1. Final Year Project Report
The Heisenberg γ ray Microscope
Name: Austin Tobin
Student Number: 58069328
Class: AP4
Supervisor: Dr. Enda McGlynn
Date: April 23, 2012
2. Declaration
I hereby certify that this material, which I now submit for assessment on the pro-
gram of study leading to the award of B.Sc. in Applied Physics is entirely my own
work, does not to the best of my knowledge breach any law of copyright, and has
not been taken from the work of others save and to the extent that such work has
been cited and acknowledged within the text of my work.
Signed (candidate’s signature):
Candidate name (typed): Austin Tobin
ID No: 58069328
Date: April 23, 2012
i
3. Abstract
The uncertainty principle is one of the fundamentals of quantum me-
chanics, and was conceived as a thought experiment by Werner Heisenberg.
This thought experiment comprises of a theoretical γ-ray microscope in an
attempt to attain the highest possible resolution of a position measurement.
What uncertainty principle states is that there is a limit to the accuracy at
which the certain measurements can be made. In this report the focus is on
the momentum and position uncertainty.
With respect to the uncertainty in momentum there is an upper limit in
terms of there only being 180◦ within which the photon can be scattered.
This means that there is a limit on the uncertainty of the photons momen-
tum. Resultingly the dominant factor in the uncertainty principle comes
from the uncertainty in position.
In this project the more rigorous Kirchoff diffraction theory, as opposed
to Fraunhofer diffraction was used to computationally model the Heisen-
berg microscope. The aim was to demonstrate uncertainty principle using
the standard deviation of the diffraction patterns generated. The results
obtained initially did not uphold uncertainty principle. Thereafter was an
attempt to find an adjustment to the model such that it modelled physical
phenomenon more correctly and thus uncertainty principle would be up-
held. This resulted in one observation that looked at the acceptance angle
of the lens, seemed to be more consistent with uncertainty principle, from
this point the model may be developed.
ii
4. Acknowledgements
I would like to thanks to those in my class who helped me with all the little
things and helpful suggestions along the way. A special thanks to Dr. Enda
McGlynn without whose help and understanding this project would never
have been completed.
iii
5. Contents
1 Introduction 1
2 Theory 1
2.1 A thought experiment to motivate the uncertainty principle . 1
2.2 Schr¨odinger’s waves and Matrix theory: the beginning of the
uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Deriving Heisenberg’s uncertainty principle . . . . . . . . . . 3
2.3.1 Uncertainty relations . . . . . . . . . . . . . . . . . . . 4
2.3.2 The Compton Effect . . . . . . . . . . . . . . . . . . . 5
2.3.3 Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . 5
2.3.4 The Kirchhoff Obliquity Factor . . . . . . . . . . . . . 6
2.3.5 Effect of the Lens . . . . . . . . . . . . . . . . . . . . . 8
2.3.6 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . 8
2.4 Standard Deviation and the Uncertainty Principle . . . . . . 9
2.4.1 Calculating Standard Deviation in this model . . . . . 11
2.5 Derivation of Equation used for the Model . . . . . . . . . . . 12
2.5.1 Differences between Kirchhoff and Fraunhofer diffrac-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Computation 14
3.1 Integration code . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Creation of the 3D diffraction pattern . . . . . . . . . . . . . 17
3.3 Testing the integration by aliasing . . . . . . . . . . . . . . . 19
4 Results 22
5 Discussion 26
5.1 Evaluation of results . . . . . . . . . . . . . . . . . . . . . . . 26
5.1.1 Use of an infinite lens varying the lens that is “seen” . 26
5.1.2 Demonstrating the Gibbs phenomenon . . . . . . . . . 27
5.1.3 Acceptance angle as a function of lens electron distance 28
6 Conclusions 33
Appendices 36
A Risk Asscessment 36
B Integration program 39
C Rotation program 41
D gibb.m and licence 44
iv
6. List of Figures
2.1.1 The basic set-up of the HM. D is the diameter of the lens,
F is the lens electron distance, β is the maximum scattering
angle. (Modified diagram based on an animation found at
reference [5]). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3.1 Diagram to show the basis of uncertainty principle from the
HM where L is the wavelength of the light. (Modified diagram
based on an animation from reference [5]). . . . . . . . . . . . 3
2.3.2 Graph of the change in wavelength due to the Compton Effect. 6
2.3.3 Diagram of point sources in an aperture to show principle of
Huygens-Fresnel principle. (Diagram is a modified reproduc-
tion of Figure 10.4 in reference [9]) . . . . . . . . . . . . . . . 7
2.3.4 Aperture A and the angles used in Equation 2.3.14 (repro-
duced from a diagram based on Figure 8.3 from reference
[9]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.5 An example of the Gibbs Phenomenon showing how with
increased orders of an infinite Fourier series produce a closer
approximation to a square wave. Also the oscillation at the
corners of the waves (which is the Gibb’s phenomenon) is
reduced.(MATLAB code used for generating these plots and
licensing appears in appendix D). . . . . . . . . . . . . . . . . 9
2.4.1 Diffraction pattern from a single slit showing some of the
features that have used to show uncertainty principle. . . . . 10
2.5.1 Diagram showing the distances needed in the derivation of
the integration to convert angles to distances. . . . . . . . . . 12
2.5.2 The origin of the ∆ in s + ∆s . . . . . . . . . . . . . . . . . . 13
2.5.3 Plot of how the multiplying factor of Kirchhoff diffraction
changes with respect to f. . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Code flow diagram of the code for the integration on which
this project is based . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Plot of the 1D diffraction pattern . . . . . . . . . . . . . . . . 16
3.1.3 Graphical representation of the integration process. The pro-
gram moves across the aperture with increments dW, sum-
ming the contributions across each dW for the coresponding
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 The use of Pythagoras’ theorem to create the Airy disk. . . . 18
3.2.2 The amplitude at the radius of the 2D diffraction pattern. . . 18
3.2.3 The Airy disk generated from the 2D diffraction pattern. . . 19
3.2.4 Code flow diagram for the rotation of the 2D diffraction pattern. 20
3.2.5 Slicing the 3D pattern of a contour-plot (top) and the quasi-
2D pattern that is generated (bottom). . . . . . . . . . . . . . 21
3.3.1 Aliasing of a sinusoidal signal. . . . . . . . . . . . . . . . . . . 22
3.3.2 Multiple plots showing aliasing the diffraction pattern . . . . 23
4.0.3 Plot of standard deviation in terms of angle verses lens elec-
tron distance (insert shows the dip as the lens electron dis-
tance comes close to zero) . . . . . . . . . . . . . . . . . . . . 24
4.0.4 Plot of ∆ x vs. lens electron distance. . . . . . . . . . . . . . 24
v
7. 4.0.5 The angular standard deviation (top) and the multiplication
factor (bottom). Comparison of the two and the correlation
between the dip in angular standard deviation and the peak
in the multiplication factor. . . . . . . . . . . . . . . . . . . . 25
5.1.1 Change in the standard deviation using a 2d diffraction pattern 27
5.1.2 Change in angular standard deviation using a quasi-2d diffrac-
tion pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.1.3 Change in the standard deviation using a quasi-2d diffraction
pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1.4 As the lens size is increased a sharper image of the diffrac-
tion pattern is produced, this is consistent with the Gibbs
Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.5 Diagram showing the acceptance angle of the lens, lens radius
and the lens electron distance . . . . . . . . . . . . . . . . . . 30
5.1.6 As the lens is brought closer to the electron the acceptance
angle is increased resulting in the standard deviation increasing. 31
5.1.7 Plot of the angular standard deviation with respect to allow-
ing the acceptance angle be dictated by the tangent of the
lens radius divided by the lens electron distance. . . . . . . . 32
5.1.8 Plot of the angular standard deviation with respect to allow-
ing the acceptance angle be dictated by the tangent of the
lens radius divided by the lens electron distance. . . . . . . . 32
5.1.9 The periodic nature of the sin in the Bessel function, also the
wave beating effects on the peaks. . . . . . . . . . . . . . . . 33
vi
8. 1 Introduction
In 1927 Werner Heisenberg published his groundbreaking paper on the un-
certainty principle. In this project the question was posed as to how is the
position of an electron measured using a γ-ray microscope which would the-
oretically give the highest attainable accuracy. At the point of measurement
of the position of the electron there would be a discontinuous change in mo-
mentum from the scattering of the photon. From this instant there begins
a discussion about the size of the electron and the concept of a probability
distribution to show its position. This thought experiment became known
as the Heisenberg Microscope (HM)[1].
