Journal of Modern Physics, 2015, 6, 2011-2020
Published Online November 2015 in SciRes. http://www.sci
How to cite this paper: Bourdillon, A.J. (2015) The Stable Wave Packet and Uncertainty. Journal of Modern Physics, 6,
The Stable Wave Packet and Uncertainty
Antony J. Bourdillon
UHRL, San Jose, USA
Received 29 August 2015; accepted 13 November 2015; published 16 November 2015
Copyright © 2015 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
The traveling wave group that is defined on conserved physical values is the vehicle of transmis-
sion for a unidirectional photon or free particle having a wide wave front. As a stable wave packet,
it expresses internal periodicity combined with group localization. An uncertainty principle is
precisely derived that differs from Heisenberg’s. Also derived is the phase velocity beyond the ho-
rizon set by the speed of light. In this space occurs the reduction of the wave packet which is
represented by comparing phase velocities in the direction of propagation with the transverse
plane. The new description of the wave function for the stable free particle or antiparticle contains
variables that were previously ignored. Deterministic physics must always appear probabilistic
when hidden variables are bypassed. Secondary hidden variables always occur in measurement.
The wave group turns out to be both uncertain and probabilistic. It is ubiquitous in physics and
has many consequences.
Stable Wave Packet, Traveling Wave Group, Quantum Mechanics, Wave Mechanics, Determini sm,
Localization, Coherence
1. Introduction
The tra velin g wa ve gr oup tha t is d ef ined on c onse rve d physical quantities is a stable wave packet. In opposition
to the Copenhagen interpretation of Heisenberg’s Uncertainty Principle, Schrödinger suggested that a particle
may be represented by a wave packet [1], thou gh t he pa cke t had been generally supposed unstable [2]. T he pur-
pose of this paper is to outline what can be learned from the simplest of all packets. We start with Popper’s pro-
pensity theory for the wave function [3]; and we are now able to consider new consequences by analyzing the
stable wave packet: we find that it has uniq ue properties; that it is the decay product of unstable wave packets;
that it contains hidden variables; that its implementation has hidden consequences; that it gives a new perspec-
tive on chronic p roblems; a nd that it gives a clear a nd precise de rivation for the Uncer ta inty Principle.
A. J. Bourdillon
Schrödinger’s equation described a wave mechanical sol uti o n fo r t he Hamiltonian equation, where the sum of
kinetic and potential energies, H = T + V was conserved. The first and chief application for the equation is
atomic physics. Working at the same time, Heisenberg developed a representation based on commuting opera-
tors. In particular he showed that any operator that commuted with the Hamiltonian operator was a constant of
motion. T he Uncer tai nty Princ iple c a me to him in a blinding flash [1] and was adopted by Bohr. Many scientis ts
came to believe that “a problem had been previously resolved which they did not care too much about” [4] [5].
To seek the context for these developments, we have to refer back to Maxwell’s electromagnetism and to
Planck’s law that described quantization in the energy of the photon in black body radiation.
In Maxwell’s electromagnetism, the energy of a wave had depended on the intensities of its component elec-
tric field E and magnetic field M1. Planck’s subsequent discovery showed that its energy depended on its fre-
quency of oscillation. The wave packet shape and amplitude were intimately constrained and quantized. The-
reafter, the wave intensities could be treated as secondary to the quantized energy that determined them. The
further discovery of Bohr orbits in the atom showed that the de Broglie wavelength was likewise constrained.
Meanwhile, the wave amplitude, developed from photons to particles, became a normalized function expressing
uncertainty in position or in time. The extent of a wave packet, in time or space, was newly described by Hei-
senberg’s Uncertainty Principle. Analytical methods were adapted for bosons and fermions. Previously, mea-
surement had always contained a probabilistic foundation due to uncontrolled secondary aspects of measurement,
even when physical laws had been deterministic, [6], (i.e. each eve nt ha vi ng o n l y one p o ss ib l e o utc o me) ; b ut the
new mec ha nics mea s ure d ne w ki nd s of o b j ec t. T hey are vector states contained in complex Hilbert space, where
measured variables help determine superpositions of elements in those states. The new type of quantized mea-
surement, by hiding physical features evident in Maxwell’s electromagnetism, posed a new issue for determin-
ism, sometimes referred to as complementarity or as the wave-particle dua lity.
