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The resolution limit interpretation of the
Heisenberg uncertainty principle This web version 2002 ABSTRACT: It is shown that the fundamental difference between classical
and quantum entities is the relative difference between the representative length scale
of the entity and its de Broglie’s wavelength. From this realization, the Heisenberg’s
uncertainty principle is derived for entities engaging in linear and spherical motions.
This new derivation leads to the conclusion that the uncertainties in position and in momentum should be
separated instead of being considered together as in existing method. It is shown that
this separation resolves all of the existing conceptual difficulties associated with
the Heisenberg’s uncertainty principle.
Physical entities and measurable effects
In the article “The Resolution Limits of Space and Time”
I have introduced the idea that there exist resolution limits in
both classical space and time. I will now use resolution limits to
re-examine the celebrated Heisenberg’s uncertainty principle, but first I must
make sure that we agree on what a “physical entity” is. I suggest we follow the
spirit of physics and simply define anything that we can measure a physical
entity. I will use the example of a droplet of water. If I’m
investigating the properties of the whole droplet, I obviously will consider
the droplet as the physical entity. However, if I conduct an investigation in
the atomic realm, I must consider the protons, neutrons, electrons, etc. as
physical entities because in this case the properties of these minute entities
are of interest to me. For differentiation purposes, I will call entities such
as the droplet macroscopic entities or classical entities. I will call protons,
neutrons, electrons, photons etc. microscopic entities or quantum entities.
There is some problem in equating “classical” with “macroscopic” and “quantum”
with “microscopic” because the distinction between classical and quantum is
process dependent; and there are cases where the quantum scales could be very
large. But since “macroscopic” and “microscopic” are popular terms and their
meanings in most cases do correspond to “classical” and “quantum”, I will
respect tradition and try to hang on to them.
Figure 1: Size difference between a
classical (macroscopic) entity (left) and a quantum (microscopic) entity. Each physical entity has a representative length scale, which I
will call Sc. Since an order of estimate is sufficient, Sc
could be, say, an “equivalent diameter” for the case of the droplet, which is
–most likely-- much larger than the spatial resolution limit L0.
Since I know from the article “The Resolution Limits of Space and Time”
that the spatial resolution limit is exactly the de Broglie’s
wavelength ld, I will
use ld in place
of L0 from now on. While it is true that the droplet is very flexible and could be
broken up into smaller droplets by certain physical processes, I can still
consider it as a “connected entity” of many elements that act more or less in
unison according to macroscopic physical laws. If the droplet as a whole is the
object of my investigation I have to use Sc as its representative
length scale. By the resolution limit postulate, in order for an entity to
qualify as “macroscopic”, its representative length scale Sc must be
much larger than the spatial resolution limit ld. To be consistent, the representative length scale of a quantum
entity must be smaller than the spatial resolution limit. This is indeed
true. De Broglie’s wavelength ld for a typical electron is about the same size as a
typical atom (of order 10-10 m). All measurements to date confirm
that the (apparent) size of the electron is much, much smaller than this. The
proton and the neutron are considerably bigger than the electron, but their
speeds are negligible, meaning that their de Broglie wavelengths are also much
larger than their sizes. Photons, the so-called “particles of light”, are
considered to be of zero size (with the understanding that this only reflects
my inability to measure them.) I have the following results: TABLE 1: Length scales of macroscopic and quantum entities Macroscopic
(classical) entities: ld <<Sc (1) Quantum entities: ld >>Sc (2) The results in table 1 seem trivial, but they are extremely
important in my effort to make sense out of the Heisenberg’s uncertainty
principle. I will return to them later. The de
Broglie’s hypothesis The de Broglie’s hypothesis was introduced to the scientific world
in 1924. It asserts the existence of the so-called “de Broglie’s wave”, whose
wavelength ld
is determined by the Plank constant h and the momentum of the entity under
consideration: ld = h/p (3) The de Broglie’s hypothesis was an ingenious postulate based on
the Plank’s formula, which historically marked the beginning of quantum physics in
1900: E = hn (4) Where E is energy of light, h the Plank constant, n the frequency of light. Several months ago I thought I had an explanation for the Plank’s
formula based on the resolution limit postulate. I later found that my argument
had been founded on circular logic. I’m still working on this problem, but for
now I can only take (4) as a starting point. This mean I will also have to
accept equation (3), i.e. the mathematical expression of de Broglie’s
hypothesis, as a starting point without being able to offer any logic for it.