Heisenberg drew his main conclusions from observing the behavior of a
single slit diffraction pattern. As the slits narrow the diffraction pattern
spreads out due to the wave nature of light instead of becoming narrower
as would classically be expected for particles. Interestingly, according to
Heisenberg the “physical interpretation of quantum mechanics (was still)
full of internal discrepancies, which show themselves in arguments about
continuity verses discontinuity and particle verses wave”[1].
In this project the use of Kirchhoff diffraction as opposed to Fraunhofer
diffraction theory was studied as a more rigorous method of generating
the single slit diffraction pattern which is the basis of the HM. A robust
method of calculating the standard deviation of the position component of
the uncertainty relation ∆x∆p ≥ ¯h
2 was used to model the HM.
The aim of this project was to model the HM using Kirchhoff diffrac-
tion which was successfully achieved. What was shown was that the main
difference between Fraunhoffer and Kirchhoff diffraction is the multiplica-
tion factor of the Bessel function that is used in the mathematical model.
What would be expected from physical observations is that when an object
is brought from infinity into focus of a lens, the image becomes sharp. Once
the object is brought inside the focal plane of a lens its image becomes less
sharp. What is shown is report is that when the object is brought closer
to the lens in the HM the standard deviation does not diverge as expected.
From this finding, the HM result does not appear to be consistent with the
uncertainty relation.
However some observations were made that could be used to develop the
model so that uncertainty principle would be upheld.
2 Theory
2.1 A thought experiment to motivate the uncer-
tainty principle
The concept of the HM was brought about as a thought experiment to mo-
tivate/demonstrate the limit to which the accuracy of both the position and
momentum of a quantum sized object can be measured. This is generally
given by what is known as the Heisenberg uncertainty principle (HUP) as
shown in Equation 2.1.1. Uncertainty relations apply in the x, y, and z di-
rections in Cartesian coordinates in 3 directions (3-D), Equation 2.1.2 refers
only to the x direction, i.e. in the one directional model (1-D). Although
there are many other uncertainty relations between conjugate variable types
1
9. Figure 2.1.1: The basic set-up of the HM. D is the diameter of the lens, F is the
lens electron distance, β is the maximum scattering angle. (Modified diagram
based on an animation found at reference [5]).
as mentioned in Section 2.3.1. The position momentum relation is the main
focus of this report.
∆Px∆x∆Py∆y∆Pz∆z ≥
¯h3
8
(2.1.1)
∆Px∆x ≥
¯h
2
(2.1.2)
Equation 2.1.1 is a mathematical, exact and formal generalization by
Kennard (1927) from the standard deviation expectation values for quantum
mechanical equations 2.1.3 with P being the momentum operator and Q
being the position operator acting on of the object wavefunction[2].
(∆P)2
= (P − P )2
, (∆Q)2
= (Q − Q )2
(2.1.3)
Figure 2.1.1 shows the basic set-up of the HM. It consists of an incident
γ ray photon colliding with a stationary electron at the focal point of a
lens which collects the Compton scatter (Section 2.3.2) photon forming an
image.
What the HM thought experiment aimed to achieve was to consider
whether or not both the position and momentum of the electron could ac-
curately be determined. The purpose of the project is to study the effects of
the position of the lens relative to the electron on the precision measurement
of the electron.
2.2 Schr¨odinger’s waves and Matrix theory: the
beginning of the uncertainty principle
Heisenberg, together with Neils Bohr, Max Born and Pascaul Jordan, pub-
lished different papers on the theory of matrix mechanics to explain the
discontinuities in quantum mechanics. It was also about this time that Er-
win Schr¨odinger published his work on wave mechanics to explain the same
subject. The main difference was that the wave theory was based on a
wave equation (Schr¨odinger’s equation) similar in philosophy to Maxwell’s
2
10. Figure 2.3.1: Diagram to show the basis of uncertainty principle from the HM
where L is the wavelength of the light. (Modified diagram based on an animation
from reference [5]).
equation for electromagnetic waves. This also tied in with the concept
of De Broglie’s particle wave theory, while matrix mechanics used matrix
multiplication which was rarely used outside mathematical concepts at the
time[3].
Heisenberg was working under supervision of Neils Bohr who had al-
ready completed significant work with regard to the hydrogen atom and
what is now considered the “old” quantum theory. When Heisenberg sub-
mitted his paper proposing the uncertainty principle and the HM, he had
not previously submitted it to Bohr for prior review. It was after publication
that Bohr read the paper and pointed out fundamental problems relating
to discontinuity in change of the electron’s momentum and energy.
Bohr also pointed to the wave particle duality, i.e. that the electron
would have a greater probability of being in a finite region in space rather
than a specific well defined position[4]. He also pointed out that the un-
certainty in observation of the position of the electron was not exclusively
due to quantum discontinuities but that the diffraction pattern would place
limitations on the position measurement of the electron[1].
2.3 Deriving Heisenberg’s uncertainty principle
Heisenberg used his thought experiment, known as the HM, to demonstrate
the uncertainty principle. He did so by looking at the resolution of a lens
in a microscope and what limitations there are. Resolving power of a lens
is given classically by Equation (2.3.1).
Dx =
L
2 sin(A)
(2.3.1)
Using Equation (2.3.1) and relating it to Figure 2.3.1 the position of the
object is given by Dx. This would normally be considered the standard
deviation error of the measurement. However, treating photons quantum
mechanically, the momentum of the photon has to be taken into considera-
tion. This is given by P = h/L. Where h is Planck’s constant.
3
11. Due to Compton scattering (Section 2.3.2), the momentum of the pho-
tons that are collected to the right and left side of the lens in Figure 2.3.1
are not equal. For the photons on the right hand side (RHS) edge of the
lens, momentum is given by Equation 2.3.2.
PRHS = px +
h sin(A)
L
(2.3.2)
Similarly for photons that are collected on the left hand side (LHS) edge,
of the lens momentum is given by equation 2.3.3.
PLHS = px −
h sin(A)
L
(2.3.3)
Maintaining conservation of momentum in the x direction, the photons
that are collected on opposite sides of the lens have the same momentum.
Thus follows Equation 2.3.4:
px +
h sin(A)
L
= px −
h sin(A)
L
(2.3.4)
If A is a small L ≈ L ≈ L, it follows:
px − px = Dpx ≈
2h sin(A)
L
(2.3.5)
Substituting Dx = L
2 sin(A) into equation 2.3.5.
Dpx ≈
h
Dx
or DpxDx ≈ h (2.3.6)
What this means is that the larger the lens, the more precise the position
measurement. Conversely resulting in greater uncertainty in the momen-
tum measurement, as the lens gets smaller. Thus, reducing the positional
measurement accuracy, reduces the uncertainty in the momentum[5].
Equation 2.3.6 that Kennard developed into Equation 2.1.2, is now the
accepted equation for uncertainty principle.
The results of matrix mechanics and Schr¨odinger’s equation were shown
to be equivalent, not only by Schr¨odinger[6], but also by Lanczos who con-
sidered them mathematically a complete equivalence by showing that matrix
mechanics was fully equivalent to using calculus operators[3].
What survives from these two theories in the modern era is most usually
Schr¨odinger’s wave equation and Heisenberg’s Uncertainty Principle.
2.3.1 Uncertainty relations
There are many uncertainty relations between conjugate variables, the fol-
lowing are some that relate to the HM in to different degrees. Relation 2.3.7
is of primary concern in relation to this project.
px − xp = −i¯h, ∆x∆p ≥
¯h
2
(2.3.7)
The x term refers to the position and p refers to the momentum.
4
12. Et − tE = −i¯h, ∆E∆t ≥
¯h
2
(2.3.8)
The E term refers to the energy and t refers to time.
Jw − wJ = −i¯h, ∆w∆J ≥
¯h
2
(2.3.9)
Here w refers to the orbiting angle of the electron in its orbit, for exam-
ple, and J refers to the unit of action[1].
2.3.2 The Compton Effect
The Compton Effect describes the scattering of a single photon by an elec-
tron. This effect takes into account the conservation of momentum, energy
and relativistic effects. If the electron is at rest the collision will result in
the deflection of the path of both the photon and the electron. The effect
is that the photon will lose energy and momentum to the electron and as
a result will have its wavelength increased in proportion to the loss of en-
ergy and momentum. Equation 2.3.10 is used to show the difference in the
wavelengths[7].
λ0 − λ =
h
mc
(1 − cos(β)) (2.3.10)
Here λ0 is the initial wavelength, β is the photon’s deflection angle, h is
Planck’s constant, m is the mass of the electron, the angle β is the angle
that the photon is deflected from its original path, and c is the speed of
light. The electron will then gain momentum from the photon consistent
with the photon’s change in wavelength. The magnitude of the change in
wavelength is given by Figure 2.3.2 as a function of deflection angle.
