[Spoiler:]

1

General Science / Re: Is 2 really prime? If so, why isn't 1?

« on: 30/04/2022 00:19:02 »

Hi.

I came across another wrinkle: what about –1?

    Your general arguments after this are reasonable.   However, I think it is again just a matter of simplicity and having a set of numbers that are useful for something.   It is possible and useful to confine your attention to what people might call the counting numbers or the Natural Numbers, so we do.   That doesn't mean that mathematicians have never considered generalising the idea of prime numbers and investigating properties like prime factorisation in a structure bigger or more abstract than just the positive counting numbers - they certainly have.

   There is already some terminology you could use to describe a set of things that behave like prime numbers but apply to a much more generalised set of objects than just the Natural numbers.   These things are called "prime elements" and the parent algebraic structure is known as a "Ring".   You seem to be interested in the Ring which is the Integers (positive and negative Naturals with 0,  under  conventional binary operations of + and x).

   See   Wikipedia entry:   Prime elements, if you're interested.   https://en.wikipedia.org/wiki/Prime_element
However, you should note that they exclude "units" which would  include -1  in the ring of Integers,   i.e.   they would directly exclude both  +1  or  -1   from the prime elements in the ring of Integers.  (For what reason?   Similar to excluding 1 from the primes,  it makes it much easier to state an equivalent unique factorisation theorem for the ring of Integers).

    Here's a quick question or puzzle, just for fun.   It relates to the idea you mentioned earlier of eliminating the number 2 from the prime numbers.   You also seemed keen to extend beyond the positive numbers and consider negative numbers but you really don't have to stop there - you can consider Complex integers.
    The Complex Integers or "Gaussian Integers"   are the  Complex number equivalents of integers.   Specifically, the Gaussian integers are the set of all complex numbers of the form   a+bi   where  a and b are integers.
    Just like in ordinary arithmetic with Natural numbers, a prime  (or prime element) of the Gaussian integers is a Gaussian integer,  p,  that is irreducible or cannot be factorised.   Specifically,   if we have  p =  q × r   (where × is just ordinary multiplication of the complex integers q and r)  then  at least one of  q or r must be a unit element.     A  "unit"  is any complex number that lies on a unit circle around 0,  so the only  units in the  Gaussian integers are    +1, -1,  +i, -i.
   The number 2   is a prime in the ordinary integers.   Is it still a prime in the Gaussian integers?   To say that another way, can you factorise the number 2 in the Gaussian integers?

Spoiler: show

  2  =   (1+i) (1-i)   = the product of two Gaussian integers, neither of which are unit elements.   So 2 is not prime in the Gaussian integers.
  As it happens, the Gaussian integers do form a Unique Factorisation Domain.   This means there is a set of prime elements often called Gaussian Primes, all the Gaussian integers can be written as a product of those prime elements and, as always, that factorisation is unique.   However, its prime elements are quite different to the prime numbers of the ordinary Integers.

Best Wishes.

[Spoiler:]

2

General Science / Re: Is 2 really prime? If so, why isn't 1?

« on: 28/04/2022 18:22:32 »

Hi.    Fantastic diagrams.   Great that you're making an effort to engage the audience with some Mathematics etc.
I wish you well.

I'll hide everything else under a spoiler because it's a bit dull and might prevent others from making their comments.

Spoiler: show

   The main reason for not counting 1 as a prime number is that most of the results we have about prime numbers, or more generally about whole numbers, won't work if you tried to state them as they are now and continued to use the term "prime number" in that statement of the result.   The decision not to include 1 as a prime number wasn't really done because of some elaborate definition or way of identifying what the primes are supposed to be.   I don't think following a pattern that emerges from dots had a lot to do with it.    Instead, it was done because it's not all that useful to have 1 included in the set.

    Another way to say this is that there's no reason you couldn't include 1 as a prime number if you want to.   You go right ahead and do that.   You don't even need to make up a good reason like drawing an arrangement of dots.   For whatever reason, you can put the number 1 into the primes if you like.   The only change that will result is that mathematicians will stop quoting their results by referring to "prime numbers" .  Instead they will identify a slightly different set of numbers, let's call them "Q-rimes" and their results will be stated with respect to that.   The Q-rimes will naturally be your Primes excluding the number 1.  So, the only thing that will have happend is that you will have changed the name we apply to describe what is currently called the prime numbers.

    I suppose to finish this I should give at least one example of a result that is useful and easily stated with reference to prime numbers (with 1 excluded but not if 1 is included).

     The fundamental theorem of arithmetic
Every counting number can be written as a product of prime numbers each raised to an appropriate (Natural number) power.   Furthermore, that respresentation is unique up to changing the order in which you perform the multiplication.

Example:     40  =  2³ x  5
   If you try to write 40 as some other product of primes, let's say you allow yourself to use three prime numbers   p, q, r    such that   40  = p^a x q^b x r^c     for some exponents  a,b,c   then you find that you can't,  there's no solution for that.     The fundamental theorem of arithmetic holds.
    However, if you allowed 1 to be a prime number then you can.....   One solution is to set p = 2,   q = 5, r = 1    and a= 3, b = 1, c= 2 .  That will be another representation of the number 40 as a product of primes:    40  =  2³ x  5  x  1²     and so the fundamental theorem of arithmetic doesn't hold.

Is 2 really a prime number?

    Actually 2 is another number that has very unusual properties even though it is prime.   It is often very useful and desirable for mathematicians to consider a subset of primes that doesn't include 2.   They call this set the "odd primes" and several theorems are stated with reference to  "odd primes" instead of just "the primes".  Alternative terms exist for this set and it's quite common not to bother naming the subset and just write a result as holding  "for all primes, p > 2".
   So 2 is a prime number but its certainly not typical of primes and there is a similar set, the odd primes, where you do just exclude it.

