Absurdities in Modern Physics:
3 - The Subjectivity of Heisenberg's
uses a relation that is interpreted as being a fundamental limit of
in physics. It is called the uncertainty relationship. This uncertainty
principle is so consequential in physics that Cramer [3.1]
important to study the reliability of this interpretation that is
to be so fundamental in physics.
"The uncertainty principle of Heisenberg is
of the most important aspects of the Copenhagen interpretation. It is
an interpretational aspect of quantum mechanics."
relationship can be written in different ways. Let us begin with the
as written by Heisenberg [3.2]:
DE = resolution in energy, Dt =
resolution in time, h = Planck constant.
let us first state the argument given by Heisenberg [3.2],
limit of resolution. The starting point is the one that
characterizes any monochromatic wave packet of frequency n.
A continuous wave is cut up into pieces by a time shutter, forming
of duration Dt. Using the Fourier
Heisenberg has observed that the spectrum of a wave packet, as given by
a spectroscope, is limited in resolution. A wave packet limited by a
interval Dt shows a wider bandwidth DE
when Dt is smaller, as given by Eq. 3.1.
equation, resulting from the use of the Planck relationship E = hn,
led Heisenberg to the uncertainty principle.
hypothesized that the wave packet described represents one single
His mathematical analysis however is solely based on the description of
a wave phenomenon. There is nothing in the wave packet model that
that it is valid only for a unique small amount of energy. It appears
to claim that a wave packet is the description of a single photon when
the results are equally applied to any number of photons. If the
uncertainty relationship characterizes only individual photons, we
expect to obtain a different resolution at higher intensity. We know
the uncertainty relationship is used to calculate the limit of
of telescopes and microscopes. However, the resolution of telescopes
microscopes is totally independent of the intensity of the signal
Even the gedanken "Heisenberg microscope" experiment uses the classical
resolution in optics Dx = 2lsin(q)
which is certainly independent of the intensity. Consequently, the
model is valid for any amplitude of the wave packet.
transformations done by Heisenberg are certainly correct. Eq. 3.1 is a
result that is a consequence of the Fourier transform of the wave
However, Heisenberg gives a physical interpretation of the result. He
that the limit of resolution predicted by the mathematical
corresponds to a physical limit. We must therefore examine whether the
limit imposed by Eq. 3.1 has physical implications or is simply a limit
of resolution of one popular particular instrument, namely the
3-2 Heisenberg's Wave
the detailed shape of a wave packet as seen on Fig. 3-A, is not
by a spectroscope. The spectroscope reveals only a fraction of the
information. It gives only the amplitude of each frequency component.
spectroscope cannot record any information about the phase
between each frequency.
aspect of the problem is illustrated in the following way. In the case
of Heisenberg's model, we consider an initial wave with an unlimited
time. Within a wave packet, coherence is maintained at one frequency as
long as the wave maintains its phase relationship. The amplitude of the
signal varies according to a time-dependent sine function, as
on Fig 3-A.
calculations are applied to the wave packet formed when the unlimited
wave is cut up by a time shutter, between t1 and t2
as seen in Fig. 3-A. Let us examine that wave packet in more detail.
there are many ways to take measurements of a wave packet but at the
Heisenberg developed his model, only one method was used. It was
spectroscopy. With the use of spectroscopy, one can easily obtain the
about the frequency and the bandwidth of the signal as illustrated on
3-B. However, a spectroscope does not reveal all the existing
contained in a wave.
We know that
different waveforms of wave packets can, when observed with a
lead to exactly the same spectrum. This can be illustrated by a wave
made from a wave with increasing amplitude or from a wave packet with
amplitude. These two different wave packets can present spectra that
in frequency and bandwidth even if the initial wave packets have
one can show that an infinite number of different wave packets can give
the same spectrum (i.e. same amplitude in the Fourier spectrum). Since
there is less information in the spectrum, it is clear that during the
conversion from the wave packet to the spectrum observed in
some information is lost. Even if many different waveforms can lead to
exactly the same spectrum, only one spectrum can result from one wave
Because the exact shape of the original waveform cannot be known from
knowledge of its spectrum, one must conclude that spectroscopy is not
best method for studying wave packets.