Based on this equation the change in wavelength due to the Compton
Effect has been ignored in the modelling of this project as the effect is
minimal and has little or no effect on wavelengths down to the level of
x-rays (where the fractional change in wavelength is only of the order of
0.01%).
2.3.3 Fraunhofer Diffraction
Diffraction is a process whereby a wave meets an obstacle and is bent around
the obstacle. The Huygens-Fresnel principle is the theory which is used to
describe this process. This principle in short, is a wave front meets an ob-
struction; the resultant propagation will be the superposition of secondary
wavelets which act as individual point sources. Figure 2.3.3 shows the ba-
sic concept that is taken into account, that is that each point source along
the aperture is slightly out of phase with any adjacent. Huygens-Fresnel
principle is a summation of each of these contributions that takes into ac-
count the phase difference and the resulting interference. What Fraunhofer
diffraction assumes is that the light incident on the aperture is in phase.
Kirchhoff further formalized the mathematical description of this process
initially described mathematically by Fresnel[8].
Fraunhofer diffraction is the result of the interference formed at a long
distance from the obstacle due to the superposition of the secondary waves.
5
13. Figure 2.3.2: Graph of the change in wavelength due to the Compton Effect.
The resultant field intensity at a given point is given by Equation 2.3.11,
where A is the area of the aperture and F(r) is given by Equation 2.3.12 for
the case of incident spherical waves.[9]
E =
A
F(r)dτ, dτ = dxdy (2.3.11)
F(r) =
A
r
eikr
(2.3.12)
Here r is the distance between the source and where E is the field intensity.
This method of finding the intensity from a given aperture can be ex-
tended for a set of apertures.
2.3.4 The Kirchhoff Obliquity Factor
The Kirchhoff obliquity factor is a feature of the simple theory above known
as Kirchhoff’s diffraction theory and is given by Equation 2.3.14 based on
Kirchhoff’s boundary condition. This equation is set up with angles as
shown by Figure 2.3.4. What Kirchhoff’s diffraction theory does is that it
takes into account a more complete picture than Huygens-Fresnel theory
and in doing so creates a more accurate picture, whereby small scattering
angles are not assumed. This more complete picture comes from taking into
account regions B and C on Figure 2.3.4 as well as the aperture A by using
equation(2.3.13). This equation is simplified to Equation 2.3.14 by letting
the radius of surface C go to infinity and ignoring normal derivatives the
terms 1
s and 1
r . Equation 2.3.14 is known as the Fresnel-Kirchhoff diffraction
formula[8].
E(P) =
1
4π A
+
B
+
C
u
∂
∂n
eiks
s
−
eiks
s
∂U
∂n
ds
(2.3.13)
6
14. Figure 2.3.3: Diagram of point sources in an aperture to show principle of Huygens-
Fresnel principle. (Diagram is a modified reproduction of Figure 10.4 in reference
[9])
Figure 2.3.4: Aperture A and the angles used in Equation 2.3.14 (reproduced from
a diagram based on Figure 8.3 from reference [9])
7
15. E(P) = −
iA
2λ A
eik(r+s)
rs
{cos(n, r) − cos(n, s)}ds (2.3.14)
Equation 2.3.14 can be simplified further by neglecting contributions
from C on W as cos(n, r) = 1 and if χ = π − (r, s) Equation 2.3.15 is
obtained (W is a portion of the aperture A).
E(P) = −
i
2λ
Aeikr0
r0 W
eik(s)
s
{1 − cos(χ)}ds (2.3.15)
This leads to Equation 2.3.16 which in more complete form is Equation
2.3.17.
dW =
−i
2λ
Aeikr0
r0
eik(s)
s
{1 − cos(χ)}ds (2.3.16)
E(P) = dW (2.3.17)
The {1−cos(χ)} is the term that is the obliquity factor that creates the
more complete picture.
2.3.5 Effect of the Lens
The net effect of the (hypothetical or perfect) lens is used to equalize the
phase of the spherical wavelet (and thus the secondary wavelets emitted)
across the entire aperture surface. It then forms an image on a screen or
a photographic plate for γ-rays. In practice the actual lens does this by
having a different refractive index compared to the external medium and
varying thickness. The result will bend the path of the photon on entry
and exit of the lens. The entry and exit angles can be calculated by Snell’s
law sin α1n1 = sin α2n2, where the angles α1 & α2 are the incident and
exit angles at a given medium boundary and n is the refractive index of
the respective medium[10]. The fact that the lens has a varying thickness
will cause there to be an optical path difference between the photon that
enters the centre and the edge of the lens. It is this variation that causes
the spherical wave to be put in phase across the lens and creates a planar
phase front. Optical path difference δ is given by δ = 2π
λ0
n(d), where d is
the thickness of the material causing the optical path difference and λ0 is
the vacuum wavelength of the electromagnetic wave[11].
For the case of the model of the HM used here it is assumed that the
lens was infinite diameter.
2.3.6 Gibbs Phenomenon
The Gibbs or Gibbs-Wilbraham phenomenon is the effect that comes from a
truncated Fourier series. In normal circumstances a Fourier series is an infi-
nite sum of periodic functions. What happens with the Gibbs phenomenon
is that higher order contributions of this series are left out and as a result
truncation occurs. Figure(2.3.5) shows this effect in terms of a signal being
converted from a sinusoidal wave to a square wave and the oscillations at
the corners of the square wave are predicted by Gibbs Phenomenon.[12] In
the case of our model this effect could be looked at in terms of when the
8
16. Figure 2.3.5: An example of the Gibbs Phenomenon showing how with increased
orders of an infinite Fourier series produce a closer approximation to a square wave.
Also the oscillation at the corners of the waves (which is the Gibb’s phenomenon)
is reduced.(MATLAB code used for generating these plots and licensing appears
in appendix D).
lens is finite in extent and thus not large enough to collect contributions
from the source at larger angles from the normal.
2.4 Standard Deviation and the Uncertainty Prin-
ciple
The standard deviation for a distribution is given as the average deviation
of results from a given value in a set of results. It is given by Equation 2.4.1,
where s2 is given by Equation 2.4.2[13].
s =
√
s2 (2.4.1)
s2
=
n
i=1 x2
i −
( n
i=1 xi)2
n
n − 1
(2.4.2)
In the case of the uncertainty principle it is not so straight forward to
calculate these values. The results are derived from diffraction patterns of
single and double slit experiments. In general the work done in relation
to this subject has been reported by J. Hilgewoord and J.B.M. Uffink. In
their work they looked at the overall width of the diffraction pattern (Wφ)
for the uncertainty in the position. The mean peak width of the diffraction
9
17. Figure 2.4.1: Diffraction pattern from a single slit showing some of the features
that have used to show uncertainty principle.
patterns (wΨ) were used as a measure of the uncertainty in the momentum.
These features are shown in Figure 2.4.1.
The region outside WΨ in Figure 2.4.1 is an area of either low intensity
or destructive interference, this is due to a phase lag where |p1 − p2|∆q =
π. What this means is that there must be two incident waves of similar
amplitude that are π out of phase with each other. There must also be
other waves that interfere destructively to reinforce this, the result being
that |P1 − P2|∆q ≥ π must be of at least width ∆p, with ∆p∆q ≥ 2π[14].,
The value WΨ of the diffraction pattern can be use as a measure of
the standard deviation where N is the smallest number N that satisfies
Equation 2.4.3.
x +W/2
x −W/2
|Ψ(x)|2
dx = N (2.4.3)
The ˘Cheby˘se˘v inequality can be used to choose N such that 0 < N < 1
where it is preferable that N be close to 1. This inequality derived by
the Russian mathematician ˘Cheby˘se˘v who devised a standard in terms of
the distribution of data in relation to its mean and whether it is within 2
or 3 standard deviations of the mean. ˘Cheby˘se˘vs rule states that 3/4 of
the data should be within two standard deviations of the mean and 8/9
of the data within three standard deviation of the mean, and there is no
particular standard of proportion of the data that should be within one
standard deviation of the mean[13].
x +n∆x
x −n∆x
|Ψ(x)|2
dx ≥ 1 −
1
n2
, (n ≥ 1) (2.4.4)
From Inequality 2.4.4 the width of Ψ(x) the wave function of the diffrac-
tion pattern has an upper bound.[15] With single slit experiments the stan-
dard deviation is given by WΨq = a/
√
3 where A is the overall width of
the diffraction pattern, for a double slit experiment ∆q ≈ WΨ∆p ≈ 1
2a with
2a being the full width half maximum of the different peaks. For peaks
10
18. of different amplitudes ∆p can be taken as the mean peak width.[14] This
leads to the generalization from single and double slit experiments shown
in Equation 2.4.5[15].
