1. Single Photon Interference
Parker Henry
May 5, 2015
Abstract
This experiment will demonstrate the ability of a single photon to
interfere with itself in a wavelike manner. This will be demonstrated
using three detectors to conditionally detect photons reaching either of
two detectors B or B’ through either of two possible paths within a small
time after detector A detects a photon. This single-photon interference
will be argued to arise due to no knowledge of which of the two paths it
took to reach either detector B or B’. This experiment is an extension
of the original experiment by Grangier et al. [2], which showed that
photons arrive at detectors B and B’ as lumps. The same result from the
experiment by Grangier et al. will be shown to continue to hold in this
experiment, thus demonstrating wave-particle duality in photons.
1 Introduction
The previous experiment ([1]) demonstrated that “photons exist.” That is to
say, it was demonstrated that photons can be demonstrated to possess particle-
like behavior, in that they can be demonstrated to have a definite position.
It is the goal of the following experiment to demonstrate wavelike behavior in
photons as well, in that a single photon, when passed through an interferometer,
can interfere with itself in a wavelike manner.
To review the previous experiment, a reproduction and slight refinement
of the original experiment done by Grangier et al. [2], it was demonstrated
that photons can possess definite position by means of three light detectors —
A, B, and B’ — and a laser of 405nm wavelength, along a path which was
subdivided via a down-converting crystal into a path toward A and a path
toward the B and the B’ detectors. The path toward the latter two detectors
was further subdivided with a polarizing crystal which would partially transmit
and partially reflect the laser toward B and B’, respectively. When detector A
was triggered by a photon, detectors B and B’ would seek photons within a
window of 4.74 nanoseconds of the detection event at A. It was argued that if
light behaved as a classical wave in this context, then both detectors B and B’
would be simultaneously triggered with every detection event at A, whereas if
light were behaving as a collection of particles called photons, then one or the
other would be triggered.
1

2. Figure 1: Diagram of the experiment. Note the polarizing interferometer (la-
beled as PI) in the dashed box. λ/2 denotes a half-wave plate, DC is the down-
conversion crystal, FFC is the fiber-to-fiber coupler, SPCMs are the single-
photon counting modules, and A, B, and B’ denote detectors A, B, and B’,
respectively [3, Fig L3.2, p.464].
To quantify the preceding discussion, let NB is the number of instances
within a time T in which detector B is triggered, let NB be the number of
instances in which detector B’ is triggered, and let NBB be the number of
instances in which both detectors are triggered within the 4.74 nanosecond win-
dow of detector A being triggered. A ratio g, which satisfies the proportionality
relation
g ∝
NBB
NBNB
was experimentally determined with computer software which counted the num-
ber of counts on the detectors. It was argued that if light was behaving as a
classical wave, then g ≥ 1, but if light consisted of photons which behaved as
particles, then g < 1. Indeed, g was experimentally measured in ([1]) to be
0.088(10), in support of the hypothesis that photons behave as particles.
This experiment will show that individual photons can interfere with each
other when passed through a polarization interferometer. The new experimental
setup is depicted in Figure (1). It will also be shown that it remains the case
that g < 1, as in the previous experiment. Thus, the following experiment will
not only be demonstrating wavelike behavior in photons via self-interference,
but it will simultaneously be demonstrating wavelike and particle-like behavior
in photons, in the truest sense of wave-particle duality.
2

3. Figure 2: Young’s Double-Slit Experiment. Feynman’s own thought experi-
ments are based on this actual experiment.
2 Feynman’s Thought Experiment on Single-Electron
Interference
In Volume III of the Feynman Lecture Series [4, Vol. III, Lecture I], Richard
Feynman proposes a series of gedankenexperiments, or thought experiments, of
three different objects – bullets, water waves, and electrons – going through
two holes, and detecting the output of the objects on a wall behind the holes.
Thomas Young did this very experiment with light, and Feynman’s gedankenex-
periments are geometrically identical in their setup, which is shown in Figure 2.
Feynman’s thought experiments, particularly the one with electrons, elucidate
the behavior in photons which we seek to experimentally demonstrate with the
following experiment.
Feynman first argues that if a machine gun of bullets is fired at the double
slits, and a detector is rapidly sliding up and down the backstop in Figure 3, then
the density P of all bullets hitting the wall, which is a function of the position
z along the wall, is simply the sum of the density functions P1 of bullets from
hole 1 on wall 2 and P2 of bullets from hole 2. This can be written succinctly
as
P = P1 + P2 (1)
Thus, the bullets exhibit particle-like behavior.
Feynman’s second thought experiment is similar to his first, though now
water waves are traversing through the two holes onto the backstop. This ex-
periment is depicted in Figure (4). The detector is detecting the intensities of
the waves striking the absorber wall. In this thought experiment, the waves will
show interference. If
ϕ1 = h1eiωt
is the wavefunction for the water waves passing through hole 1, for a complex
3