It is well
known that a spectroscope is an instrument giving the total amplitude
the Fourier transform. The phase relationships between the frequency
are not measured by a spectroscope. Consequently, the spectroscope is
experimentally the same amount of information as when neglecting the
in the Fourier transform. The real as well as the imaginary parts of
Fourier transform are necessary to reconstitute the information about
phase. Therefore the spectroscope is a useful but limited method of
wave packets, since much information is necessarily lost.
giving incomplete information are very common in physics. If we
a signal by the description of its derivative, we clearly lose the
constant. Another example occurs when we consider the energy given up
a wall that absorbs nitrogen molecules (from air). If we can make a
measurement of the energy absorbed by the wall, following the
of a nitrogen molecule, we cannot deduce what the velocity of the
molecule was before it hit the wall. The reason is that we leave out of
account what the rotational and vibrational energy of the molecules
before they hit the wall. The method of measurement of energy
to the wall gives an incomplete account of the kinetic energy of
3-3 Static Analysis.
the following example, illustrated on Fig 3-C. Let us generate a sine
at frequency f1, having amplitude A1,
during one cycle and a quarter (between 0 and 450 degrees).
to a continuous wave packet, with a cut such that only one and a
cycles are saved. Such a description is perfectly clear and complete,
anybody can reproduce such a wave packet from that description without
In order to
study the wave packet, let us consider taking numerous samplings of the
amplitude at different times of the signal presented on Figure 3-C.
these data the original waveform can be faithfully reproduced.
the exact original frequency can be calculated by fitting (mathematical
fitting by computer) the observed data between t1
and t2 with a sine wave. That
fitting must be able to take into account three parameters:
the frequency, b) the phase and g)
the amplitude. The fitting is made on the original undisturbed wave and
not on the sharp cut induced by the experimenter. It is that sharp cut
that widens the spectral lines in spectroscopy.
of such a fitting leads to the exact frequency of the wave packet and
accuracy achieved can be as good as noise and technology will permit.
fitting will not give a wide bandwidth as it does in the case of the
transform. Using the frequency, the phase and the amplitude obtained by
the fitting, the wave packet can be reproduced without Heisenberg's
analysis of the wave packet gives an incomplete description. It is
impossible to reconstitute the same initial wave packet from the
spectrum. There is no way to recover the lost phase. The only way not
lose information is by not using the spectroscope.
nothing fundamental in physics about losing information when that loss
of information is solely the consequence of the use of a mathematical
- the Fourier transform. Giving a fundamental meaning (like the
principle) to the limitation of the Fourier transform is like inventing
another fundamental principle in physics to describe the information
based on losing the integration constant when taking the derivative of
a signal. There is no physics involved, only properties of mathematics.
We will now study another method that, contrary to the use of the
transform, will not lose the information (the phase) contained in the
3-4 Characteristics of
Phase-Sensitive Frequency Meters.
the static analysis mentioned in section 3-3, to explain basic
fast dynamic frequency meters using phase-sensitive detection have been
developed in different frequency ranges. For example, an instrument
an automatic wave meter [3.3], uses
detection to measure accurate frequencies during short periods of
The instrument uses the fringe-counting Michelson
interferometer technique and can count fringes while taking phases
into account. The instrument typically counts electronically one
fringes in one second and can distinguish 1/50th
of a fringe. Consequently, the instrument has an accuracy of 2×10-8
while counting only 106 fringes. This
(±0.02 fringe) is superior to the one set by the uncertainty
The uncertainty principle allows only the detection of 1/2p
of a fringe.
involves the use of phased-locked loops [3.4].
the principle involved, we give here a description of the more
phased-locked loops method to measure frequencies. Phased-locked loops
circuits are commonly used in all color TVs, to adjust the phase
the color on every sweep of the electronic beam on the TV screen.
3-5 Basic Principles
in Phase-Locked Detectors.
involved in phase-locked detectors corresponds to the following
procedure. Let us consider the wave packet described in Fig. 3-A. As
as the electromagnetic signal enters the system at time t1,
the incoming wave is detected and coupled with a reference signal
at about the same frequency. When the leading part of the wave-packet
at the detector, the phase is, of course, completely random with
to the reference local oscillator. However, during the first few cycles
of the wave-packet, an active electronic circuit changes the phase of
local oscillator until it matches the phase and the frequency of the
wave packet. The electronic circuit maintains the matching of the phase
until the wave packet ends at t2. The
of cycles starts automatically, at the first zero crossing, after time
t1, when the phase lock action
the two signals has been achieved. Then the phase difference between
wave packet and the local oscillator is about zero. The counting
is typically one million fringes. The counting stops, just before time
t2 at the moment another signal
the coming end of the wave packet (through a delay line technique). The
electronic control stops the counting, at the moment that the last zero
phase is crossed, just before t2. An
clock can measure very accurately the time elapsed during the integer
there are a few cycles that are wasted during the initial period of
phase adjustment but this is negligible compared with the total of one
million cycles usually counted. That knowledge of the number of cycles,
with an accuracy of ±0.02 cycle, provides much more accurate
about the wave parameters than can be detected by a spectroscope,
the phase relationship is now known and the detector is not handicapped
by a simple spectroscope. One of those instruments [3.3]
give reproducible results within ±0.02 cycle. This is smaller
1/2p, contrary the Heisenberg uncertainty
It is smaller than 1/15p.