Since there is no real explanation for (4) available in the literature today, I
don’t think I’m doing worse than anyone else in physics because of this
drawback. Resolution
Limit derivation of the Heisenberg’s uncertainty principle I will start a thought experiment with the quantum world inside
the water droplet that I mentioned earlier. In the quantum world I’m dealing
with quantum entities such as photons, neutrino, electrons, neutrons, and
protons. They are either of zero size or very small size. Conceptually I can think of quantum entities as “things” localized
in space. For argument’s sake I will assume that I have a perfect measurement
tool at my disposal, and a perfect procedure to measure the position of a
particular quantum entity. I already know from an earlier analysis that my
ability to perform this task will be compromised by the resolution limit of the
space associated with the quantum entity, which is de Broglie wavelength ld. This
means that when I measure the position x, the measurement error Dx that I certainly will commit could range anywhere
from -ld/2 to +ld/2. In
the more familiar language I would say that the uncertainty in my measurement
is ld/2. In reality there is no perfect tool and there is no perfect
procedure. Therefore I have to assume in the general case that the uncertainty
in my measurement is equal to or larger than ld/2. I express this mathematically as follows: Dx ³ ld/2 (5) Next, I look at momentum. In order to measure the momentum of a
quantum entity I must first detect it. This is easy in the macroscopic world, but not
so in the quantum realm, because I cannot resolve length scales smaller than
the wavelength, and the effective size of the quantum "thing" is much smaller
than its wavelength. There are only two cases, with nothing
in between: Either my measurement tool detects or fails to detect the quantum
entity. For momentum, a detection failure will result in a zero reading instead
of p. Thus, the measurement error in momentum measurement Dp is exactly p (here I don't have the luxury of cutting the
total error in half.) Again, because my tool is not perfect
and my measurement is not perfect I have to say that Dp is equal to or greater than p: ∆p ≥ p (6) With de Broglie’s hypothesis, namely: p = h/ld (7) I can combine (5) and (6) to get: ∆x∆p ≥ ∆x(h/ld) ≥ (ld/2)(h/ld) (8) Which simplifies to: ∆x∆p ≥ h/2 (9) Strictly speaking, inequality (9) is applicable only to quantum
entities engaging in linear motion. For a quantum entity in a
spherical environment (e.g., an electron in an atom), the circumference of its
idealized orbit has to fit an integer number of ld (otherwise ld would not be the resolution limit.) This allows me to
write: 2p r = nld (10) Thanks to
the spherical symmetry inherent with atomic structure, position for me means
the radial distance r instead of the cell length ld. Thus, the measurement error in position is: ∆x =
∆r ≥{∆(nld)}/(2p) (11) Since the
increment for n is 1, the error in measuring nld could be as large as 1x ld = ld. By
referencing to the middle of the wavelength, I can cut the error in half: ∆x
≥ ld/(4p) (12) The error in
momentum is again: ∆p ≥ h/ld (13) By combining (12) and (13), and use the short hand notation ∆x∆p ≥ I notice immediately that inequalities (9) and (14) look exactly
like the Heisenberg’s uncertainty principles for linear and spherical
environments, respectively. After searching for and finding nothing that can
give me the same inequalities, I’m forced to conclude that they are indeed the
linear and spherical expressions of the Heisenberg’s uncertainty principle!!! Thus, starting with the resolution limit postulate and de
Broglie’s hypothesis, simple logic has led me to this celebrated principle,
which has been responsible for most, if not all of the so-called “quantum
weirdness”. The incompleteness of the Heisenberg’s uncertainty principle Then how come there is nothing “weird” in my derivation of the
Heisenberg’s uncertainty principle? It does not take me long to realize that
there is a fundamental difference between how Heisenberg and I have arrived at
the same inequalities. I started with the resolution limit postulate, therefore
inequalities (9) and (14) were natural consequences of the de Broglie’s
hypothesis. Heisenberg, on the other hand, started with the assumption that
space and time are indefinitely divisible (i.e., the point assumption) and came
up with his principle as the necessary (and ingenious) patch to reconcile the
point assumption with de Broglie’s hypothesis.
I will put myself in Heisenberg’s position to rationalize the
existing interpretation of quantum mechanics, despite my disagreement with it.