(2∆p)−1 < w(Ψ) ≤ w(|Ψ|) ≤ W(|Ψ|) = W(Ψ) ≤ 2∆x (2.4.5)
Equation 2.4.5 is also another interpretation of the uncertainty principle
related to single and double slit experiments[14].
For the mean peak width w the auto-correlation function given by Equa-
tion 2.4.6. This function is usually used with random signals of infinite
duration. In this case the modulus is squared to give g(w) and set to equal
M2 where M is less than 1 but close to 1. The mean peak width can be
defined as the smallest w that can be used to satisfy Equation 2.4.7[14].
f(q) = Ψ∗
(q )Ψ(q − w)dq (2.4.6)
g(w) = Ψ∗
(q )Ψ(q − w)dq
2
= M2
(2.4.7)
Taking N from equation (2.4.3) and M from equation 2.4.7 it can be
shown that M2 + N2 ≥ 1, using WΨ ≤ 2(1 − N)−1/2∆q, inequality 2.4.8 &
2.4.9 can be shown[14].
1 − M2/∆p < wΨ ≤ WΨ ≤ 2∆q/
√
1 − N (2.4.8)
1 − M2/∆q < wφ ≤ Wφ ≤ 2∆p/
√
1 − N (2.4.9)
If M ≥ 2N − 1 then it follows that the uncertainty principle can be
shown by inequalities (2.4.10) & (2.4.11), as if exterior uncertainty in Ψ is
reduced, then the interior uncertainty in Ψ consequently increases with the
opposite happening if the uncertainty in Ψ increases, the uncertainty in Ψ
will be reduced.
wΨWφ ≥ 2 arcsin
M + 1 − N
N
(2.4.10)
wφWΨ ≥ 2 arcsin
M + 1 − N
N
(2.4.11)
2.4.1 Calculating Standard Deviation in this model
Figure 2.4.1 shows the pattern from a single slit diffraction. While most
methods work on finding standard deviations of Gaussian distributions, they
don’t work as well on diffraction patterns.
The basis of the method for finding standard deviation in this project is
given by Equation 2.4.12, this method has previously been used in reference
[16]. In effect, what this says is that the standard deviation is the square
root of the expectation value of x2 minus the expectation value of x.
∆x = x2 − x 2 (2.4.12)
Expectation value E[X] can be considered to be a weighted mean of
an event occurring at a particular location, it is found by using Equation
2.4.13, similarly E[X2] is found using Equation 2.4.14[17].
11
19. Figure 2.5.1: Diagram showing the distances needed in the derivation of the inte-
gration to convert angles to distances.
E[X] = ΣiXi × P(Xi) (2.4.13)
E[X2
] = ΣiX2
i × P(Xi) (2.4.14)
In this model the standard deviation is found using this method, as the
diffraction pattern is an even function E[X] equals zero. As a result the
standard deviation is equal to E[X2]. There are a number of variations
on how E[X2] is calculated, these variations are all based around how the
function is normalized to make a probability mass function. The two meth-
ods should take the area under the curve and either normalize the function
elementally (Equation 2.4.15) or to find the expectation value and divide it
by the area under the curve (Equation 2.4.16).
σx = Σi
xi
ΣP(xi)
(2.4.15)
With Equation 2.4.15 the function is first normalized before computing
E[X] or E[X2], this can result in rounding errors.
σx =
∞
−∞ P(x)x2 dx
∞
−∞ P(x) dx
(2.4.16)
The second method using Equation 2.4.16 produces less rounding errors
and as a result is better in this situation.
2.5 Derivation of Equation used for the Model
The model used in this project is derived from Fresnel-Kirchhoff diffraction,
Section 2.3.4. It is from equation 2.3.17 that the derivation begins. Figure
2.5.1 shows the set-up of the lens and points of reference to set up the
integration. From this diagram Equation 2.5.1 is easily shown.
12
20. Figure 2.5.2: The origin of the ∆ in s + ∆s
cos(n, s) =
f
p 2 + f2
(2.5.1)
From this the diffraction equation can be written as:
E(P) = −
i
2λ
Aeikr0
r0 A
eiks
r
{1 + cos(n, r)}ds (2.5.2)
With eikr0 = 1 it can be ignored, as the integration moves to more
extreme angles s becomes s0 + ∆ where ∆, is given by p sin(n, r) cos(φ) as
shown in Figure 2.5.2 with the cos(φ) coming from the rotation shown in
Figure 2.5.1.
From this the equation becomes:
E(P) = −
i
2λ
Aeikr0
r0 A
eiks
r
f
p 2 + f2
+ 1 ds (2.5.3)
Using Pythagoras theorem it can be shown that r is equal to p 2 + f2
and so the equation becomes:
E(P) = −
i
2λ
Aeikr0
r0 A
eikp sin(n,r) cos(φ)eiks0
p 2 + f2
f
p 2 + f2
+ 1 ds (2.5.4)
With ds = p dp dφ equation 2.5.4 becomes:
E(P) = −
iA
2λ
eikr
s
feiks
A
eikp sin(n,r) cos(φ)
p 2 + f2
f
p 2 + f2
+ 1 p dp dφ
(2.5.5)
Including the limits of integration:
E(P) = −
iA
2λ
eikr
s
feiks
R
0
2π
0
eikp sin(n,r) cos(φ)
p 2 + f2
f
p 2 + f2
+ 1 p dp dφ
(2.5.6)
13
21. On integration with respect to φ the equation becomes a Bessel function[18]:
f2 + f2 + p 2
f2 + p 2
J1(k sin(n, r)) (2.5.7)
Where,
Jm(u) =
i−m
2π
2π
0
eimν+u cos(ν)
dν (2.5.8)
2.5.1 Differences between Kirchhoff and Fraunhofer diffrac-
tion
Fraunhofer diffraction is intended to look at diffraction of plane waves which
has a source that is sufficiently far away from the obstacle causing the
diffraction that the plane waves are effectively parallel. In the case of a
Kirchhoff diffraction, the equation takes into account contributions of the far
field source as well as reflections from around the aperture. This is done by
taking into account the distance from which the source is from the obstacle
using the multiplying factor in front of the Bessel function on which both
diffraction patterns are based. Equation 2.5.9 shows the equation integrated
for a Fraunhofer diffraction pattern, the difference between it and Equation
2.5.10 is the multiplying factor that is Equation 2.5.11.
dw = ΣR
p dp k2
sin2
(n, r)J1(k sin(n, r)) (2.5.9)
dw = ΣR
p dp
f + f2 + p 2
f2 + p 2
J1(k sin(n, r)) (2.5.10)
f + f2 + p 2
f2 + p 2
(2.5.11)
This multiplying factor has the shape shown in Figure 2.5.3 which is non
linear.
3 Computation
This Project was carried out using MATLAB or MATrix LABoratory. This
computational environment was initially was written in FORTRAN by nu-
merical analyst Cleve Moler in the late 1970s. As this environment became
popular it evolved into a commercial software package written in C[19].
The main equation used in the code is Equation 2.5.8, this was integrated
by summing each of the angular contributions at each point across the
aperture (lens). From this a diffraction pattern was built up. Section 3.1
explains the workings of this part of the program, from this the subsequent
parts of the program were built. The next part of the experiment was to
build up a 3D diffraction pattern or Airy disk. This is explained in Section
3.2, also within this program there is a routine where the 3D image is split
into slices which are individually binned to produce a quasi-2D pattern. It
is this quasi-2D pattern from which the standard deviation is calculated.
The standard deviation is generated using Equation 2.4.12, the fact that
E[x] is equal to zero, it is only E[x2] requires calculation. The problem that
14
22. Figure 2.5.3: Plot of how the multiplying factor of Kirchhoff diffraction changes
with respect to f.
was found with calculating the standard deviation of a diffraction pattern of
this type is that the numbers for each increment of the diffraction pattern are
very small. For this reason equation (2.4.16) was settled on for its robustness
in calculating expectation values. It was found from testing on Gaussian
distributions (both normalized and non-normalized) that expectation values
can be accurately calculated so long as the increment is no smaller than the
square of the expectation value. If the increment is 10−5 for example, as
long as the expectation value is not smaller than 10−10, the expectation
values can be considered accurate. For dealing with large numbers, it was
found that there were small enough increments the expectation value could
be easily calculated. More often MATLAB would have problems of array
sizes exceeding memory limits. For this project only the lower extreme of
the expectation values were of concern.
