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Saturday, May 14, 2011
Saturday, January 26, 2013 7:34:11 PM
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Last 10 Posts
Saturday, January 26, 2013 4:58:48 PM
This is one example showing that decimal representations are not unique, viz., that 1 and 0.999... represent the
real number. Formally, this is easy to see by, e.g., setting
x = 0.999...
and subtracting this from
10 x = 9.999...
and then solving the resulting equation
9 x = 9
One has to keep in mind that a decimal representation of a (non-negative) real number x is actually a limit of a series, i.e., it is defined as
x = n_0.n_1 n_2 n_3 ... = limit_(N->Infinity) Sum_j n_j / 10^j
, where the sum is taken from
j = 0
j = N
stands for 10 power j, and where the allowed integer numbers n_j range between 0 and 9 if j > 0.
The analogous considerations hold, of course, also for 9.999... and 10, and 99.999... and 100, etc.
was bedeutet das?
Friday, November 9, 2012 4:27:49 PM
The German expression
"Tomaten auf den Augen haben"
can perhaps be related to
the English idiom
"to be as blind as a bat"
and it means that one lacks the capability
to see or recognize something that is completely obvious.
Saturday, August 4, 2012 5:59:36 PM
From a rigorous point of view, unfortunately this is not a good example, since when x tends to zero (in symbols: x -> 0) the expression 1/x² diverges. Or, one may also say that the function f(x) = 1/x² has no limit when x -> 0. If one wants to include infinity into the formalism, one needs to to work in the extended complex plane (or Riemann sphere).
To become familiar with the English terms, a better example would be
lim 1/x² = 0 as x -> ∞
, in words:
The limit of one over x squared equals zero when x tends to infinity
(where the latter means that for any given natural number n one may take a x greater than n, x > n).
Thursday, June 14, 2012 12:06:40 PM
"Knowledge is priceless: that's why it is inappropriate to lock it away and make it available only to the well-heeled."
In contrast to this assertion and what some previous postings claim,
knowledge is free.
(i) As a rule, the email address of the authors (or the corresponding author) is displayed in the journal (some journals, e.g. Nature Medicine, provide forms to contact the authors); if no email address is given, usually it is a matter of two minutes to find the electronic coordinates of the authors by a search engine. When asked by email, every author will be happy to provide an electronic (or even hardcopy) reprint (or preprint) of his/her work. Some publishers even provide authors with corresponding PDF files to distribute them this way or to display them on their homepages. I frequently get demands from colleagues (not only from developing countries) for articles (also old ones, published more than a decade ago) to which they have no access (because the respective journal isn't available to them), and I never failed to provide the requested material (and always received articles that I had asked for).
(ii) Nowadays, the majority of papers in natural science is uploaded to preprint servers (e.g., the arXiv.org e-print archive) before publication in a journal, and they remain there for a free download also after the publication.
Of course, it would be much more convenient to download a dozen papers just by pressing a button instead of having to write twelve emails. But everyone who is really interested in a specific scientific paper has possibilities to get it without paying for it. Actually, the situation today is much better than two or three decades ago. E.g., at the ICTP (International Centre for Theoretical Physics in Italy, an institution that focuses upon inviting scientists from the Third World for some weeks to profit from a "First World" working environment), in the pre-electronic publishing area there always were long lines in front of the copy machines of the ICPT library, so that the copying time per visitor had to be restricted to 20-30 minutes a day. Nonetheless, at the end of their visit, most scientists had collected so much papers that they couldn't carry them in their luggage; whence, the ICTP also paid for mailing a certain amount of printed matter to the home countries of the visitors. Nowadays, there is almost no copying anymore at the ICTP libraries because it is much easier to acquire the material via electronic channels.
"Dire qu'il y a pis de voler" Définition?
Thursday, June 14, 2012 9:41:21 AM
"Dire qu'il y a pis de voler"
Selon le contexte, je le traduire comme "To say that there is (something) worse than stealing", ou peut-etre, "To say what is worse than stealing".
Il ya un proverbe
Donner et roter,
C'est pis que voler.
("Giving and withdrawing,
is worse than stealing")
What is matter?
Tuesday, April 24, 2012 3:45:26 PM
Are you saying that energy comes into existence from nothing, then goes back out of existence? If so, it’s the first time I have heard of this. That strains my imagination to the maximum trying to comprehend how that can occur. Does anyone have any idea how this occurs?
To explain these spontaneous pair creation and annihilation processes, sometimes (even in physics classes) the wrong picture is used that for a short time some energy is "borrowed" from the universe to create the pair, and then "given back" when the pair annihilates. A more correct explanation rests upon a proper use of the uncertainty relation between time and energy: the energy of the vacuum is only on the average equal to a constant (e.g., zero), and it always fluctuates around this value. For sufficiently short time intervals this fluctuation may be large enough to provide the energy required for the creation of a particle - antiparticle pair.