Mathematically the point assumption is the limit of the resolution postulate
when the spatial and temporal resolution limits approach zero. By following the
exact procedure that I have used earlier all I get would be: ∆x ≥ 0 (15) This does not lead me anywhere, although it is qualitatively consistent
with de Broglie’s hypothesis. Since my goal is to obtain quantitative
information, I’m forced to consider not just ∆x, but ∆x in
combination with something else1. After considerable manipulations I will find
that the appropriate combination is ∆x∆p; and in order for this
combination to be consistent with de Broglie hypothesis I will arrive at either
(9) or (14) depending on the type of problem I am investigating. I have just outlined the reason why the Heisenberg’s uncertainty
principle -as we know it- is in the combined form of ∆x∆p. Because
(15) is the only known criterion, the possibility of ∆x =0 cannot be
ruled out. It follows from (9) or (14) that absolutely precise measurement of
the position x will come at the price of complete uncertainty in the momentum
p. By symmetry the reverse also has to be true. My point is that, given the information available at the beginning
of quantum physics any reasonable scientist would have interpreted the Heisenberg’s
uncertainty principle exactly the same way Heisenberg did. Today’s standard interpretation of Heisenberg’s uncertainty
principle is still that one can measure either position or momentum as
accurately as the measurement tool allows, even down to zero error; but an
accurate measurement of position implies that the measurement error in momentum
is arbitrarily large, and vice versa. This interpretation is scientifically
unsatisfactory because it implies, for example, that the sole electron in a
Hydrogen atom may be light years away from the location where its momentum is
successfully measured to absolute precision. This is known as the “infinity
problem” of the Heisenberg’s uncertainty principle. This problem, and many
others similar to it, is a direct interpretation of the original form of the
Heisenberg’s uncertainty principle; in which the uncertainties in position and
momentum are combined into a single product as a consequence of the point
assumption. With the resolution limit postulate in mind, I must therefore
conclude that the Heisenberg’s principle, as it is today, is incomplete! Revised
implications of the Heisenberg’s uncertainty principle I will now show that the separation of ∆x and ∆p in
the new derivation that I have just given will resolve several serious
controversies associated with the original Heisenberg’s uncertainty principle. 1. Implications on single
and multiple measurements of quantum properties If the measurement error of a quantity is ∆s, its
actual value S0, its measured value S, then the following holds
true: S-∆s £ S0 £ S+∆s (16) It can be seen from (16) that ∆s is the maximum possible error that could be committed, not the error certainly will be committed.
Thus, it is hypothetically possible for the measured value S to be exactly the
actual value S0. Of course this very special case only has
hypothetical meaning, because even if I have achieved a perfect measurement, in
my mind I still think that I could be off by as much as ∆s. The point,
however, is that the Heisenberg’s uncertainty principle has nothing to do with
the actual result of a single measurement. I will assume for argument’s sake that I have achieved a perfect
measurement in x (without knowing that it is perfect, of course), by the same
argument there is nothing that prevents my measurement in momentum to be also
perfect (thought I would not know this either). Thus, regardless of how large
∆x∆p is, perfect measurements of both x and p are
hypothetically possible, just that there is no way for me to know if they are indeed perfect.
This is contrary to an interpretation popularized by many trade books on
science, which claims that a perfect measurement in x would have such a
detrimental effect on the “quantum wave” that the error in momentum measurement
would approach infinity. What if I measure the same property (x or p) multiple times? The
Heisenberg’s uncertainty principle implies that my x measurements cannot be the
same each time (i.e.,
it is impossible for me to duplicate my x measurements.) For this reason, I
will end up with a range of values for x. The same with momentum measurements.
If I make many many measurements to cover all possible cases and call the x
measurement range ∆x, the momentum measurement range ∆p, then they
must obey the Heisenberg’s uncertainty principle. 2. On the myth that
∆x or ∆p could be infinity For easy reference I will restate (14), which is the
Heisenberg’s uncertainty principle for a quantum entity in a spherical
environment: ∆x∆p ≥ Recall that the inequality sign is needed in (17) only because I
know that my measurement tools cannot be perfect, and my measurement procedures
cannot be perfect either. If I had perfect tools and perfect procedures I would
have written (17) as an equality. Since the errors associated with my tools and my procedures must
be finite, I can rewrite (17) as: ∆x∆p = In fact, I can rewrite (18) as: ∆x∆p = K (19) Where K is defined as the sum While it is true that if the uncertainty in position ∆x is
zero, the uncertainty in momentum ∆p will be infinity; but can ∆x
be zero? Recall from (2) that ∆x cannot be smaller than ld/2, the
only way for ∆x to be zero is that ld is zero; but since ld is the resolution limit of space, ld = 0
implies that space has no resolution limit, which is the point assumption.