A wavelength of 500 nm was used to make the computation experimen-
tally verifible
3.1 Integration code
Equation 2.5.10 is in the form of a Bessel function which is non-trivial to
integrate analytically. As a result, numerical methods are used to integrate
this equation. Figure 3.1.1 shows a code flow diagram, with the initial
flow of the basic code with the integration and checks on the left and the
optimizations on the right. In its most basic form the code sums all the
contributions from each portion of the lens (dW) at the coresponding angles.
The main for loop merely moves the integration across the different portions
dW of the lens. A graphical representation of the integration process is
shown in Figure 3.1.3. The code that is referred to here is given in appendix
B.
The integration code has built in checks to see if the algorithm correlates
with already existing models of diffraction patterns modeled using Fraun-
hofer diffraction and a Bessel function. The results shows high correlation
between the models indicating that the model derived using the Kirchhoff
15
23. Figure 3.1.1: Code flow diagram of the code for the integration on which this
project is based
Figure 3.1.2: Plot of the 1D diffraction pattern
16
24. Figure 3.1.3: Graphical representation of the integration process. The program
moves across the aperture with increments dW, summing the contributions across
each dW for the coresponding angle.
obliquity factor may be correct.
This part of the code offered a degree of optimization. First because the
Bessel function is even and as a result one side of the diffraction pattern is
calculated the other side can simply be copied across as a mirror image. A
second optimization resulted from the fact that only one part of the equation
that is integrated has the lens electron distance as an input. This allowed
this part to be calculated once before integration loop and the result used in
following iterations of the loop. Figure 3.1.2 shows the diffraction pattern
generated using the integration code.
3.2 Creation of the 3D diffraction pattern
The use of a circular lens offered an advantage for the creation of a 3D model
of the diffraction. Because the lens is geometrically symmetric radially from
its centre, a 2D diffraction pattern can simply be rotated about its central
axis to produce the desired 3D pattern or Airy disk[18]. The method of
generating the Airy disk was done as follows: Figure 3.2.1 shows an array
whose centre can be considered the centre of the Airy disk, the distance
from the centre can be found using Pythagoras’ theorem and the boxes of
the array as a counting units. By matching this distance and the distance
from the centre of the 2D diffraction pattern shown in Figure 3.2.2, the
amplitude of the 2D diffraction pattern can be copied to the Airy disk array
to produce a 3D plot shown in Figure 3.2.3.
This rotation is done using embedded for loops within the program and
the use of Pythagoras theorem. The advantage of not using the MATLAB
airy function is that the Kirchhoff diffraction pattern can be more easily
manipulated than the MATLAB airy function.
The MATLAB code that is used to generate this rotation is given in
Appendix C. Figure 3.2.4 shows the code flow diagram for this code. It
is built from the integration code, but with the diffraction pattern being
17
25. Figure 3.2.1: The use of Pythagoras’ theorem to create the Airy disk.
Figure 3.2.2: The amplitude at the radius of the 2D diffraction pattern.
18
26. Figure 3.2.3: The Airy disk generated from the 2D diffraction pattern.
rotated on the XY array by nested for loops.
The next part of the program is the binning of the 3D pattern, this is
done by taking slices parallel to the edge of the 3D pattern (Figure 3.2.5 top).
These slices are then summed to create a quasi-2D pattern (Figure 3.2.5
bottom). From this quasi-2D pattern the standard deviation was calculated.
With the rotation there was an optimization, by generating a quarter
of the 3D image and careful binning of the values to create one side the
quasi-2D pattern that could subsequently be copied across to complete the
quasi-2D pattern.
The whole process is in a nested loop to allow the measurement of stan-
dard deviation as a function of the lens electron distance as in Figure 3.2.4.
3.3 Testing the integration by aliasing
Aliasing is a form of sampling a signal where part of the original information
is lost leaving a partial or distorted picture of the original information[20].
It is achieved by periodic sampling of the original signal, generating a signal
that is of a lower frequency. The common example of aliasing is shown in
Figure 3.3.1. Here the original (red line) sine wave has a higher frequency
than the aliased signal (green line), the place where the samples are taken
is at the intercepts of the two sine waves[20].
In testing the integration aliasing was used. If the integration was work-
ing correctly the minima (or maxima) would change position in proportion
to the change in increment of the lens used. These aliasing characteris-
tics were observed indicating that the integration was working correctly.
Figure 3.3.2 shows aliasing in the 2D diffraction pattern. Aliasing was im-
plemented by varying the increment (dW) by which the integration of the
lens was taken.
19
27. Figure 3.2.4: Code flow diagram for the rotation of the 2D diffraction pattern.
20
28. Figure 3.2.5: Slicing the 3D pattern of a contour-plot (top) and the quasi-2D
pattern that is generated (bottom).
21
29. Figure 3.3.1: Aliasing of a sinusoidal signal.
4 Results
This project aimed to look at uncertainty principle and relate it to standard
deviation of a diffraction pattern. What was expected is that the standard
deviation would decrease as the distance between the lens and electron re-
duce, until at some point the standard deviation would tend towards infinity.
The first thing that was looked at was the angular position of the peaks
and troughs and whether they changed with the lens electron distance.
What transpired was a straight line plot parallel to the horizontal axis.
This showed that they did not move leading to questions about what was
happening in the model.
What was then examined was the angular standard deviation as this
could easily converted to positional standard deviation ∆x. It is shown
that the multiplication factor is not the main factor in the variance of the
standard deviation.
What was found was that the expected behavior was not followed. The
standard deviation was found to reduce linearly with the lens electron dis-
tance and thus that the uncertainty principle was not upheld. This sug-
gested that the model may need alteration to correctly show what is physi-
cally known.
Using the program described in Section 3.2, standard deviations were
calculated over a range of lens electron distances. This range was from
10−15m to 2m as this would allow for an electron to be resting on the lens.
The angular standard deviations were calculated, the results of which are
shown in Figure 4.0.3. The unexpected feature of this is the drop off in
standard deviation as the lens electron distance is about 10−5m. This dip is
22
31. Figure 4.0.3: Plot of standard deviation in terms of angle verses lens electron
distance (insert shows the dip as the lens electron distance comes close to zero)
Figure 4.0.4: Plot of ∆ x vs. lens electron distance.
shown in the insert. This dip corresponds to the fact that standard deviation
cannot be calculated at zero as it is a non-physical location to measure the
position of the electron.
The standard deviation in terms of the angle was then converted into
uncertainty in position. This was achieved by using the f × tan(θ) where f
is the lens electron distance. A plot of the result is shown in Figure 4.0.4.
What followed was an investigation of the causes of the features. For
this a plot of the multiplying factor and the angular standard deviation
were compared. The result is that the dip in standard deviation roughly
corresponds to the peak in the multiplication factor. This is shown in Figure
4.0.5, the top plot is of the standard deviation in angle with the lower plot
the multiplication factor.
24
32. Figure 4.0.5: The angular standard deviation (top) and the multiplication factor
(bottom). Comparison of the two and the correlation between the dip in angular
standard deviation and the peak in the multiplication factor.
25
33. 5 Discussion
5.1 Evaluation of results
The standard deviations from the 2D and quasi-2D diffraction patterns fol-
lowed the same trends. The main difference being that the quasi-2D angular
standard deviation was about 2 orders of magnitude greater than the 2D
angular standard deviation. This is something that could be expected when
from dealing with uncertainty principle. That is that if uncertainties are
looked at in multiple directions, the uncertainties are multiplied and can
often cause an increase in uncertainty. This would obviously lead to a re-
duction of uncertainty in the momentum of an object.
The result displayed in Figure 4.0.4 shows the positional standard devi-
ation reducing with the lens electron distance. This is not consistent with
the uncertainty principle and that with this model the uncertainty principle
is not upheld. In this model the standard deviation ∆x did not follow what
would be physically expected from real diffraction patterns, this indicated
that the model might be incomplete.
What followed this was the exploration of what may be used to correct
the model. The first was to consider a theoretical infinite lens and let the
small angle approximation limit how much of the lens would be “seen”. This
approach was followed by varying the size of the lens within the program,
the results of which can be described using the Gibbs phenomenon.
Another possibility came from examining the effect of changing the lens
size as while keeping the lend electron distance constant. This resulted
in the width of the diffraction pattern changing. This observation can be
explained using Gibbs Phenomenon
If one considers the idea of using the Kirchhoff Obliquity Factor, it seems
strange that the acceptance angle (shown in Figure 3.1.3) of order 0.01
radians, that is 1.4 degrees. The question that was considered was what
happens if the lens size is kept constant, letting the acceptance angle be
dictated by the tangent of the radius of the lens divided by the lens electron
distance. The result of this was that the standard deviation diverged as
the the lens electron distance became smaller. This observation ignores the
Gibbs phenomenon and can be explained by the fact that the diffraction
pattern becomes wider.