To address complementarity anew, in the light of the stable wavepacket, an initial statement is needed about
suppositions that have been thoroughly discussed in the literature and that are accepted for the present purpose:
1) Realism: the world existed before my short life started, will continue after it is ended, and will do so
whe t her I observe it or not.
2) Theory: scientists ar e free to invent a ny theor y that expla ins physical pheno mena or that simplifies a p rior
explanation. A th eory must be discarded when falsified [7].
3) Comple me ntarity was the term used by Bohr to describe the two aspects of particle behavior that is both
corpuscular and wave -like. The former corresponds to our wave group envelope; the latter to its carrier wave,
including interference and superposition. We shall see t hat in a n event, t he larger the unc ertaint y in mo ment um
component the shorter is the wave packet in space; while the larger the uncertainty in energy, the smaller is the
time taken by the wave packet to pass. T he stable wave packet adds precision by showing how mome ntum a nd
locality are complementary; as are energy and time.
4) Measurement: elimination of observer bias has always been a core concern in scientific measurement, and
is not ne w in qua ntum mec hanics. In particular it has always been the case that a measurement can alter proba-
ble outcomes in subsequent measurements. An example, in quantum theory, is the impossibility of measuring
simultaneously two observables represented by non-commuting variables. However measurement in quantum
mechanics was new because quantization required measurements of superpo sitions composed from elements of
state vectors. Measurement required identification of those state vectors and determination of the relative coeffi-
cients (occurrence) of their elements.
5) Statistica l interpretation : T he wave f unctio n, whic h is consi dere d bel ow in furthe r det ail, has been und ers-
tood in various ways. Among them are:
a) Random variations in measure ments of determined properties due to uncontrollable experimental features.
Typically re sults follow a normal distribution whose mean is the most probable value to b e measured. Thi s dis-
tribution implies hidden variables that are less significant than the quantity measured. For a sufficiently large
sample, the accuracy of the mean depends on the square root o f the number of measurements made, and this is
not represented by the Uncertainty Principle described below. T he Principle applies to single measure ments and
is derived, as below, from the wave function.
b) I n statistical mechanics, p robabilities ar e randomized, for example the number of atoms in a given volume
1In electromagnetism, E and M
are both real, but they can be represented by a single complex exponential function that expresses their rela-
tive phase relationships by separating mathematically real and imaginary parts. The
complex exponential conserves energy in the photon
over a peri od of osc illation, as it does in other free bosons and fermions, where the imagin ary components a r e not real.
A. J. Bourdillon
element, because practical measurements cannot give the locations and velocities of each atom. The statistical
inter p re ta tio n s up p os es t ha t w ith ful l k nowle dge , t he wave func t io n wou ld gi ve wa y to d et e rministic corp uscular
c) Even so, some values, including gas pressure, result directly from atomic collision due to randomized mo-
tion. They are therefore independent of knowledge deprivation.
6) Action at a distance: T hough the well known thou ght e xper iment d ue to E instei n, P odol ski and Rose n, [8]
[9], was described to de monstrate incomplete ness, it also treats localiz ation. Give n two pa rticles A and B, o rigi-
nally in the same state, can a measurement on A at a later time give instantaneous information abo ut the state of
B? If, at the time of measureme nt, the separa tion is su f ficiently large, the in formation would contradict a tenet of
special relativit y. Bohr argued against any information without measurement; Einstein believed the information
is localized. E xp eriments support localiz a tion [10]-[16], so does the stable wave packet.
7) Propensity is the theory for the wave function expounded by Popper [1]. It is agre ed tha t its Her mitia n pro -
duct gives the probability for finding a particle or photon. Wha t then i s the wave functi on? P opper compared it
to Aristotle’ s potential as a po wer to exist at a certain place or time. Pr opensity is the basis for the stable, mean-
valued, wave packet.
8) Completeness claims under the Copenhagen interpretation [17]—continued eve n after the d iscover y of t he
positron, the neutron, the neutrino, and clusters of bosons, hadrons, leptons, quarks, and now even the Higg’s
bosonare sure ly d ubious [1].
9) Quantum mechanics: Measurements determine coefficients for linear superpositions of elements of state
vectors in complex Hilbert space. A wave function describes those states, typically through the vector that de-
termines their coefficients. It can also be expressed in canonical variables by summation of elements multiplied
by their coe fficients. T he elements may be found as solutions to state equations as classical analogues, or as ob-
servables determined by commutation relations given by the rules of quantum mechanics. The elements are
quantized, eithe r through constraint s in herent in st anding waves, or throu gh ot her quantal p roper ties.