This picture is analogous to the uncertainty relation for canonically conjugate operators, like, e.g., position and momentum (but not quite the same, since a time operator doesn't exist in quantum mechanics). The uncertainty between position and momentum means that if one would try to localize a particle within an ever smaller region in space, this would cause ever larger fluctuations of its momentum (and thus a corresponding increase of its kinetic energy); hence, a precise localization (determination of the position) of the particle is impossible.
Now I’m really baffled as to how an atom can “spontaneously” jump to an excited state. I would have thought there would have to be some kind of interaction with another force of some kind.
The interaction occurs between the electron and the vacuum, or, more precisely, between the electron of the atom and the electron-positron pairs that are spontaneously created in the vacuum.
Now my mind is buzzing with questions about what, exactly, a charged particle really is or even electromagnetic energy itself. But it’s too deep for me. It’s extremely interesting, but I can’t even come up with a pictorial allegory to use for it. And a pion that can change a proton into a neutron and vice-versa, ….getting way-y-y over my head there.
No, there is no neutron involved in the hypothetical decay of protons; (one of) the proposed decay process(es) is like this:
proton --> positron + neutral pion --> positron + 2 gamma
where gamma stands for (high energy) photons.
Well, the question
what is exactly an electron (or an elementary particle)
isn't yet settled. From the point of view of string theory, elementary particles like electrons and quarks are quantum excitations of more fundamental objects, viz., strings and p-branes.
What is matter?
Monday, April 23, 2012 2:23:37 PM
While I didn't understand completely some of what you said, such as virtual photons, etc., I see now that the pictorial allegories of which you spoke are all I have to work with, and they are woefully insufficient to the task. Mathematics is, indeed, the language of physics, and explains why I am so ignorant on the subject. If I were twice as knowledgeable as I currently am, I would still be ignorant in physics. So I should quit while I am ahead, or behind, as the case may be. Thanks again. A very interesting read.
Sorry, dear FounDit, if by pointing out the limited value of allegories I discouraged you from continuing to think about physical issues; this was definitely not my intention. Actually, pictorial allegories are quite common in physics, and scientists use and rely on them in their everyday work and thinking. A scientist familiar with the field, however, usually knows the limits of such pictures, or, if he/she employs them and builds something new upon them, then constructions based upon over-interpreted pictures will sooner or later lead to troubles with experiments or other theories, thus revealing the invalid aspects of the used allegory. This is one way how learning and research is proceeding. So, rather than to quit thinking and enquiring, it is much better to continue with allegories even if sometimes they may lead one astray.
As you have posted above an interesting question, viz.
"if an atom is unstable and gives off energy to become stable, then how long can it remain stable?"
, let me try to comment on it.
For simplicity, consider an isolated hydrogen atom, and, for the moment, neglect the internal structure of the nucleus, i.e., of the proton. Quantum mechanically, a stable hydrogen atom (i.e., where the electron is bound to the nucleus and not ionized from it) can only exist in certain states with
energy levels. The state associated with the lowest possible energy is called the
, the other states are the
and have an energy that lies above the ground state energy. Suppose that initially the hydrogen atom is in one of these states. Then, without an interaction with its environment, it will remain in the same state forever. However, this is an idealized situation, as there always will be interactions of the atom with its environment, even if it is just put into the physical vacuum. The latter, in fact, is not really a void, rather, the vacuum "boils", which means that spontaneously pair creation and annihilation processes occur all the time, e.g., an electron - positron pair is created and, after its creation, almost immediately it again disappears by annihilation. Nonetheless, by interacting with these
, the electron of the hydrogen atom achieves the possibility (even if isolated from perturbations by external radiation) to change its state, where the probability to change from an excited state to the ground state is much larger than the opposite direction. Therefore, with overwhelming probability, a hydrogen atom in vacuum ends up in its ground state after a short time; although then spontaneous jumps into excited states cannot be excluded, they will be extremely rare.
Another part of the question is:
"How stable are the constituents of the hydrogen atom, i.e., the electron and the proton?"
For the electron, all current knowledge predicts the electron to be stable, i.e., there are no processes known by which an isolated electron can decay. The stability of the proton, on the other hand, is still an unsolved problem in physics and has attracted significant theoretical and experimental interest since many decades. GUTs (grand unified theories, i.e., theories that attempt to unify the electromagnetic, the weak, and the strong interaction into a single force) predict that a proton may decay, e.g., into a positron and a neutral pion (where the pion itself is unstable and rapidly decays into photons). Despite of many efforts, until today such decay events have not yet been observed experimentally. The lower limit of the lifetime of protons extracted from these experiments is larger than 10^30 (ten power 30) years (cf., e.g.,
) and thus much much larger than the age of universe.