4. Figure 3: Feynman’s thought experiment with bullets. Note that the density
of the bullets striking the backstop (c) due to the contributions of both holes is
the sum of the individual densities (b). [4, Vol. III, Lecture I].
amplitude h1, and if
ϕ2 = h2eiωt
is the wavefunction for the water waves passing through hole 2, for a complex
amplitude h2, then the total wave function is given by
ϕ = (h1 + h2)eiωt
If hole 1 is blocked off, then the intensity I1 detected is proportional to |h1|2
.
If hole 1 is blocked off, then the intensity I2 detected is proportional to |h2|2
.
If both holes are open, then
|h1 + h2|2
= |h1|2
+ |h2|2
+ 2|h1||h2| cos δ
where δ is the phase difference between h1 and h2. Then the intensity I is given
by
I = I1 + I2 + I1I2 cos(δ) (2)
Thus, I is not simply I1 + I2. This is what Feynman means if the waves show
interference. The interference is constructive if δ = 0, and destructive if δ = π.
Feynman denotes the last term in equation (2) as the interference term.
If the intensity in the second thought experiment is regarded as a density
function of the detection of water waves, then we may take the density to exhibit
particle-like behavior if there is no interference, and to exhibit wavelike behavior
if there is interference.
In Feynman’s third gedankenexperiment, electrons are fired by an electron
gun, as in Figure 5. A mobile Geiger counter goes along the back wall to measure
the rate of electrons striking the backstop. If detectors were to be placed at each
hole, then a single electron would trigger one or the other, never both at the
same time; “each electron either goes through hole 1 or it goes through hole
2 (not both)” [4, Vol. III, Lecture I]. Thus, one concludes that the electrons
arrive in lumps on the backstop.
4

5. Figure 4: Feynman’s thought experiment with water waves. Note that the water
waves are exhibiting interference [4, Vol. III, Lecture I].
Figure 5: Feynman’s thought experiment with electrons. Though electrons can
be measured to have definite position, they interfere with each other. [4, Vol.
III, Lecture I].
5

6. However, if P1 is the clicking rate along the back wall of the electrons when
hole 2 is closed off, and P2 is the clicking rate when hole 1 is closed off, and if
P is the clicking rate when both are open, then
P = P1 + P2
Indeed, if ϕ1 is the probability density amplitude for hole 2 being closed, which
is in general a complex number, then
P1 = |ϕ1|2
and if ϕ2 is the probability density amplitude for hole 1 being closed, then
P2 = |ϕ2|2
The total probability density P is then
P = |ϕ1 + ϕ2|2
= |ϕ1|2
+ |ϕ2|2
+ 2|ϕ1||ϕ2| cos(δ)
where δ is the phase difference between ϕ1 and ϕ2. Hence,
P = P1 + P2 + 2 P1P2 cos(δ)
This is mathematically identical to the water-wave intensities from the second
thought experiment, yet the electrons arrive in discrete lumps, unlike the water
waves. This is an instance of wave-particle duality.
Feynman argues that the electrons’ nonclassical behavior is connected to
knowledge of which path the electrons took. Indeed, he argues that if any sort
of detectors are placed at hole 1 or hole 2, then the interference effect would
disappear, and the electron probability densities would add as if they were simple
bullets from the first thought experiment! Furthermore, if hole 1 is blocked off,
then the experimenter knows that the electrons went through hole 2, and vice
versa. The electrons still collect as lumps, and so this result is identical to the
bullet result. It seems from this gedankenexperiment that any sort of knowledge
of which path the electrons take destroys the interference effect of the double-slit
phenomenon. Feynman goes so far as to posit that all of the other mysteries of
quantum mechanics could be reduced to this particular quantum phenomenon
[4, Vol. III, Lecture I].
Feynman’s first two gedankenexperiments are relatively easy to implement.
His third experiment, he cautions, would require slits on an experimentally un-
feasible scale to carry out that exact experiment. However, versions of this
thought experiment have been implemented in spirit by various actual exper-
iments. The following single-photon interference experiment with lasers, pho-
tons, and photon detectors is one implementation of Feynman’s thought exper-
iment. This experiment will seek to demonstrate the same behavior in photons
as that of the electrons in Feynman’s third thought experiment.
6