The stable wave packet that is discussed here is wave mechanical. It c ontains consta nts of motion. We b egin
by using the packet to understand Uncertainty. With the consequent knowledge, further anomalies lead to further
conclusions [18].
2. The Stable, Mean-Val ue d, Wa vepacket
Dirac wrote [2] that the wave packet is unstable, but failed to note that without the packet there is no wave. The
traveling wave group for a free particle or photon:
( )
, ,,exp2
kx tAX
φω σ
, with
( )
Xi kxt
= −
, (1)
defined o n val ues o f mean wave vector
and mea n a ngular fre quenc y
is a stable wave packet on a wide
trans verse wa ve. Those two variables are themselves stable not only because they are mean values in a symme-
tric wave function; but their stability is guaranteed by respective conservation of momentum and energy; and
triple guaranteed by symmetry in space-ti me. In one spatia l dimensio n for simplicit y, X is an imaginary variable
that cause s
to oscillate. If the real part represents the electric field in an electromagnetic wave, the imaginary
part represents the relative phase of its (real) magnetic force field. T he deno minato r
is particular because it
depends on initial conditions, but it is stable during propagation in free space as a consequence of Newton’s first
law of motio n. T he variable determines the width of the wave packet, in either space or time, and can be defined
as coherence. The normalizing amplitude A depends on
and has corresponding stability. The envelope,
2), d ep e nd s on the sq uar e o f ima gi nary X, itself a function of four variab les. T wo are already described,
so we are left with the variables x and t that describe the pro file. Since the other variables are all stable, this pro-
file is also stable. With a stable wave gro up, we can progress to new p hysics. The one dimensional equation is
extensible to four space-time dimensions by appropriate substitution of the scalar product kr for kx. Exceptional
about this wave function is t he fact that it is def ined b y mean val ues while imp lyi ng a co nt inuo us ra nge o f co m-
ponent energies and momenta that are accessible by Fourier transforms. Ba ck transfo rms on the Gaussia n wave
packet prove it stable. Moreover, beyond the plane wave approximation, edge effects are minimized as illu-
strated be low. In Equation (1) and hencefor t h we us e simpl if i ed units for t he reduced Planck constant and fo r the
speed of light
1c= =
. The stable wave packet is a solution to the wave equation [18].
A. J. Bourdillon
To express the variables
and k implied in the equation, find the Fourier transform F for the wave group.
Write the pr obability:
()( )( )
( )
P Ftt
ω φφ
= ⋅
ignor ing the constant normali zing amplitude A, and evaluat e:
( )
exp .
σ ωσ
= −
Then proceed to write the uncertainty in angular freque ncy and energy as double the root of the argument to
the exponential func tion in Equation (3), at the point where the p robab ility fall s to half maximum.
4 0.832
. (4)
The Fourier back transform of Equation (3) gives the expected
. Similarly
( )
= −
. (5)
Then, using the same criterion as for Equation (4),
2 0.832t
∆= ×
. (6)
Equations (4) and (6) yield
5.538 2π0.882t
∆⋅∆ ==⋅
and, tra ns l ating to energy E:
0.882Et h∆⋅∆ =
⋅∆ =∆
when the unit of energy is generalized by the reduced Planck constant. Equation (8) is
const ant wit hin t he o ne-di mensional stable wave packet described , and is more precise than Heisenberg’s value.
Similar expressions can be found for
etc., and for the momentum, component px:
px h
∆⋅∆ =
When the wave packet is not symmetric, the number may be larger, as in Heisenberg’s equation; but the wave
decays typically towards the stable packet as shown below in Section 3.3. The new der ivati on o f the U nc er ta i nt y
Principle in the free p a rticle makes it a consequence of more fundame ntal wave mec hanics. The derivati on i s not
only well defined; it is also mathematically precise to any number of figures. Beyond the free particle, bound
states are quantized by standing wave interference as in the Bohr atom, the Schrödinger equation, wave me-
chanics etc. Those are the states that quantize the photon. T he precise result and clear definitions in Equations (8)
and (9) raise several questions that we will come to after some minor illustrative observations. For the stable
wave packet, the equality sign holds where Heisenberg’s Principle has a limiting valuation. The well-defined
product for the unidirectional wave is about 6 times greater than values given by Heisenberg for three dimen-
sional systems.