5.1.1 Use of an infinite lens varying the lens that is “seen”
If we consider using an infinite lens, only a certain portion of the lens pro-
duces the image that is produced. If the small angle limits the size of the
lens that is “seen”, the result is that the aperture will vary with respect to
the lens electron distance. This experiment was carried out over a range
of 0 to 8 meters. The results come in three forms, the angular standard
deviation from the 2D diffraction pattern(Figure 5.1.1), angular standard
deviation from the quasi-2D pattern (Figure 5.1.2) and standard deviation
in terms of position (Figure 5.1.3).
What would physically be expected is that as the lens is moved from
infinity closer to the lens, the standard deviation would reduce and then at
some point start to diverge again towards infinity as the object lens distance
became smaller. There are some discontinuities with this plot but the basic
26
34. Figure 5.1.1: Change in the standard deviation using a 2d diffraction pattern
Figure 5.1.2: Change in angular standard deviation using a quasi-2d diffraction
pattern
trend is that the standard deviation converges as the object comes in from
a large distance and then diverges again.
5.1.2 Demonstrating the Gibbs phenomenon
The model used is based on a Bessel function and as a result the width of
the wave function is directly dependent on the inputs to the Bessel function.
With no change in the inputs to the Bessel function, the standard deviation
changed almost linearly with the change in lens electron distance.
This led to a question of what happens if the lens electron distance is kept
constant and the size of the lens is changed. Figure 5.1.4 shows graphically
the effect of varying the lens size. Table 5.1.1 shows the angular standard
deviations corresponding to the individual lens radii. What Figure 5.1.4
and Table 5.1.1 together show is that the main factor effecting standard
deviation is the shape of the diffraction pattern. That is as the lens gets
27
35. Figure 5.1.3: Change in the standard deviation using a quasi-2d diffraction pattern
Figure Lens radius Angular standard
reference (meters) deviation (Radians)
5.1.4a 1.0 × 10−5
31.5910×10−6
5.1.4b 2.5 × 10−5
21.9555×10−6
5.1.4c 5.0 × 10−5
12.3164×10−6
5.1.4d 7.5 × 10−5
9.1886×10−6
5.1.4e 1.0 × 10−4
7.5723×10−6
5.1.4f 2.5 × 10−4
4.3131×10−6
5.1.4g 5.0 × 10−4
2.9134×10−6
5.1.4h 7.5 × 10−4
2.3351×10−6
Table 5.1.1: Table of angular standard deviations as function of lens radius corre-
sponding to the sub figures in Figure (5.1.4)
larger the diffraction pattern gets narrower and more defined, conversely
so does the standard deviation. This observation corresponds to the Gibbs
phenomenon (Section 2.3.6) where a truncated Fourier series produces a less
well defined wave function.
5.1.3 Acceptance angle as a function of lens electron dis-
tance
If one considers the concept of Kirchhoff Obliquity Factor, that is taking into
account all the contributions from the from the extreme angles of the lens.
The question is asked posed as to what effect this could cause. The way
this was implemented was to allow the acceptance angle for the integration
of the model be dependent on the tangent of the lens radius divided by
the lens electron distance. Figure ?? shows multiple plots of diffraction
intensity patterns at different lens electron distances, Table 5.1.2 shows the
corresponding angular standard deviations of this observation.
As this observation was conceived in the last weeks of the project, it
was not extended to create an airy disk and so there are no results from the
28
36. (a) Lens Radius 1 × 10−5
m (b) Lens Radius 2.5 × 10−5
m
(c) Lens Radius 5 × 10−5
m (d) Lens Radius 7.5 × 10−5
m
(e) Lens Radius 1 × 10−4
m (f) Lens Radius 2.5 × 10−4
m
(g) Lens Radius 5 × 10−4
m (h) Lens Radius 7.5 × 10−4
m
Figure 5.1.4: As the lens size is increased a sharper image of the diffraction pattern
is produced, this is consistent with the Gibbs Phenomenon.
29
37. Figure 5.1.5: Diagram showing the acceptance angle of the lens, lens radius and
the lens electron distance
Figure Lens electron Angular standard
reference distance (meters) deviation (Radians)
5.1.6a 0.1 7.663×10−3
5.1.6b 0.2 1.936×10−3
5.1.6c 0.3 0.808×10−3
5.1.6d 0.4 0.449×10−3
5.1.6e 0.5 0.282×10−3
5.1.6f 0.6 0.193×10−3
5.1.6g 0.7 0.151×10−3
5.1.6h 0.8 0.105×10−3
Table 5.1.2: Table of angular standard deviations as function of lens electron
distances corresponding to the sub figures in figure (5.1.6)
quasi 2D pattern.
When examined over a larger lens electron distance it was found that
the angular standard deviation diverged as this distance got smaller. This is
shown in Figure 5.1.7, this angular standard deviation was then converted
into the actual standard deviation which is shown in Figure 5.1.8. Both
these observations are consistent with the uncertainty principle. However
this method of creating the diffraction patterns does not take into account
the Gibbs phenomenon.
What effectively happens is that the diffraction pattern becomes wider,
the acceptance angle is fed into a sin function within the Bessel function.
This causes the resultant diffraction intensity pattern to be locally periodic
as shown in Figure 5.1.9. It is also clear from this that there are wave
beating effects that decay with increasing angular obliquity. That is the
interference of two or more slightly out of phase signals resulting in varying
constructive or destructive interference of the resulting signals[21], in this
case the signals are wave fronts.
This set of arguments leads to the conclusion that the inputs into the
Bessel function are critical for the development of this model. The integra-
tion is designed such that the inputs to the Bessel function are the variation
in the angle, the size of the lens and the increment by which the lens is in-
tegrated. For the development of the model, the observations show a clear
30
38. (a) Lens electron distance 0.1m (b) Lens electron distance 0.2m
(c) Lens electron distance 0.3m (d) Lens electron distance 0.4m
(e) Lens electron distance 0.5m (f) Lens electron distance 0.6m
(g) Lens electron distance 0.7m (h) Lens electron distance 0.8m
Figure 5.1.6: As the lens is brought closer to the electron the acceptance angle is
increased resulting in the standard deviation increasing.
31
39. Figure 5.1.7: Plot of the angular standard deviation with respect to allowing the
acceptance angle be dictated by the tangent of the lens radius divided by the lens
electron distance.
Figure 5.1.8: Plot of the angular standard deviation with respect to allowing the
acceptance angle be dictated by the tangent of the lens radius divided by the lens
electron distance.
32
40. Figure 5.1.9: The periodic nature of the sin in the Bessel function, also the wave
beating effects on the peaks.
direction. That is to vary the acceptance angle as a function of the lens
electron distance in combination with the Gibbs phenomenon.
6 Conclusions
The HM was born at a time in the infancy of quantum mechanics. It
was conceived by Werner Heisenberg, a post-doctoral researcher working
for Neils Bohr. Bohr had himself made a considerable contribution to the
model of the Hydrogen atom and the “old” quantum mechanics. Heisen-
berg’s uncertainty principle has survived to this day as a fundamental of
the “modern” quantum mechanics.
This project focused on the momentum position uncertainty, this was
something that Heisenberg drew some of his conclusions from. Classically
as a bigger lens is used there should be a greater certainty in the position
measurement, however this is now known not to be true. This project turned
from an original focus on quantum mechanics to becoming optics focused.
This made the main focus of this report the manipulation of a diffraction
pattern model.
With the uncertainty in momentum there is an upper limit coming from
Compton scattering and the maximum angular scattering that can occur.
The limit is produced by the maximum angular uncertainty in the path
of the photon is falls below 180◦. This results in a maximum uncertainty
in momentum of 6.556×10−33 N for a 500 nm photon. This shows that
the model is limited by the standard deviation of the diffraction pattern.
Using positional standard deviation, the standard deviation can be found
for diffraction patterns. The model did not show the expected results, the
angular standard deviation of the diffraction patterns did not converge to a
point and diverge to infinity as the lens electron distance was brought from
infinity towards the lens. The results lead to an examination of the func-
tioning of the model. How would the model successfully demonstrate what
is physically known in terms diffraction patterns? Fraunhofer diffraction
only differs to Kirchhoff diffraction by a multiplication factor in front of the
33
41. Bessel function. This Bessel function effectively equates to the Fraunhofer
diffraction model.