7. 3 Connection of Feynman’s Thought Experiment
to Single-Photon Interference
As will be seen, the single-photon-interference experiment is a version of Feyn-
man’s thought experiment which uses photons rather than electrons. This ex-
periment will demonstrate single-photon interference using a polarization inter-
ferometer to vary the lengths of two paths along which a photon may travel.
The description and experimental setup are from Beck, Quantum Mechanics:
Theory and Experiment [3, Lab 3, p.463-474].
In this experiment, the photons in the beam, which has been shown to consist
of single photons under the conditional detection mechanism with detector A,
will be shown to interfere with themselves. In this experiment, what that means
is that a photon will be able to travel down either of two paths toward detectors
B and B’. These paths will be created with a pair of birefringent crystals, as will
be discussed in the experimental setup. The idea is that if it is not known which
path the photon took to get to the detectors, then the photon will interfere with
itself in the sense that if N1 is the number of detection events at one detector
due solely to hole number 1 being open to the photon, and N2 is the number of
detection events at that same detector due solely to hole number 2 being open,
then N, the number of detection events when both holes are open, will not be
N1 + N2:
N = N1 + N2
This assertion is consistent with Feynman’s third thought experiment involving
electrons. The following experiment will demonstrate this assertion.
4 Classical Theory of Polarization
To describe the effect of the polarization interferometer on the photon, we first
describe the classical theory of wave polarization. Suppose a wave is propagating
in vacuum in the z-direction, with wave vector k = kuz, where uz is the unit
normal vector in the z-direction, and k is related to the wavelength λ, frequency
f, and angular frequency ω by
k =
2π
λ
=
2πf
c
=
ω
c
One can write the electric field E as
E = Exux + Eyuy
If φ is the phase difference between Ex and Ey, E0x is the amplitude of the wave
in the x direction, and E0y is the amplitude of the wave in the y direction, we
can rewrite E as
Ex = E0x cos(kz − ωt)
Ey = E0y cos(kz − ωt + φ)
7

8. Figure 6: Diagram of birefringence in a crystal. Note the transmitted ordinary
ray, which is perpendicular to the optic axis, and the transmitted extraordinary
ray, which is parallel [5].
E can be written in terms of complex exponentials, where it is agreed that the
real part of the complex exponential is taken whenever the physical value of the
field is desired:
E = E0xei(kz−ωt)
ux + E0yei(kz−ωt+φ)
uy
If we let
E0 = E2
0x + E2
0y
be the amplitude of E and
ε =
E0x
E0
ux +
E0y
E0
eiφ
uy
be the normalized polarization vector, then E can be written succinctly as
E = E0ei(kz−ωt) E0x
E0
ux +
E0y
E0
eiφ
uy = E0ei(kz−ωt)
ε
Optical detectors respond to the power incident on the detector from the electric
field. This power is proportional to the intensity I, given by
I = |E|2
= E∗
E = E2
0
5 Theory of Birefringence
The above theory of classical polarization was derived in vacuum. If a wave
propagates in a linear medium, however, then it has an index of refraction n
which determines the speed v = c/n of the wave through the medium. The
value k becomes
k =
ω
v
=
n2π
λ
A birefringent material has two indices of refraction, which are distinguished
by its internal axis of polarization. Light which is solely along this axis, which is
called the axis of rotation, is refracted by an index n1, and light which is normal
to this axis is refracted by an index n2. This can be seen diagrammatically in
8