Since I have ruled out the point assumption, I conclude that ld cannot
be zero2. Thus, it is impossible for ∆p to be infinity. Next, since ∆p cannot be smaller than p, the only way for
∆p to be zero is that p is zero; but if p is zero the velocity of the
entity must be zero (it cannot exist in the physical sense and have zero mass,
otherwise it would move at the speed of light and hence possess momentum). If
the velocity is zero, the entity either does not exist (in the physical sense)
or it sits still in space (v=0). Here I will rely on a result established by
thermodynamics and claim that v can never be exactly 0 because the absolute
temperature would have to be 0oK, which is not realistic. However,
the case of very small momentum is real. It implies that ld could be
very large, and this is why sometimes the quantum scale is not “microscopic” (I
have acknowledged this fact earlier.) 3. On the myth that quantum
rules are applicable to macroscopic objects Serious
problems with the Heisenberg’s uncertainty principle arise when one attempts to
apply it to the macroscopic world. I will look
at the problem with the error in momentum ∆p first. Since the error in
position ∆x is directly related to ld/2, and ld is vanishingly small for a typical macroscopic
entity, it follows from the original Heisenberg’s uncertainty principle that
the measurement error in its momentum, ∆p, is extremely large; which is
in conflict with reality. This is resolved by recognizing that the measurement
error in momentum, as implied by the Heisenberg’s uncertainty principle, is
caused by the inability (of a classical tool) to determine the exact location
of the quantum entity within its de Broglie’s wavelength. This problem is
caused by the fact that
ld >>Sc
which is particular to quantum entities and does not apply to classical
entities, where ld<<Sc.
Figure 2: Because of the difference
in relative size and measurment scales, quantum effect of positional
uncertainty is dramatic (as if the entities has
moved), whereas its classical effect is negligible (right picture has been
exaggerated to illustrate the point.) Next, I will
look at the problem with the error in position ∆x. The mass of a
macroscopic entity is much larger than that of a quantum entity, and by the
same thermodynamic requirement I’m certain that it is always moving, at least
at the quantum level. This makes its de Broglie’s wavelength, and hence
∆x, vanishingly small. Some may argue that if a classical entity is
sitting still, its de Broglie’s wavelength should be the average wavelength of
all quantum entities belong to it. But even if this case is true (and I’m not sure
it is) the de Broglie’s wavelength of the classical entity still ends up being
much smaller than its representative length scale Sc, as I have established
at the beginning of this article. I will use
figure 2 to visualize the different effects of the Heisenberg’s uncertainty
principle on a quantum entity and a macroscopic entity. When I measure a
quantum entity, a positional error gives me the eerie feeling that the entity
has moved from one position to the other (which, by the way, explains
superluminal motion in the quantum realm.) On the other hand, my inability to
measure the position of the macroscopic droplet exactly will at most
give me the feeling that the droplet is vibrating. In
realistic cases, this “vibrating motion” is too small for the classical scale.
This is why I don’t see the effect of positional uncertainty in the macroscopic
world. Conclusion The Heisenberg's
Uncertainty Principle rocked the world of science when it was first announced in 1927.
It was promoted by best selling science trade books in the 1970's and 1980's as conclusive
evidence that the line between science and mysticism had become fuzzy. But as you can see
from this article, the Heisenberg's Uncertainty Principle is
simply a natural consequence of the resolution limits of classical space and time.
In fact, operationally it is no different than an engineering measurement problem.
I hope I have been successful in bringing this celebrated principle back to
common sense.
Your criticism will be helpful and is deeply appreciated. First written
for the Web on September 15, 2002 ÓDangSon Tran Notes 1Heisenberg focused on position and momentum, therefore he obtained the ΔxΔp version of the principle. Had he focused on time and energy we would have obtained the less well known (but possibly more important) version of ΔtΔE. 2This point only has academic value since if λd=0 the process would be classical, and there would be no problem locating it. The Heisenberg's uncertainty principle therefore does not apply. Now that you have read the article, it is my turn to ask you to contribute your wisdom: -Did you find this article stupid, useless, a waste of time, or having some other negative effect? If so you can help the world by e-mailing me a wake-up message. Who knows? You may be able to wake me up from my delusion so that I won't keep on with this kind of frivolous activities. That would help me personally and spare the internet world of some annoying cyber-pollution. -Do you have any other feedback for me? Like criticism, encouragement, ideas, opinions, etc. Please let me know by email. -Did you find this article good, useful, valuable, or having some other positive effect? If so please help spread the word through your family, your neighbors and friends, your work place, educational and/or academic institutions, internet user groups, etc., so that this article will be read by more people.
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