3-6 Confusion between
Instrumentation and a Basic Phenomenon.
was misled because he assumed that the only way to measure a wave
was by measuring its spectrum. He even dared to claim that no other
than that obtained with a spectroscope can even exist. Heisenberg's
is compatible with the Berkeley-Copenhagen interpretation on
According to the Berkeley-Copenhagen interpretation, the wave packet
the photon) is created at the collapse of the wave function and does
exist independently of the observer. Heisenberg never tried to measure
nor suggested measuring the frequency by using a more sophisticated
than the spectroscope.
suggested by Heisenberg can now be interpreted either as an
limit due to the particular use of a spectroscope or as a mathematical
limit due to the information lost while using the Fourier transform
the imaginary part. In the case of a phase-sensitive detector, there is
no fundamental limit. With a phase-sensitive detector, the limit of
depends only on the limit due to noise. It is unfortunate that
could not distinguish between the loss of information given by a poor
and a fundamental principle.
3-7 Hidden Criteria.
We have seen
that the Heisenberg uncertainty relationship is not fundamental in
We will now see that the amount of information lost, described as h/2p,
is not absolute but quite arbitrary. It depends on the units chosen.
Let us go
back to Heisenberg's model to describe the wave packet, as illustrated
on Fig. 3-A. We will show how one can obtain three different limits of
resolution, using a method compatible with the one used by Heisenberg.
The results will be discussed afterwards.
3-A, let us consider that we are counting the number of cycles, in the
wave packet starting at time t1. Let
assume that the phase is zero when we start counting. When the
time Dt has elapsed, we have counted x
A fraction of a cycle cannot be counted. However, after counting the
cycle, an unknown fraction of another cycle has started before the end
of the time interval Dt. Therefore the
number of cycles is a fraction between number x and x+1. Then, the
possible frequency is:
The error in
due to the random phase at cutting time t2
sides by h and using the Planck relation E = hn,
means the unit cycle, multiplied by the units of h.
a form of uncertainty when cycles are counted. Before concluding, let
examine the case of radians.
described, instead of counting cycles, the detector could monitor the
of the incoming wave, counting an integer number of radians. The
are recorded and counted until the time Dt
elapsed. It is found that y radians are counted at the end of the time
interval Dt. Again, after the last radian
been counted, an unknown fraction of a radian has started before the
of Dt. Therefore, the number of radians in
wave packet is between number y and y+1.
cycle equals 2p radians, the maximum
The error in
due to the random phase at the cutting time t2is:
sides by h and using the Planck relation E = hn,
might appear repetitive, let us finally examine what happens in the
of counting individual degrees. Using the same method, we can find the
maximum possible frequency when we count z degrees during the time
The error in
due to the random phase at the cutting time t2 is:
sides by h and using Planck relation E = hn,
we find that:
equations have been obtained from that method. They are Eq. 3.6, 3.11
3.16. How can they be different? Since the equations are different, one
must search for a hidden parameter. The difference between these three
equations comes from that we have arbitrarily set up an
error of respectively one cycle, one radian and one degree when
to the resolution, the method of counting an integer number of radians
is equivalent to the Fourier transform. The Fourier transform however
supplementary information because one can get the shape of the
while the method of counting the integer number of radians (or any
angle) gives an average value of DE. Other
such as the lock-in amplifier can show the frequency distribution
Of course, the mathematical distribution given by the Fourier transform
has an absolute shape (Fig. 3-B). It is the absolute shape of the
given by a spectroscope.
resolution is mathematically identical to the result obtained when we
counting an integer number of radians (Eq. 3.11), we must conclude that
the resolution of one radian corresponds to Heisenberg's criteria of
This resolution of one radian is exactly equal to the loss of
that is intrinsic to the use the Fourier transform when the phase is
In other words, Heisenberg's criteria of resolution has been set to be
mathematically equal to the loss of information when we use a
It is certainly arbitrary in physics to decide that the exact
of information loss due to the use of a particular instrument (a
should become a fundamental principle. Furthermore, we know
such an information loss does not exist when we use a different
method like the phase-lock detector.