In the simple one-dimension case, normalization gives
xx x
, leading, in three dimensions, to
where i = x, y, z. Excepting the plane wave,
i is often three dimensional. A depends on
, since
φφ σ
, so that, in one dimension,
12 14
. In Bohr’s quantum mechanics A and
suppressed together because they depend on the single quantized energy
as they do on
, through X. The
is in principle observable and can be measured. It depends on initial conditions which may not
for m the Gaus sian. If it were quantized, a s by an initial emitting ato m,
could not decay and the packet would
be perfectly stable. Experimental data described below in Section 3.3 show ensembles decaying.
Meanwhile the expected value for the packet energy is derived from the gradient with respect to time,
( )
( )
1it Xi
φσ ωφ
−∂∂ =+−
. Since X|x=0 is a first ord er function of time, t he integral on this sym metric group
A. J. Bourdillon
envelo pe va nishes l eavin g
. Similarly
, as expected. Operations on the Gaussian are compara-
tively easy.
The ideal stable wave packet is unidirectional, with a wide tran sverse field. However the theoretically stable
Gaus sian wave p a cke t i s no t t he o nl y wa ve p ac ke t tha t is p o s sib le . Initial c onditio ns de ter mine a p artic ular wave
packet, but many examples in Fresnel diffraction [19] show that unstable packets rapidly lose their shapes be-
cause of edge effects. T hese cause decay to the relatively stable Gaussian, as will be discussed further in Section
3.3. When t he Gaussia n wave pa cket bro adens in e xtent b y an incr ease i n
, its momentum becomes more cer-
tain according to the equality sign. This may not be the case in unstable wave packets that supposedly hold to
Heisenberg’s limitation. The transverse momentum becomes uncertain when a packet passes through a trans-
verse restriction in its path such as a slit.
Notice moreover that Equations (2)-(9) are consistent with t he applica tion of arbitrar y realism. Suppose that a
const ant phase
is added to Equation (1) by initial conditions:
( )
, ,,exp2
kxtAX i
φω σ
= ⋅++Φ
. (10)
The phase
expresses the relative displacement of phase in the carrier wave from the peak amplitude of
the gro up (Figure 1). In Equation (2) it disappears, but might have an effect elsewhere as an example of a hid-
den variable. In Fresnel diffraction through a slit, the phase is known to lag [19]. In quantum mechanics, the
phase is often included as a complex constant ci that includes an exponential function, exp(iΦ), for example in
the linear superposition of states A and B:
RcA cB= +
. We leave for another time the superposition of
two states having different coherence.
These minor observations on the stable wave packet raise so me general questions. The f undamental p ostulates
of quantum mechanics are not affected: the linear superposition of states in complex Hi lbert space s; quantizat ion
of standing waves; commutation relations between observables and operators; measurement of states that are
superposed but independent in orthogonal sets, etc. The questions we raise are as follows:
1) The Uncertainty Principle lacks precision and definition. Can it be improved through the example of the
simple and stab le wave packet, and through the application of wave mechanics?
2) The packet introduces new variables. What new information do they provide?
3) Can the ne w information al te r probab ility measurements to make them more deterministic?
4) How, physically, are free particles quantized?
3. Probability vs Det erminism
Any deterministic system containi ng an unknown hidden varia ble will appear probabi listic. Neverthel ess, though
Figure 1. Real part of phase carrier wave (b lack), retarded by phase Φ = π/4 with re-
spect to its wave group (red). The group velocity is vg; the carrier veloci ty vp = c2/vg.
retard pi / 4
-5-4 -3-2 -101 23 4 5
A. J. Bourdillon
probability theory has always held a part in measurement (Section I.5a above), the method of physics from ear-
liest times has bee n to isolate sin gle variables and measure them as determined quantities. This req uires eli mina-
tion of hidden variables wherever they might be significant and where they can be eliminated. Mean while, a suf-
ficient condition for a new variable to ensure determinism is that there is one-to-one correspondence between
each value of the variable and each experimental outcome. This c orrespondence is still lacking.
3.1. Probabilism in the Transverse Plane
The stable wave packet that has been discussed so far is defined in the direction of propagation. T hen the trans-
verse plane is unrestricted. There are important features in both cases: unrestricted and restricted.
3.1.1. The Unrestricted Transverse Plane
In the direction of propagation, motion is subject to all the restrictions of relativity, for example, in particle ki-
netics with gr oup veloci ty vg < c; in electro dynamics; in solution s to the relativistic Klei n-Gordon equation, etc.