9. Figure 7: Birefringence in calcite. Note that the calcite makes two images of
the same × appear from the black paper [5].
Figure 6, and an example of this phenomenon occurring with calcite can be seen
in Figure 7.
The discussion of the theory of polarization shows that any light wave can be
decomposed into two orthogonal components: a component with polarization
orthogonal to the optic axis, and another component with polarization parallel
to the optic axis. In the birefringent medium, the component of the beam
orthogonal to the optic axis becomes the ordinary ray, and the component
parallel becomes the extraordinary ray (see Figure 6).
6 The Polarization Interferometer
Other optical apparatuses in this experiment include the linear polarizer and
the half-wave plate. The action of the linear polarizer is to project the incident
wave along its optic axis and transmit that projection, i.e. if the optic axis
makes an angle θ with the x-axis, and the incident wave is
Ei = E0ei(kz−ωt)
ε
then the transmitted wave is given by
Et = E0ei(kz−ωt
(ε · uθ)uθ
The action of the half-wave plate is that if its axis of polarization is at an angle
θ to the polarization of the incident wave, then it rotates the polarization by an
angle 2θ.
These actions can be represented in terms of linear algebra, using basis
components ux and uy, as follows: if a linear polarizer is at an angle θ to the x-
axis, and a light beam E is incident on it, then the action of the linear polarizer
on the beam is described by JθE, where
Jθ =
cos2
θ cos θ sin θ
cos θ sin θ sin2
θ
9

10. Figure 8: A simple polarization interferometer, which induces a phase difference
φ between the ordinary and the extraordinary rays [3, p.36].
Figure 9: A polarization interferometer, which induces a phase shift proportional
to φ, and which will be able to show coherence length [3, p.37].
projects the beam along the axis of θ. Similarly, if a light beam is incident on
a half-wave plate with axis of polarization at an angle θ to the x-axis, then the
action is given by Jλ/2,θE, where
Jλ/2,θ =
cos(2θ) sin(2θ)
sin(2θ) − cos(2θ)
A simple polarization interferometer, with parallel axes of polarization, such
as the one shown in Figure 8, induces a phase shift
Jp,φ =
eiφ
0
0 1
between the ordinary and the extraordinary rays. It also projects the top ray
along the x-axis, projects the bottom ray along the y-axis, and then recombines
them, so its total effect on the incident beam is
J = Jπ/2 + Jp,φJ0 =
eiφ
0
0 1
However, we use a modified version of the polarization interferometer shown
in Figure 8, because the coherence length is so short that to see any interference,
the lengths of the top and bottom paths must be nearly equal. We thus use a
polarization interferometer as in Figure 9. The axes of polarization of the two
birefringent materials are again parallel. The action of this device is given by
J = Jp,φJλ/2,45J0 + Jλ/2,45Jπ/2 =
0 eiφ
1 0
10

11. Figure 10: Diagram of the polarizing interferometer. λ/2 denotes the half-wave
plates, BDP denotes a beam-displacing polarizer, and PBS denotes a polarizing
beam splitter. Note the angles θ1, θ2 (which is fixed at 45 degrees throughout
the experiment), and θ3. B and B’ refer to the detectors B and B’ [3, Fig L3.1,
p.464].
If φ is set to 0, then this apparatus is simply a half-wave plate. If φ is not zero,
then the path length of the two branches differs, then this apparatus acts as a
half-wave plate, with an additional phase shift. This phase shift, however, will
alter the path length of one path, making single-photon interference observable.
7 Experimental Setup
Much of the setup of the experiment is identical to Lab 2 of Beck, which the
experimenter carries out in [1]. Therefore, the theory of photon detection by the
detectors remains the same as in Lab 2, so we do not repeat the full discussion
here. As in Lab 2, there are two detectors B and B’ at the outputs of a polarizing
beam splitter, and a detector A for an idler beam. Detector A is triggered by
a photon, which triggers the detection event in which detectors B and B’ seek
photons within a 4.7 nanosecond interval of detector A having detected a photon.
The polarizing crystal will transmit the light if its polarization is normal to its
optic axis, and reflect it if the light’s polarization is parallel to the optic axis.
It was shown in Lab 2 of Beck that this beam consists of single photons.
The new feature of this experiment is the addition of the polarization inter-
ferometer (the PI in Figure 1) A calcite crystal is used to create birefringence.
As discussed in Section 5, If the incident light is decomposed into two orthogonal
components: one parallel to the axis of polarization of the calcite, and the other
orthogonal to it, then the orthogonal component of the incident beam becomes
the ordinary ray, and the parallel component becomes the extraordinary ray.
The polarization interferometer in Figure 10 is a modification of the one
described in Section 6. First, we add a linear polarizing plate to polarize the
light incoming from the down-converting crystal to 0 degrees. We add two
half-wave plates at varying angles of polarization (θ1 and θ3 in Figure 10) to
polarize the light beam at varying angles before and after being acted on by
the polarization interferometer from Figure 9. A half-wave plate fixed at a 45
degree polarization angle corresponds to θ2.
The optical equipment in Figure 1 is aligned by placing an 810 nm laser
11