What is matter?
Sunday, April 22, 2012 4:25:30 PM
It is nice to see the many postings in this thread and to witness the great interest that these physical issues attract. Unfortunately, there are some fancy ideas around which are rather "incompatible" with the current knowledge that physics provides. Mathematics is the language of physics and it allows the precise statements needed there. Words or pictures fail to be precise enough to furnish an adequate description. While pictorial allegories may reflect properly some aspect of a physical fact, they can become grossly misleading if taken too literally. Even if the very doubtful assertion that
matter is energy slowed down
is replaced by one like
matter is condensed energy
that comes closer to the truth, also the latter picture is still incorrect and, e.g., may lead to the confusion of "matter" with "mass";
are not the same. As I already wrote in a previous post, matter can be characterized as a
system that contains particles with non-zero rest mass
. Taking the simplest atom, viz. the hydrogen atom, as an example, this consists of a proton (that itself enjoys an internal structure by being composed of quarks that are held together by the exchange of gluons) and one electron, both interacting via the exchange of virtual photons. The latter are the messenger particles that convey the electromagnetic force; photons have zero rest mass, while the electron and the proton enjoy a nonzero rest mass. The mass of the latter is about 1840 times the mass of the former, so that most of the rest mass of the atom is concentrated in its nucleus.
Let me try to provide a bit more information to some statements in postings above:
Matter can neither be created or destroyed
No, that is not true. If matter and antimatter collide, e.g. hydrogen and antihydrogen (consisting of a (negatively charged) antiproton and a (positively charged) positron), then both annihilate into radiation, i.e., photons. However, the assertion above becomes correct if "matter" is replaced by "mass", so:
Mass can neither be created or destroyed
It is the motion of the electrons that creates the illusion of solid matter
. Well, this "motion" must be regarded as being very different from the orbiting of, e.g., planets around a central star. If the electrons in an atom were really moving, then (since they are charged particles) this would imply that they would constantly produce electromagnetic radiation, so atoms would lose energy by radiating all the time and thus couldn't be stable. Actually, the probability distribution of electrons in an atom is static and thus does not change in time (unless perturbed by "external" effects). It is the complex phase of the electronic wavefunction that oscillates in time. On the other hand, it is true that the kinetic energy of the electrons is required to render atomic systems stable. Without this kinetic energy, the electrostatic attraction between the positive nucleus and the negatively charged electrons would make the latter to collapse into the nucleus. The structure of atoms and molecules and thus of "ordinary" matter is also significantly influenced by the fact that electrons are fermions and thus the Pauli exclusion principle holds which forbids any two electrons to occupy the same quantum state. If electrons were bosons, they could occupy the same quantum state which would result in very a different "chemistry" and very different properties of normal matter.
Electromagnetism is mostly influential at the chemical or molecular scale. At larger or smaller scales it is not all that significant, but rather follows the dominant forces
. At larger scales, this is not the result of another force taking over (like it is for smaller scales), but a consequence of the neutrality of (most of) ordinary matter. If for atoms the net effect of positive nuclear and negative electronic charge wouldn't compensate "outside" of the atom to render it neutral, then the electromagnetic force would also dominate on astronomical scales and the gravitational force would play almost no role. Again, this would lead to a world rather different from that we are in now.
What is matter?
Thursday, April 19, 2012 2:26:27 PM
Although there is no "strict" definition of
in physics, one may consider a system as
if it contains particles with nonvanishing
; there is no unique convention whether there must be more than one particle or whether one already may regard, e.g., a single electron as
. If one takes the electron as a point particle and requires that matter should also occupy some volume in space, then a single electron falls outside the definition of matter.
is the mass of the particle in the inertial system where it is at rest. Photons, the quanta of the electromagnetic field, always move with the speed of light in any inertial system, there is no Lorentz transformation that can bring them to rest; their rest mass is zero. So they (and thus electromagnetic waves or radiation) aren't matter according to the definition above.
is as real as
is. According to special and general relativity, they cannot be separated, but must be considered together as
is not independent of its content; the Einstein field equations (basic to general relativity) relate the structure of space-time with the so-called
. The latter not only comprises the matter density, but also the density of energy as well as its flux. One may think about a generalization of the notion of
by speaking of
as soon as the energy-momentum tensor doesn't vanish (which would imply that then also photons may be regarded as matter), but probably this will more confuse than enlighten the discussion.
jedoch vs dennoch
Wednesday, March 7, 2012 4:13:42 PM
Hi. Please let me know if there is particular difference between jedoch and dennoch
Please explain me in English...
but in Deutch also appreciate ;-) of course
This is similar as the difference between
) in English.
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