Operation of the last equation on Equation (1) yields, as output for a particle in free space without magnetic
fields, an algebraic equation in second order:
2 222224
kc mc
= +
. (11)
This is the same as is obtained from Einstein’s well known relativistic formula by substituting for energy
and for momentum
. The equation can be simplified with appropriate units
1c= =
. Diffe-
rentiation then gives a new result in re lativity, for the product of group velocit y2 vg and p hase velo city vp:
. (12)
In the l ine of prop agation the group velocity is regular and relativistic; on the transverse p lane, the group ve-
locity approximately zero, and the carrier phase velocity, vp = c2/vg, tends to infinity. The phase velocity de-
scri bes wa ve mot ion be yond the e vent ho rizo n of c. The speed of light functions as the Schwarzschild radius in
cosmology, because the faster carrier wave is elastic, does not transport energy, and is not measured in special
relativity. Though not directly measured in the conventional way, we know what it is: the inverse of the group
velocity which is observable, and also the ratio of energy to momentum, each of which is an observable constant
of motion in the free par ticle. When phase velocity beyond c is accepted, multiple consequences follow. Its ref-
erence frame is the same as applies to the group velocity as described in special relativity for real space.
Firstly, for the plane wave, time in the transverse plane is Newtonian. Secondly, phase velocity provides a
new concept for probability or propensity: propensity results from the instantaneous addition of wave ampli-
tudes in the transverse plane. This explanation overcomes problems in a supra-relativistic spread of phase in-
formation when, at measurement, the wave packet is reduced: an example is the applicati on of Fresnel i ntegrals
in diffraction; another is Bohr’s concept of wholeness, which receives new life, though only within the cohe-
rence of the wave group. Thirdly, for antiparticles, a ne w set of dynamics is found as solutions to fundamental
equa tions. The set is intro duced through p rior anomali es. Since the stable wave packet has been overlooked for
so long, there should be no surpr ises:
a) Dir ac did not think his calculation for the velocit y of the electr on in relativistic theor y was ano malous. His
application of Heisenberg’s commutator formalism gives a value, for its velocit y equal to c [20], which is ind eed
a constant of the motion. This is the geometric mean of the group and phase velocities and represents an instan-
taneous wobble. However this observation of the geometric mean becomes acceptable only after space beyond
the c-horizon is acknowled ged. The true speed, as measured in vacuum tubes, is given, as a classical analogue,
by the group velocity of the stable wave packet; the wobble is a theoretic curiosity. No tice that part of his wob-
ble must be faster than c.
b) More seriously, Dirac discovered the negat ive ener gy solution for a ntiparticles b ut insi sted that the mass is
positive. The anomaly is arithmetical: at rest, a positive m0c2 cannot equal a negative E. His analogy of hole
states in semiconductor valence bands makes proble ms for g ravitation that we attempt to a void.
c) Likewise he insisted that the kinetic energy of the antiparticle is positive. This later raised a problem for
Feynman and Stueckelberg, whose principle states, “An antiparticle traveling forwarding time doesn’t exist”
are st able, th e differen tia l
has no value, but dω/dk follows the relativistic constraint in Equation (11) and d
term in es the group velocity.
A. J. Bourdillon
[21]-[23]3. However, another consequence occurs for the c-horizon: as k m0,
. The
former result contradicts a tenet of relativity; neither result has been reported in experiments. These anomalies
all disappe a r when, for the ant iparticle, both mo and k are negative.
Further consequences occur in other areas of physics: en tangle ment at a distance (since the wave packet is lo-
calized by its coherence); the wave function of a particle-antiparticle pair and their possible application to su-
per conduc tivity, etc. However a more fundamental concern is the core of the argument for probabilism in quan-
tum mechanics. This is normally illustrated by diffraction phenomena [1], so we shall consider what effect a
new hid d en va ri ab le migh t have o n tho se p henome na. Moreover diffraction theory has significant details that we
shall also consider.