A look at the multiplying factor as a function of lens electron distance
shows a peak as the function approaches zero. This peak results in no diver-
gence of the angular standard deviation which is what would be expected.
When the angular standard deviation is converted into the positional stan-
dard deviation there is no divergence, as could happen with changes in the
angular standard deviation. This showed that uncertainty principle was not
upheld in this model. Any fluctuation in the standard deviation was due
to the multiplication factor, caused by the obliquity factor. This in its self
proves that the obliquity factor has an effect on the standard deviation, even
if it does not cause the expected convergence and subsequent divergence in
standard deviation.
The next question was what exactly affected the wavefunction that might
hold the key to adjusting the model to make it fit the expected results?
The standard deviation seen to converge and then subsequently diverge.
The first attempt to simulate this was to use a theoretically infinite lens,
letting the small angle approximation dictate the size of the lens that is
“seen”. The results of which demonstrate the convergence and subsequent
divergence as the lens electron distance decreases.
If one considers the focal point of a lens, it is the point at which the most
optical information can be gathered to form an image. At this point most
optical information about the object is collected. If the object is outside
the focal plane of the lens, information is lost at the edges of the lens.
This can be considered a truncation in the Fourier series that makes up the
final image. This can be attributed to Gibbs phenomenon where smaller
lenses collect less of the available information, thus truncating the complete
picture.
If the acceptance angle is allowed to vary as the tangent of the lens radius
divided by the lens electron distance. If this is done closer the electron
is to the lens the greater the acceptance angle. This results in a wider
diffraction pattern and so an increasing standard deviation similar to that
which would be physically expected. With this observation uncertainty
principle is upheld which would suggest that it may be in some degree
correct.
This set of arguments leads to the conclusion that the inputs into the
Bessel function are critical for the development of this model. The integra-
tion is designed such that the inputs to the Bessel function are the variation
in the angle and the size of the lens.
If one considers the focal point of a lens, at this point the most optical
information can be collected to form an image. If the object is outside the
focal plane of the lens, information is lost at the edges of the lens. This
can be considered a truncation in the Fourier series that makes up the
final image. Within the focal plane there may be too much information
collected causing saturation in the image, this may cause a distorted image
to be formed by the lens. For the development of the model, the direction
that is suggested from the observations, is to vary the acceptance angle
as a function of the lens electron distance in combination with taking into
account Gibbs phenomenon.
34
42. References
[1] Heisenberg W. A translation ofҬUber den anschulichen Inhalt der
quantentheoretishen Kinematik und Mechanik”. Zeitschrift f¨ur Physik.
1927;43.
[2] Hilgevoord J, Uffink JBM. More certainty about the uncertainty prin-
ciple. Eur J Phys. 1985;6.
[3] Mehra J, Rehenberg H. The History and development of Quantum
Theory;.
[4] Tanona S. Uncertainty in Bohrs response to the Heisenberg micro-
scope. Studies in History and Philosophy of Modern Physics. 2004;5
Part 2(35):483507.
[5] http://www aip org/history/heisenberg/p08b htm. The Gamma-Ray
Microscope;.
[6] http://plato stanford edu/entries/qt uncertainty/. The Uncertainty
Principle; 2001.
[7] Kenyon IR. The Light Fantastic A Modern Introduction to Classical
and Quantum Optics. Ch. 9 Page(s) 237-252. University of Illinois at
Chicago, USA: World Scientific Pub Co Inc; 2005.
[8] Born M, Wolf E. Principles of Optics 7th edition;.
[9] Hecht E. Optics. Sec. 10.2.1. Addison Wesley;.
[10] Hecht E. Optics. Sec. 4.4.1. Addison Wesley;.
[11] Hecht E. Optics. Sec. 7.1.1. Addison Wesley;.
[12] Jerri AJ. The Gibbs Phenomonen in Fourier Analysis, Splines an
Wavelet Approximantions. Mathematics and its applications. Kluwer
Academic Publishers;.
[13] McClove JT, Sinclair T. Statistics 8th Edition;.
[14] Uffink JBM, Hilgevoord IJ. Uncertainty Principle and Uncertainty
Relations. Foundations of Physics. 1985;15(9).
[15] Uffink JBM, Hilgevoord IJ. Overall Width, Mean Peak and the Un-
certainty Principle. Physics Letters;95A.
[16] Uffink JBM, Hilgevoord IJ. Overall Width, Mean Peak and the Un-
certainty Principle. Physics Letters;95A.
[17] Wackerly DD, Mendenhall W, Scheaffer RL. Mathematical Statistics
with Applications. Duxbury Advanced Series. Duxbury Publishing;.
[18] Hecht E. Optics. Sec. 10.2.5. Addison Wesley;.
[19] Higham DJ, Higham NJ. MATLAB Guide Second Edition. page xix.
Siam Publishing;.
[20] Robers RA, Mullis CT. Digital Signal Processing. Addison Wiley series
in Electrical Engineering. Addison Wiley;.
[21] Rossing RD, Moore FR, Wheeler PA. The science of Sound;.
35
43. A Risk Asscessment
Page 1 of 3
Physics 4th
year project RISK ASSESSMENT form
Student Name:
Student Number:
Austin Tobin 58069328
Project Title: The Heisenberg gamma ray microscope
Main Project Working Location: N201/N222
Supervisor Name: Dr. Enda McGlynn
Brief Listing of All Risks Associated with Project (bullet point format)*:
None
*3 categories of project are envisaged:
1 – Theoretical/computational projects – in these cases you can write “None” above and proceed to the signature
page (but be sure to confirm this with your supervisor and remember that there can be manual handling risks even
with lifting computers, printers, monitors in addition to ergonomic issues such as seating etc.).
2 – Experimental work where no very significant or meaningful risks are identifiable, e.g. if one is working with very
standard “off the shelf” equipment modules, with no accessible high voltages, radiation exposure dangers, manual
handling, chemical, physical or other potential hazards – in these cases you should list above the potential risks and
the safety feature which renders them not significant (e.g. all high voltages such as mains voltage fully enclosed in
insulating or earthed container, all laser radiation fully enclosed in opaque container etc.) above and proceed to the
signature page – but be sure to confirm this with your supervisor.
3 – Experimental work where some significant or meaningful risks are identifiable – in these cases you should list
above the potential risks above. Each risk listed above should also have a risk assessment page of the type on the next
page of this document completed. This page should also include controls and precautions and other measures to
reduce the risk level and these sections must be completed and then proceed to the signature page – but again be sure
to confirm this with your supervisor.
36
44. Page 2 of 3
Risk (from list above – use a separate page with the format below for each risk):
Context
(Note: Context means issues such as the location / piece of equipment / type of task & how often completed / reason
considered a hazard and how harmful the outcome of a potential mishap could be)
Current Controls / Precautions
List any precautions, methods of work etc currently used to prevent accidents/injuries (bullet points):
Risk Assessment
(Use matrix below and tick or
highlight the category)
Probability Rating Severity Rating Risk Rating
Slightly Harmful
Harmful
Very Harmful
Unlikely
Trivial Risk
Acceptable Risk
Medium Risk
Likely
Acceptable Risk
Medium Risk
Significant Risk
Very Likely
Medium Risk
Significant Risk
Intolerable Risk
Further Measures Required To Reduce Risk
Date Complete
Residual risk
(when above measures
have been implemented)
Probability Rating Severity Rating Residual Risk Rating
Residual risk
acceptable?
Last Revision Date
37
45. Page 3 of 3
Signature page:
Student:
I have attempted, to the best of my ability, to assess the risks associated with my
project and to suggest appropriate controls and other measures to reduce the risk level.
I have also consulted with my supervisor in performing and completing this risk
assessment. I will implement these controls and other measures to reduce the risk
level consistently throughout my project and make my supervisor aware if at any
stage I feel the current risk assessment is insufficient as the project evolves*.
Student Name: Austin Tobin
Student number: 58069328
Signature:
Date:
Supervisor:
I confirm that the student has undertaken a risk assessment of their project work and
that they have consulted with me on this matter. Furthermore, the assessment above
appears thorough and I will inform the student if at any stage I feel the current risk
assessment is insufficient as the project evolves**.
Supervisor Name: Dr Enda McGlynn
Signature:
Date:
*The implementation of controls will be monitored throughout the year by Dr. Enda
McGlynn and Mr. Patrick Wogan.
**This risk assessment document should be revised by the student in the early part of
semester 2, and a brief discussion of this should form part of the mid-term meeting on
progress at that stage.
