Research Unit in Physics Education – Physics Department - University of Udine
WORKSHEET_MQUD_ConF
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Workshop MQUD_ConF– Quantum particles and trajectories
We would like to consider here the interaction of photons with ideal birefringent crystals (every
incident photon is thus transmitted). We would like to explore the possibility of associating a
trajectory to a photon, taking advantage of the correlation between the polarization of photons
and the path of the beams.
1.Recognition of the phenomenology of birefringence and probability interpretation
A_ The beam of a laser trainer with vertical polarization strikes a facet of a birefringent crystal
(calcite), oriented such that the ordinary beam is polarized at 45°.
A.1 What do we observe?
A1.1. Description in words
A1.2 Complete the diagram
calcite
A.2 If the ordinary beam is polarized at 45°, what polarization does the extraordinary beam have?
______
A3.Fixed position of the crystal:
A3.1 Do the polarization of the ordinary beam and that of the extraordinary beam depend
upon the polarization of the incident beam?
Yes
No
Explain: ______________________________________________________________
A3.2 Does the intensity of the ordinary beam and that of the extraordinary beam depend
upon the polarization of the incident beam? Yes
No
Explain: __________________________________________________________
A4 If the direction of polarization of the incident beam forms an angle of
polarization of the extraordinary beam, we have:
45 with the
I str
 _________
I ord
B_A beam of low intensity of vertically polarized N photons strikes a facet of an ideal birefringent
crystal (calcite). Corresponding with the exits of
Rs
both beams, ordinary and extraordinary, we
V
place two photon detectors.
Ro
B1. For each incident photon:
calcite
detector
each time the single detector will find reveal a
photon
both detectors with find a photon
each time only the detector placed at the exit of the beam______ with find a photon
B2. For each incident photon:
B2.1. the detector placed at the exit of the ordinary beam in____% of cases will find a photon.
Research Unit in Physics Education – Physics Department - University of Udine
WORKSHEET_MQUD_ConE
B2.2 the detector placed at the exit of the extraordinary beam in______% of cases with find a
photon.
B3. If we call the detector Rs, the detector Ro goes off (finding a photon):
for each incident photon
on average only in 50% of cases, by chance
V
alternatively i,e. yes for one photon and no for the next
for no incident photons.
Rs
Ro
calcite
detectors
B4. If we call the detector Ro, the detector Rs goes off (finding a photon):
for every incident photon
on average only in 50% of cases, by chance
alternatively i.e.yes for one photon and no for the next
Rs
V
Ro
calcite
for no incident photons.
detectors
B5. If we set up both detectors, each one activates:
for every incident photon
on average only in 50% of cases, by chance
alternatively i.e. yes for one photon and no for the next
for no incident photon.
Rs
V
Ro
calcite
detectors
Research Unit in Physics Education – Physics Department - University of Udine
WORKSHEET_MQUD_ConE
2. Polarization and beams transmitted in a crystal of calcite
A. We place two calcite crystals in a line, one direct and the other inverse. A beam of N photons
strikes the first crystal. Emerging from the second crystal we see a single beam given by the
combination of the ordinary beam and the extraordinary beam. The incident beam is vertically
polarized, the ordinary beam is polarized at 45°, the extraordinary one is polarized at 135°.
135°
V
Direct calcite
45°
Inverse calcite
A1. What is the polarization of the single beam emerging from the second crystal?___________
A2. What photons will emerge from the second crystal? _________________
B. Given that the ordinary beam is always polarized at 45° (photons with property) and the
extraordinary one is always polarized at 135° (photons with property:), there is a close
correlation between the path of the light and polarization.
Is it possible to maintain that single photons will follow one of the two paths?
This question is equivalent to reconsidering, in the case of the birefringent crystals, hypothesis A,
explored previously in the case of the interaction of photons with polaroids.
Let us see what consequence will be added to the concept of trajectory.
On two crystals of calcite, one the inverse of the other, strike vertically polarization N photons
(with property ). We may think that the photons travel along one of the two trajectories (that of
the ordinary beam or that of the extraordinary beam)?
This is equivalent to asking: 
a question to which the reply is already given, but will be reconsidered here to explore the
consequences and their possible association with a photon trajectory.
C. Let us complete the figures below:
Research Unit in Physics Education – Physics Department - University of Udine
WORKSHEET_MQUD_ConE
Group of photons with vertical
polarization (proprietà )
Experimental situation
C1.1

C1.2
……..
C2.1

……..

……..
C3.2
……..
C4.1

……..

C2.1
C3.1

Group of photons, half with
polarization at 45° (proprietà ) and
half at 135° (proprietà ) – Ipotesi A

……..
C4.2
……..

……..
D. On the basis of the comparison between experimental situations and results based upon
hypothesis A:
D1. May we hypothesize that half of the photons follow the ordinary path and half follow the
extraordinary one? Yes
No
Explain _____________________________________________________________
D2. May we say that the photons follow both paths? Yes
No
Explain_____________________________________________________________
D3. May we maintain that the photons follow paths that are different from those that are ordinary
or extraordinary? Yes No
Explain_____________________________________________________________
D4. Can we attribute a trajectory to a photon? Yes No
Explain
______________________________________________________________________
______________________________________________________________________