3.1.2. Diffraction in the Restricted Transverse Plane
Following Newton, light was rectilinear, and more strictly now, relativistically geodesic, though locally Carte-
sian. This propagation provides the deterministic method of ray tracing that is applied, by approximation, in both
light optics and in electr on optics. T he light bends sharply at a dielectric interface. When lig ht passes thr ough a
narrow slit it diffracts so that some rays are observed to bend outside the aperture provided by the slit. T he p he-
nomena were consistently analyzed by Fresnel, using wave theory [19]. In particles, the same diffraction occurs,
where the phenomena are analyzed in the same way after adjusting for corresponding wavelengths. The phe no-
meno n i s commo n l y used to demonstrate probabilism in physics. Be yond the near field, th e illuminatio n on a far
screen will show an intensity distribution described, at least approximately, by a Gaussian. The bell shaped
curve that may be measured is a probability distribution due to the product of the wave function with its com-
plex c onjugate at a gi ven dif fr action pl ane. I t represents scattering of an ensemble. S uppose the i ntensit y of the
light bea m is so reduced that light stri kes the slit, o ne photo n at a time. I n the mec ha nics of Sc hrö dinge r o r He i-
senberg or Fresnel, there is no way of predicting beforehand, what path the light will take, nor where it will
strike the screen on a selected diffraction plane.
3.2. Flexible Ray Tracing in Fresnel Diffraction
The straight line ray tracing that represents rectilinear theory must be made flexible to accommodate the wave
theory. A method can be adapted to the Fresnel diffraction simulated in Figure 2. This sho ws scatter ing due to
the emergence of a plane wave from a narrow horizontal slit . A cut of the calculated intensity on the plane at the
Critical Cond ition [24] [25] is shown in t he inse t. At this condition4 the slit is o ptimally foc used by Fresnel di f-
fraction for high re s olution proximi ty p rinting. There are two p o ints to notice about these si mulations. Firstl y the
progression o f the diffractio n i s illustra ted succe ssi vely fro m plane to plane. Knowing t ha t int e nsity is c o n ser ved ,
it is possib le to co nstr uct fle xing r a ys that follo w the li ght tr ans mission fro m the sl it to a g iven dif fractio n p lane.
Secondly, we ma y ask a gain a questi on about the ray path t ake n by a part icular photon. Suppose that the incident
beam intensity is red uced until light is emitted a nd absorbed one photon at a time. When the photon is detected
at a point on the diffraction plane, can we say which ray it followed, and can we predict the ray path in the future?
The answer has to be, “No” to both questions for the following reason. The exp erimenta lly verified [26] proba-
bility profile s hown in the inset is ca lculated fro m all ra ys emergi ng fro m the slit; no t just one o f them. The d e-
termination of where, on the diffraction plane, the photon is detected is not determined by the variables known
in the wave propagation; but supposedly includes both emitter and absorber states and phases. Inclusion of all
variables is, for sound experimental reasons, bypassed i n t he route taken by probabilism.
Notice the example that is given in the figure, of the decay of the transverse rectangular wave packet from the
slit, to a Gaussian in far field . The decay to the Gaussian, owing to edge effects, is typica l in Fresnel diffractio n.
The eventual divergence is itself an example of the Uncertainty Principle in the unstable wave packet in two
spatial dimensions. A minimal prediction due to the Heisenberg Uncertainty Principle is represented by the
dashed line in Figure 2. Before the slit, the horizontal momentum py
0. At the slit, py depends on the Prin-
pp w=∆≥
. So the minimal uncertainty scattering angle:
() ( )()
qpp whw
= ==
where px is given by the de Broglie wavelength. The dashed li ne in the figure shows the prediction for increasing
Uncertainty displacement, with downward passage, owing to the required minimal transverse momentum.
Anyone can see, in bubble chamber photographs [23], that antiparticles travel forward in time, but for an illustration of interactions see
[18] .
4In proximity X-ray Lithography, the condition is governed by wavelength, mask feature size and mask-
wafer gap. That scattering amplitude
is greatest for which the vec tor join s the extremities of Cornu’s spiral.
A. J. Bourdillon
Figure 2 . Simulated Fresn el pattern due t o X-rays of wa velength λ = 0.8 nm, focused
from a slit of width w = 150 nm in cross-section. The inset at right shows the intensity
profile at the Critical Gap Gc = w2/3λ [24]-[ 26], where the intensity of the incident
plane wave was 2. The abscissae are to scale and the slit width is 2.4 dimensionless
unit s (see Ref. [ 19]). The dashed line includes, according to Heisenberg’s uncertainty
Principle, the effect on beam spread of the minimal transverse momentum introduced
by the slit. The transverse momentum is greater in Fresnel diffraction where it is first
convergent, and divergent after focus. Fresn el’s method is more informative (see t ext ).