38
46. B Integration program
function int = integration()
clc;
L = 0.002; %Lens radius is L
thetalimit = 0.01; %thelimit is the largest angular extent of diffraction pat-
tern
p = 2e-4; %distance from the object to the lens
k = 2*pi/500e-9; %wave number
inc = 0.01*L; %increment of ldash
thetainc = thetalimit*0.0001;
Ldash = 0:inc:L; %integration variable re radius
thetarange = -thetalimit:thetainc:thetalimit;
s = size(thetarange);
integralrange1 = zeros(1,s(2));
integralrange = zeros(1,s(2));
%bess exp = zeros(1,s(2));
count = 1;
for theta = -thetalimit:thetainc:thetalimit
integrand1 = inc*(Ldash).*besselj(0,k*sin(theta).*Ldash);%Fraunhoffer
integrand = ((2.5e-13/1.32e-6)ˆ0.5)*inc*(((p+(((pˆ2)+(Ldash.ˆ2)).ˆ(0.5))).*Ldash)./((pˆ2)+(Lda
%(p+((pˆ2)+(Ldash.ˆ2))ˆ(0.5)))/(((pˆ2)+(Ldash.ˆ2)))
integral1 = sum(integrand1);
integral = sum(integrand);
%integralrange1(count) = real(integral1).ˆ2+imag(integral1).ˆ2;%testing
the significance of the real +imag not just the real
%integralrange(count) = real(integral).ˆ2+imag(integral).ˆ2;
integralrange1(count) = (integral1.ˆ2);%array for the storing of the dif-
ferent areas of slices
integralrange(count) = (integral.ˆ2);%squaring the wavefunction
bess exp(count) =(L/(k*sin(theta)))*besselj(1,k*sin(theta)*L);%bessel func-
tion solution to fraunhoffer
count = count+1;
end
intensity = (real(integralrange)).ˆ2+imag(integralrange).ˆ2;%abs(integralrange);
%intensity = (integralrange).ˆ2;
%intensity1 = (real(integralrange1)).ˆ2+imag(integralrange1).ˆ2;
%intensity1 = (integralrange1).ˆ2;
%intensity2 = (abs(bess exp)).ˆ2;
%intensity2 = (bess exp).ˆ2;
39
48. C Rotation program
function int = rotate2()
clc;
%dec = 1000; %no need to round in this case
format long;
L = 0.02; %Lens radius is L
thetalimit = 0.01; %thelimit is the largest angular extent of diffraction pat-
tern
k = 2*pi/500e-9; %wave number
inc = 0.001*L; %increment of ldash
thetainc = thetalimit*0.001; %thetainc = 1e-5 =¿ accurate to 1e-9
Ldash = 0:inc:L; %integration variable re radius
thetarange = -thetalimit:thetainc:thetalimit;%the angular range
[a s]= size(thetarange);
integralrange = zeros(1,s);
a = (s/2+0.5);%point in the middle of thetarange setting up loops ets
%**************************************************************************
p = 0:1e-14:1e-11;
[l M]= size(p);
const = zeros(1,M);
const1 = zeros(1,M);
%multiply = zeros(1,M);
for m = 1:1:M
mult = inc*(((p(m)+(((p(m)ˆ2)+(Ldash.ˆ2)).ˆ(0.5))).*Ldash)./((p(m)ˆ2)+(Ldash.ˆ2)));
%multiply(1,m) = mult;
%no use doing that too many times count = 1;
for theta = -thetalimit:thetainc:0%computing the first half of diffraction
integrand = mult.*besselj(0,k*sin(theta).*Ldash);%kirchoff
integral = sum(integrand);
integralrange(count) = abs(integral);%squaring the wavefunction
count = count+1;
end
for i = a:1:s%loading second half of array with values from the first half
integralrange(i) = integralrange(2*a-i);
end
%figure;
%plot(thetarange,integralrange);
%computating the expectation value for the 1D diffraction pattern
41
49. Ix = integralrange.*thetainc;%integralrange.*thetainc;
x2 = thetarange.ˆ2;%thetarange.ˆ2;
denom = sum(Ix);
num = sum(x2.*Ix);
expvalue = (num/denom);
%fprintf(’expectation value 1 2d for xˆ2 is:%dn’,expvalue);
const(1,m) = expvalue;
%**************************************************************************
a = (floor(s/2)+1);%parameter to set centre of the array
XY = zeros(s,s);%2d array for the 3d plot
XY(a,:) = integralrange;%loading the first set of values
%biulding up the 3D diffraction pattern
for i = 1:1:s%cycling through the rows
for j = 1:1:s%imbedding to cycle through the columns
R = floor(sqrt((a-i)ˆ2+(a-j)ˆ2)+0.5);%radius from the centre
if R>a%stop error radius larger than half the total width
XY(i,j) = 0;
else%else load the value at the distance from the centre
XY(i,j) = real(XY(a,(a-R+1)));
end
end
end
%figure;
%surf(XY);%3d plot of the pattern
%shading flat;
%camlight;
%**************************************************
clear i;%can’t have i starting at s
for i = 1:1:s%binning the 3D Airy disk
out(i) = sum(XY(i,:));%sum up the respective columns, aka binning the
values
end
X = -floor(s/2):1:floor(s/2);%set up the x axis
X = X.*thetainc;
%figure;
%plot(X,out);%plot the outcome see does it make any sense
%***********************************************************
Ix = out.*thetainc;%computing expectation value
x2 = X.ˆ2;
denom = sum(Ix);
num = sum(x2.*Ix);
expvalue2 = (num/denom);
%fprintf(’expectation value quazi 2d-xˆ2 is:%dn’,expvalue2);
const1(1,m) = expvalue2;%storing the expectation values in the global
array
%************************************************************
end
figure;%plots of the standard deviations over a range of electron lens dis-
tance
42
50. plot(p,const(1,:),’r’);
figure;
plot(p,const1(1,:),’b’);
Z = [p; const(1,:)];%outputting data to files
fid = fopen(’2d p0-1e-14-1e-11 L=2e-2-1-3.txt’,’w’);
fprintf(fid, ’%6.15f %12.15fn’,Z);
fclose(fid);
Z = [p; const1(1,:)];%; const2(1,:)];
fid = fopen(’quazid p0-1e-14-1e-11 L=2e-2-1-3.txt’,’w’);
fprintf(fid, ’%6.15f %12.15fn’,Z);
fclose(fid);
43
51. D gibb.m and licence
% Source: http://www.mathworks.com/matlabcentral/fileexchange/21291-
demonstration-of-gibbs-phenomenon/content/gibb.m
%Copyright (c) 2008, RAVEENDRANATHAN KALATHIL CHELLAP-
PAN
%All rights reserved.
%Redistribution and use in source and binary forms, with or without
%modification, are permitted provided that the following conditions are
%met:
%
% * Redistributions of source code must retain the above copyright
% notice, this list of conditions and the following disclaimer.
% * Redistributions in binary form must reproduce the above copyright
% notice, this list of conditions and the following disclaimer in
% the documentation and/or other materials provided with the distribution
% * Neither the name of the Government Engineering College Bartonhill
nor the names
% of its contributors may be used to endorse or promote products derived
% from this software without specific prior written permission.
%
%THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS
AND CONTRIBUTORS ”AS IS”
%AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT
NOT LIMITED TO, THE
%IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
FOR A PARTICULAR PURPOSE
%ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
OR CONTRIBUTORS BE
%LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
EXEMPLARY, OR
%CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
TO, PROCUREMENT OF
%SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
PROFITS; OR BUSINESS
%INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
LIABILITY, WHETHER IN
%CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLI-
GENCE OR OTHERWISE)
%ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE,
EVEN IF ADVISED OF THE
%POSSIBILITY OF SUCH DAMAGE.
%%% MATLAB script to demonstrate the
%%% Gibb’s phenomenon..
t=linspace(-2,2,2000);
u=linspace(-2,2,2000);
sq=[zeros(1,500),2*ones(1,1000),zeros(1,500)];
k=2;
N=[1,3,7,19,49,70];
44
52. for n=1:6;
an=[];
for m=1:N(n)
an=[an,2*k*sin(m*pi/2)/(m*pi)];
end;
fN=k/2;
for m=1:N(n)
fN=fN+an(m)*cos(m*pi*t/2);
end;
nq=int2str(N(n));
subplot(3,2,n),plot(u,sq,’r’,’LineWidth’,2);hold on;
plot(t,fN,’LineWidth’,2); hold off; axis([-2 2 -0.5 2.5]);grid;
xlabel(’Time’), ylabel(’y N(t)’);title([’N= ’,nq]); end;
%%% End of gibb.m
45