12. into the optical equipment, to verify that the light beams indeed travel on the
correct paths for the light from the 405 nm laser source to be detected by the
detectors. The position of the polarization interferometer such that the path
lengths are equal, and hence there is no phase shift between the two paths, is
found. We do not go into detail about either of these parts because they were
laborious and they were carried out by the lab instructor prior to entering the
laboratory.
8 Experimental Procedure
First, the room must be completely dark so as not to damage the photodetection
equipment.
A 405 nm laser is turned on while θ1 and θ3 from Figure (1) are varied at 0
degrees and at 22.5 degrees. At these varying angles, the counts AB and AB’
are determined.
Having obtained these counts, the procedure in the preceding paragraph
is carried out again, though this time one of the beams in the polarization
interferometer (the PI in Figure (1) is blocked. The AB and AB’ counts are
again determined, with θ1 and θ3 being either at 0 degrees or at 22.5 degrees.
Finally, the coherence length of the photon is determined by varying the
vertical orientation of the first birefringent crystal. This is done by altering the
number of divisions on a dial, which allows the crystal to point at a different
angle, and thus create a different path length in the extraordinary branch of
the polarization interferometer in Figure 10. The ordinary branch length is
kept constant, however, leading to a difference in the path lengths between the
ordinary and extraordinary branches. The dial is started at 40 divisions, and
incremented by 0.5 divisions, until it reaches 130 divisions. If φ is the number
of divisions, then we seek the number
δ(φ) = NAB −
2
3
NAB
NAB is multiplied by 2/3 because the B detector is only 2/3 as efficient as
the B’ detector. If δ(φ) is high in magnitude, then this means that a photon
traveling along this path has high self-interference, whereas if it is low, then it
is not exhibiting self-interference.
9 Results
The results of detecting the beam at varying angles θ1, θ3 are summarized in
Table 1.
These results may be explained in terms of the polarization axes of the
polarization interferometer, the half-wave plates, and the polarizing crystal.
The polarization interferometer and the polarizing crystal were both set up
with optical axes at 0 degrees, and were fixed on these axes throughout the
experiment.
12

13. Table 1: Table of counts obtained at angles θ1, θ3
θ1 (degrees) θ3 (degrees) Is one beam blocked? NAB NAB NAB/NAB g
0 0 No 195 53119 0.00367 0.58(21)
22.5 0 No 21952 25626 1.01 0.121(5)
0 22.5 No 20838 27653 0.754 0.093(9)
22.5 22.5 No 1589 50983 0.0312 0.163(40)
22.5 22.5 Yes 11543 13960 0.827 0.056(63)
When θ1 = θ3 = 0 degrees, then the entering light wave at a polarization of 0
degrees remains at 0 degrees polarization after leaving the first half-wave plate,
and then it is almost only transmitted as an extraordinary beam in the first
birefringent crystal. The fixed half-wave plate then rotates the polarization by
45 degrees, so that it becomes the ordinary component of the second birefringent
crystal. Since this component dominates the extraordinary component, and it
is parallel to the polarizing crystal’s own optic axis, the polarizing crystal will
almost entirely reflect the light beam to the B’ detector, and so we see in the
table that NAB is much less than NAB .
When θ1 = 22.5 degrees, and θ3 = 0 degrees, then the entering light, starting
at a polarization of 0 degrees because of the linear polarizer, is rotated to a
polarization of 45 degrees. Then the ordinary and the extraordinary components
at the first birefringent crystal are approximately equal in magnitude. The beam
is projected onto either the ordinary or extraordinary axis with equal probability.
The second half-wave plate makes the ordinary component extraordinary, and
the extraordinary component ordinary. With θ3 = 0, the incident beam to the
third half-wave plate is not rotated at all, so the polarization of the entire beam
remains 45 degrees. The polarizing crystal then, in approximately equal parts,
transmits and reflects the light wave into detectors B and B’. Thus, the ratio
NAB/NAB is measured as 1.01.
The case θ1 = 0 degrees, θ3 = 22.5 degrees has a similar result, though this
time it is because at θ1 = 0, one path of the polarizing interferometer — the
path which is initially extraordinary, which becomes ordinary when leaving —
dominates the other component. However, θ3 = 22.5 degrees rotates this beam
by 45 degrees, so that the polarizing crystal will, in roughly equal parts, transmit
and reflect the beam toward detectors B and B’. Thus, NAB/NAB = 0.754
With θ1 = θ3 = 22.5 degrees, there are two different trials based on whether
one path was blocked, or both paths were left open. As in the θ1 = 22.5
degrees, θ3 = 0 degree case, the photons are projected with equal probability
onto the ordinary or the extraordinary axes. However, with θ3 = 22.5 degrees,
these photons are rotated in polarization by 45 degrees. What is observed is that
NAB/NAB = 0.0312. However, when one beam is blocked in the θ1 = θ3 = 22.5
degree case, the ratio changes entirely: NAB = NAB = 0.827.
As we see from the table, all of the g values are less than 1. As it was argued
13