Thi s makes a n inter estin g compar ison with the expe riment all y verifie d Fres nel pa ttern s hown i n the Figure 2
in gra ysca l e. At first sight, it seems—since the Fresnel diffraction focuses the slit so that the beam shrinks to one
third of its width at the slit—t hat the Uncertainty Principle is defied5. Ho wever closer inspection shows that the
conflict can be re-construed. When the Fresnel image shrinks, the divergence angle increases to roughly three
times the angle of the minimum uncertainty scattering. Then there is a crossover at focus and the beam passes
from convergence to divergence, again at an angle greater than the minimum that transverse momentum requires.
Beyond 60 µm from the slit, the dashed line crosses the edge of the diverging Fresnel bea m, so that diffraction
continues to scatter with a transverse momentum greater than the minimum allowed by the Principle. T he Fres-
nel beam in fact has an uncertainty closer to
p phw=∆=−
(adjusting conveniently 0.881 1 for the
rectangular transverse wave packet), consistent with the image width y w/3. This illu strates an uncertain ty in
the definition of “position” in the Principle. According to the experimental data, the uncertainty of position
could be roughly defined at the point of focus; not by the slit width! In summary: near field d iffraction shows a
beam width less than expected from Uncertainty based on the slit width. This is inconsistent with the limiting
Principle; far field diffraction shows a momentum uncertainty that is larger than the slit width dictates, consis-
tent with the limiting Principle—but systematically underestimated.
Moreover the focusing due to a top hat wave packet is yet more pronounced [19] [27]: the increased focusing
induces increased uncertainty in the plane normal to the direction of propagation. T he Principle therefore unde-
restimates momentum uncertainty in the transverse plane due to diffraction from in-plane apertures, when fo-
cusing typically occurs. Of course, the image in Figure 2 does not represent a true stable wave packet because
the transverse plane is restricted. This wave is more stable along the line of propagation, at least where λ < w.
5The convergent beam suggests that p
< 0 ; or that Δp
= −Δp
; or that the limiting sign in the Principle is reversible; or that the Principle
lacks of definition.
A. J. Bourdillon
4. Discussion
The purpose of this discussion section is to provide perspective; not to repeat arguments already made. We set
out originally to find the best wa ve function that would si mulate the Uncerta inty Principle . T his t urned out t o be
symmetric and Gaussian, so this was built into the wave function for a unidirectional plane wave. Because of
minimal edge effects it was stable in the direction of propagation and approximately stable in the transverse
The corresponding uncerta inty is more precise than Heisenb erg’s Principle which in fact co nflicts with estab-
lished data. His Principle is confusingly and improperly defined. It needs supporting detail from wave mechan-
Part of the detail is surpr ising: the carrier wave exists elastically in space beyond the speed of light, but with-
out transporting energy. It provides an explanation for the reduction of the wave packet during quantum me-
chanic a l mea s ureme nt.
There are other surprising consequences. Antiparticles occur as solutions to equations as they did for Dirac.
The wave packet suggests conflicts in his solution and finds other wave mechanical solutions for particles and
Such a fundamental concept invites a new examination of modern science. After Newton, science was con-
structed as a deterministic theory; but modern quantum theory is imbedded with wave mechanics when in mea-
surements on single particles, outcomes are typically unpredictable, as in diffraction. These outcomes are un-
derstood through probabilities and propensities. There are hidden variables [28] in the stable wave packet, but
they are not suf ficient to change the prob a ble nature in i ndi vid ual measurement.
The uncertainty in a unidirectional wave packet is larger than in a two dimensional system such as a Bohr
atom where the path is longer. The difference is a factor representing the ratio diameter/circumference, so that
p xh
∆⋅∆ ≥
. In three dimensions, the apparent factor is again smaller by a factor about
. The combina-
tion is usuall y written
p xh∆⋅∆ ≥
5. Conclusions
Heisenberg’s Uncertainty Principle conflicts with established experimental data and needs support from wave
mechanics. This comes from the stable wave packet that is the principle vehicle for unidirectional photons or
free particles with wide, planar, wave fronts. It is Ga ussia n, with mi nimal ed ge effects, a nd is t he decay product
of unstable packets. It is also well behaved, being differentiable and integrable; moreover it can be Fourier
transformed and used with quantum operators. From this packet is derived a precisely defined uncertainty prin-
ciple. The definition is fundamental and ideal; rather than over-simple and semi-empir ic a l—perhaps even un-
exa mined.