of 1 radian, 1/2p cycle or 57.3 degrees,
by Heisenberg is certainly not an absolute or physical criterion. He
have decided to count cycles (or any other unit). In that case, the
fundamental uncertainty relationship would be different as shown in
3.6 and 3.16 In fact, it was neither rational nor coherent for
to use the unit radian since in the same equation, he uses E = hn,
n is expressed in cycles and not in
interpretation is, the relation should be written:
where k is
arbitrary criterion of resolution corresponding to many cycles, one
or a fraction of a cycle that the detector can resolve. The units of k
are cycles. Heisenberg (consciously or not) assumed that k = 1/2p
to fit the resolution of a spectroscope. The units of k are compatible
with the dimensional analysis of E = hn that
shows that the units of h are equal to Js/cycle. Using a
detector, the coefficient k can be experimentally as small as permitted
by technology. In fact, there is no fundamental limit.
relationship claimed is uncertain for another point of view.
scientists like Schiff, [3.5] state
that the relation
is good "within an order of magnitude". Furthermore, Schiff
the equation as an approximation using the symbol ~ giving:
Van Name [3.6] gives a different
amount of uncertainty
using the relation:
It is clear
all those values correspond to a rough estimate of a practical limit
using the Fourier transform or a spectroscope to analyze data. Clearly,
the uncertainty relationship has no fundamental meaning.
3-8 Irrelevance of the
Constant h to Photon Energy.
that following Eq. 3.1, the Planck constant determines the amount of
of energy of the photon. On the contrary, one can show that the
is not a function of the Planck constant. Let us divide both sides of
3.1 by h and let us substitute E by its value hn.
a division on both sides of the equation does not
the nature of the equation. This gives:
3.21 show that the uncertainty relationship does not change at all even
if the Planck constant h has a different value. We must conclude from
above that the Planck constant is fundamentally irrelevant to the
relationship, even if we are accustomed to believing that h appears to
determine the amount of uncertainty.
3-9 Hidden Units.
is related to hidden units. In practice, the unit cycle is
The absence of the unit cycle is confusing because,
as seen in the case of Heisenberg's uncertainty relationship, it makes
it impossible to detect mixed units, such as cycles and radians, when
are used in the same equation.
Eq. 3.11, we can see now that the units of DEDt
h/2p are: "radian·Joule·s."
and not simply "Joule·s." To avoid confusion, it is essential to
indicate clearly that the units are "radian·Joule·s".
is a hidden and arbitrary criterion of resolution in Heisenberg's
relationship which means that there is no fundamental physical
involved. Heisenberg's relationship (Eq. 3.11) which corresponds to a
of one radian, is just as arbitrary as Eq. 3.16, which corresponds to a
criterion of one degree (or 1/57.3 radian), as Eq. 3.6, which
to a criterion of one cycle. Since the Heisenberg uncertainty
is an arbitrary criterion that is one of the most important aspects of
the Copenhagen Interpretation
[3.1]we must realize that
it is necessary to reevaluate the significance of Quantum Mechanics [3.1].
3-10 The Relationship
We will show
units in this relationship are not coherent. We know that the units of
E are Joules. The units of h are Joules-second. The units of frequency
are "cycles per second". Therefore E = hn
we have the relationship:
sides of 3.23 by [Cycle]. This gives the energy of the photon per
After simplification, we get:
(Joule/cycle) = Joule. That equation does not make sense because the
are not coherent. "Energy per Cycle" cannot be equal to "Energy". This
shows the incoherence of the units in the relationship E = hn.
The missing "cycle" corresponds to quantizing the energy per cycle.
will be discussed later.
3.1 Cramer, John G., "The
Interpretation of Quantum Mechanics", in Reviews of Modern Physics,
3, 1986, p. 658.
3.2 Heisenberg, Werner, Physics
and Philosophy, the Revolution in Modern Science, New York, Harper
and Row, 1966, 213 p.
3.3 Lew, H., Marmet, N., Marshall,
M. D., McKellar, A. R. W., Nichols, G. W., "A Compact Automatic
for Use with Tunable Infrared Diode Lasers", in Applied Physics B,
3.4 Hall, J. L., Lee, S. A.,
Real-time Display of CW Dye Laser Wavelength with Sub-Doppler
in Applied Physics Letters, Vol. 29, No. 6, 1976, p. 367.
3.5 Schiff, Leonard I., Quantum
Mechanics, Toronto, McGraw-Hill, 1955, 417 p.
3.6 Van Name, F. W. Jr., Modern
Physics, Englewood Cliffs (N. J.), Prentice-Hall, 1962, p. 117.