorgente e il rilevatore la particella si trovi
una situazione di sovrapposizione di tutti gli stati possibili e che essa (la
particella) precipiti in uno stato ben definito solo in seguito ad un’interazione con
un altro corpo, ad esempio il rilevatore stesso
Dafne: In meccanica quantistica non è possibile attribuire una traiettoria alla
particella. Di una particella si può solo conoscere lo stato iniziale e quello finale
mentre non è possibile stabilire ciò che avviene in mezzo, e più in particolare non
si può stabilire con certezza quale percorso abbia seguito la particella. Con la
misurazione pratica, sperimentale si può stabilire la posizione di una particella; ma
nel momento in cui si effettua tale rilevazione si altera il sistema, e la particella
modifica le sue proprietà. Di conseguenza se ne deduce che una particella, tra la
sorgente ed il rilevatore, può seguire qualsiasi percorso, quindi non è possibile
attribuirle una traiettoria precisa, e nel tentativo di determinarne la posizione
(nell'atto sperimentale) vengono modificate le sue proprietà.
A causa di questo indeterminismo è logico supporre che la particella, tra sorgente
e rilevatore, si trovi in una sovrapposizione degli stati possibili e che sia
l'interazione stessa con il rilevatore a determinare lo stato finale con cui la
particella viene rilevata.
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CQ index (Michelini, Stefanel 2006a) build according to the
C-index proposed by Müller, Wiesner (2002).
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7. Conclusions
• Personal learning paths and the construction of independent
lines of thinking are activated when students are actively involved
in hands-on/minds-on activities, in a specific explorative contex,
producing a solid understanding of phenomenology and the
capability of rigorous argumentation (R4).
(see also Stefanel 2001; Michelini et al 2001),
The evolution of conceptual schemes of reference (R2, R3) happens
not only towards typically quantum ideas, but also, for a non
negligible fraction of students, towards conceptions in which the
evolution of a system is described on the basis of hidden
variables.
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• Conceptual nuclei (R1) about an alternative interpretation of
QM, constitute the initial ideas with which some students start
studying QM.
- In several cases these nuclei can become stronger and lead
logically to develop conceptions coherent with alternative
approaches to QM
or
to play the role as a bridge toward a quantum orthodox vision.
• Tendency in the evolution of students ideas, from a completly
classic conception of phenomena, to a hidden variables ideas, to
a quantum vision (R3).
This process is not linear and regards part of the students of the
current study (R2).
Thank you
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Quantum ideas - Michele:
Q2 he chooses B (Results of measurements are
predictable only in probabilistic terms)
in both cases, without motivations;
Q3 he chooses A (intrinsic indeterminism in QM
measure processes) in both cases motivating in
the Post-test:
"the probability of transmitting a photon with
vertical polarization which passes through a
polaroid at 45º is 1/2";
Q4 his answer changes from C to E, without
motivations;
Q6 his answer changes from A to D motivating his
choice with the "uncertainty principle"
According to Michele:
the central knot for the quantum phenomena understanding is the
uncertainty principle which takes the role of cognitive organizer.
It is important to be noted that here this principle has been introduced underlying
the role of incompatible observables.
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 Hidden variables. Federica:
Q2 (measurement ) her answer changes from A to B motivating her new choice
with the fact that "measurements are predictable only in probabilistic terms
because the trajectory of quantum particles is not predictable";
Q3 her answer changes from C to A (intrinsic stocatic nature of measure
process)
 "the phenomenon of the photon transmission has a probabilistic nature,
because it is not possible to predict the trajectory (we have experimentally
demonstrated this fact: the number of photons passing through a polaroid is not
always the same, but it varies constantly, and half of them pass on average)";
Q4 the Pre-answer is E, the Post-answer is C (it is possible to attribute a
trajectory only when a position measurement is performed) motivating her
choice by saying that "this is due to the fact that the measurement of a physics
quantity is predictable only in terms of probability";
Q6 her answer changes from A to B including the comment: "That is why we
can describe the physics observable in terms of probability"
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According to Federica:
the central knot is the impossibility to predict the trajectory of a
quantum system. It is implicit that this trajectory exists, but it is not
accessible to us. That is way it is necessary to make probabilist
predictions. This is the typical vision of a hidden variables theory, in
which the hidden variable is precisely the position.
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