The packet differentiates, in wave mechanics, the transverse wave from its direction of propagation. In the
transverse plane where the group velocity tends to zero, the phase velocity tends to infinity. Phase velocities
beyond the horizon of the speed of light are used to explain the reduction of the wave packet for the case of
Fresnel diffractio n of a single photon. The packet thus contains hidden variables, but these are not sufficient to
completely determine the reduction of the wave packet on detection of a single photon in a diffraction plane. In
consequence, the site at which a single photon is absorbed must be treated with conventional probabilism. It re-
mai n s in principle possible, by identifying hidden variables and excluding them experimentally, to recover de-
terminism in physics. Most likely they would include phase infor mation in both emitter and absorber. Whether
or not determinism will ever be achieved, many physical and kinetic consequences follow the stable wave pack-
et. Every phot on t hat is ever measured is packaged [29].
[1] Popper, K.R. (1982) Quantum Theory and the Schism in Phys ic s . Hutchison, 18.
[2] Dirac, P.A.M. (1958) The Principles of Quan tum Mechan ics. 4th Edition, OUP, Oxford, 121.
[3] Popper, K.R. (1982) Quantum Theory and the Schism in Physics. Hutchison, 22.
[4] Popper, K.R. (1982) Quantum Theory and the Schism in Ph ys ic s . Hutchison, 10.
[5] Gel l -Mann, M. (1979) In: Huff, D. and P rewett, O., Eds., The Nature of the Physical World, 197 6 Nob el Conference,
A. J. Bourdillon
New Yo rk, 29.
[6] Bohm, D. and Bub b, J. (1966) R eviews of Modern Physics , 38, 453-475.
[7] Popper, K.R. (1959) The Logic of Scientific Discovery. Hutchinson.
[8] Einstein, A., Podolski, B. and Rosen , N. (1935 ) Physical Review, 47, 777-780.
[9] Popper, K.R. (1982) Quantum Theory and the Schism in Ph ys i c s . Hutchison, 15-22.
[10] Aspect, A. (2007) Nature, 446, 866-867.
[11] Aspect, A. , Grangier, P. and Roger, G. (1982) Physical Review Letters, 49, 91-94.
[12] Aspect, A. , Dalibard, J. and Roger, G. (1982) Physical Review Letters, 49, 1804-1807.
[13] Longdell, J. (2011) Nature, 469, 475-476.
[14] Clausen, C., Usmani, I., Bussieres, F., Sangouard, N., Afzelius, M., de Riedmatten, H. and Gisin, N. (2011 ) Nature,
469, 508-511.
[15] Saglamurek, E., Sin clair, N., Jin, J., Slater, J. A. , Oblak, D., Bussieres, F., George, M., Ricken, R., Sohler, W. and Tittle,
W. (2011) Nature, 469, 512-515.
[16] Faipour, A., Xing, X. and Steinberg, A. (2011) Physical Revi ew Letters, 107, Article ID: 133603.
[17] V on Neumann, J. (1932) Die Mathematischen Grundlagen der Quantenmechqanik.
[18] Bo urdillon, A.J. (2015) Journal of Modern Physics, 6, 463-471.
[19] Jenkins, F.A. and White, H.E. (1981) Fundamentals of Optics. International Edition, McGraww-Hill, New York, 390.
[20] Dirac, P.A.M. (1958) The Principles of Quantum Mechanics. 4th Edition, Oxford University Press, Oxford, 261.
[21] Feynman, R.P. (1949) Physical Review, 76, 749-769.
[22] Stueckelberg, E.C.G. (1941) Helvetica Physi ca Acta, 14, 588-594.
[23] Littlefield, T.A. and Thorley, N. (1963) Atomic and Nuclear Physics and Introduction. Van Nostrand, 234.
[24] Bourdillon, A .J. and Vladim irsky, Y. (2006) X-Ray Lithography on the Sweet Spot.
[25] Bo urdillon, A.J. , Boothroyd, C.B., Kong, J.R. and Vladimirsky, Y. (2000) Journal of Physics D: Applied Physics, 33,
1-9. /17/ 307
[26] Jenkins, F.A. and White, H.E. (1981) Fundamentals of Optics. International Edition, McGraww-Hill, New York, 399.
[27] Bo urdillon, A.J. and Boothroyd, C.B. (2005) Journal of Ph y s ics D: Applied Physics, 38, 2947-2951.
[28] Bell, J.S. (1966) Reviews of Mod ern Physics, 38, 447-452.
[29] B ourdillon, A.J. Uncertainty and th e Stabl e Wave Packet.