14. Figure 11: The difference between the AB and two thirds of the AB’ counts as
a function of the difference between path lengths.
Figure 12: The difference between the AB and two thirds of the AB’ counts
as a function of the number of divisions passed. Note that 1 wavelength is
approximately 2.5 divisions.
in the previous experiment [1], this implies that the photons have a definite
position when they are detected by the detectors B and B’. The result from
that experiment, that light consists of individual photons which can behave as
particles, then carries to this experiment.
Finally, the coherence length of the photons is determined by altering the
path length of the extraordinary path of the polarizing interferometer. The
results of altering the number of divisions is shown in Figure 11.
We use the conversion factor of 2.5 divisions to one wavelength, after look-
ing at the original graph in Figure 12 and finding approximately when the wave
repeats itself. The wavelength is 810 nanometers (the down-converting crystal
halves the wavelength of the incoming photon, in order to preserve energy con-
servation), and then we convert nanometers to millimeters with 1000 nanometers
to 1 millimeter. Having converted divisions to millimeters, we arrive at Figure
14

15. 11.
Because δ = NAB − (2/3)NAB oscillates rapidly, yet the overall amplitude
seems to decrease as a function of the distance, one can conjecture that the data
may fit a damped sine curve. However, an attempt to fit this data to a damped
sine curve was unsuccessful, due to the wild oscillations of the data, so this is a
very crude approximation. Still, one can see from Figure 11 that the distance at
which the “amplitude” reaches 1/e of the initial value is 16 millimeters. Thus,
the coherence length of the photon is approximately 16 millimeters.
10 Conclusion
What could cause the difference between the runs in which θ1 = 22.5 degrees,
θ3 = 0 degrees, and in which θ1 = θ3 = 22.5 degrees, with one path blocked or
no path blocked? One possible explanation comes from quantum theory, and it
lies in knowing which path a photon took. In the θ1 = 22.5 degree, θ3 = 0 degree
case, the photons which trigger detector B were initially of the extraordinary
component of the polarization interferometer, and the photons which trigger
detector B’ were initially of the ordinary component. With θ1 = θ3 = 22.5
degrees, and one path blocked, it is again known which path a photon took,
simply because one of the paths was not blocked. With one path blocked,
the photons that come through the other path will have, before reaching the
polarizing crystal, a polarization of 45 degrees, plus a multiple of 90 degrees,
because they have been projected onto either the ordinary or extraordinary axis.
Thus, the photons trigger detectors B and B’ in equal measure.
However, when neither path is blocked in the θ1 = θ3 = 22.5 case, then
no knowledge is available of which path a photon reaching the detectors took
to get there. One would assume, if the photons were behaving in a particle-
like manner `a la Feynman’s first thought experiment, that the counts on the
detectors would simply add if both paths were opened. Both paths are open, yet
this does not happen; the detection counts of the photons are adding as waves
do, in the manner of Feynman’s second thought experiment. However, the value
of g is less than 1 throughout the experiment, which implies that the photons
are collecting in lumps at the detectors. In conclusion, this experiment shows
that photons follow the paradigm of Feynman’s third thought experiment in
which electrons experience wave-particle duality: not only are photons neither
waves nor particles, but they are both.
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