Best Wishes.

3

Just Chat! / Re: Are The Naked Scientists Listeners open to interviews?

« on: 19/04/2022 10:23:24 »

Hi.

   How short?

   How regularly do you need to listen?   I listen to about 1 in 5 podcasts when I see them on the front page on this website.  I've only been doing this for a few months.

   What security or safety steps do you plan to offer the participants?  For example,  most people using this website are not going to want a face-to-face meeting with a complete stranger.   Some people won't even want you to get their email address or similar information like that.   Can answers be written to questions directly on this forum?

    Provided it's safe and not too intrusive, the staff and moderators of this site probably won't mind this thread being used to write the answers or replies in  -  But I am not a moderator and you probably should check with someone first.   

Best Wishes.

4

New Theories / Re: A message to the University of Cambridge

« on: 15/04/2022 21:41:48 »

Hi again.

    It looks like you @Yahya A.Sharif  have had replies about your new theory - it seems to have been in the "new theories" section under the title   "The theory of the human body special mass".    They may not have said exactly what you wanted to hear and some of them weren't even all that friendly - but that's a slightly different issue. 
Your new idea and your time and effort to write it down wasn't  just ignored. 
    I'm sorry if the comments weren't what you may have wanted to hear.  Most new proposals are challenged, quite often a new proposal does not survive this sort of scrutiny and the work is never published.

Best Wishes.

5

Guest Book / Re: Introduce yourself and say hi.

« on: 11/04/2022 19:09:18 »

Hi @kaskas and welcome,
   

I'm a wannabe polymath on endless quest to learn more. And Oh boy is there a lot.

   Sounds good.

I was looking for a place to talk about publications, metabolism and climate science in a kind of research-centered fashion, so, here I am on this forum.

  At least some of this should be possible.   I'm not staff and I don't set the rules, however, reasonable attempts to do these things seem to be acceptable:
   1)   Ask questions   or   answer and contribute to existing posts.
   2)   Seek genuine discussion.
 
    Officially, all posts in the main sections should be phrased as questions.   Even if you're just seeking discussion it helps to do this because it opens the door for discussion and shows that you're not just creating a monologue.

   3)  If you really want to create a monologue, the "new theories" section and/or the "just chat" section have different standards and are much more tolerant.   Although you could also just put it on another website entirely.  This is a forum, so if you don't want discussion what's the point in putting an article in here anyway?

  Conversely, things that won't be very popular and might be stopped by the moderators immediately include advertising.   You said "I was looking for a place to talk about publications..."  if that is just your own publications or articles then that could be considered as advertsing.   If you are seeking opinions on your own articles or very new ideas that aren't currently established as main stream science then that is almost precisely what the "New Theories" sections is intended for.   However, if you're seeking discussion based on existing and well established science then the main sections might be a better place.

   Finally you mentioned discussion  "...in a kind of research-centered fashion....".   I would not want to put you off, the forum could certainly use some additional members of all abilities.    However, one of the moderators recently advised me that their average level of answer or response is aimed at an audience who are not assumed to be graduates in any sort of Science.   Some of the users will engage at a higher level but not all.

Best Wishes.

6

General Science / Re: At what gravity does a person run the fastest?

« on: 06/04/2022 16:45:48 »

Hi.

Found this:
https://journals.biologists.com/jeb/article/221/3/jeb162024/20344/Reducing-gravity-takes-the-bounce-out-of-running

A scientific study, with actual data collected, about the running stance and gait adopted in simulated low-G environments.

Their conclusions:  Mainly that the gait is adjusted so as to keep the centre of mass extremely flat and level, i.e. almost all bounce is removed.  Their models suggest this is energy efficient (they use "an impulsive model of running" developed by Rashevsky and Bekker - although these people contributed at different times and not collaboratively).
    There is no comment or investigation about the maximum speeds attainable, sorry.  However, if this low bounce method is more energy efficient you would have thought that a runner can sustain a higher maximum speed.

Noteable limitations
   They didn't seem to have a wind fan or anything to re-create the effect of air resistance.  However, you would have thought this would only further reduce speed while in the air and unable to provide propulsion with your feet, so that it would only increase the desirability of maintaining a low bounce running style.
- - - - - - - - - -
We've already mentioned that accelerating from a standing start is a completely different thing to sustaining a high top speed.   Just for amusement, here is Usain Bolt trying to sprint in low G:

Best Wishes.

7

New Theories / Re: what is temperature?

« on: 03/04/2022 01:07:34 »

Hi.

   I really don't know why burgers have become important.  I was just going to back up to some earlier posts:

Originally a quote from @hamdani yusuf , re-used in a reply from Bored Chemist:

We know that a system can have many forms of energy. They are often classified as kinetic and potential energy. Which category does temperature fall into?

 Bored Chemist replied:

Neither.
That's why we call it thermal energy.

   That's old style.   Presumably you went to school roughly when I did but you (bored chemist) presented the article about re-defining temperature, so you've got to play fair here.
    The modern definition of temperature (in Kelvin post 2019) is trying to remove the need for such oblique references as calling something "thermal energy".  It does not step around declaring what that energy is at a microscopic scale and leave it open to all sorts of possible interpretations.   Instead it embraces the notion of microscopic statistical mechanics head on.

Alancalverd replied:

Temperature is a measure of the mean kinetic energy of the molecules inside an object. It is not a measure of the kinetic energy of the whole object, or the potential energy of any stresses within it. If you input energy in such a way as to increase the mean kinetic energy of the molecules inside an object, you will increase its temperature. If you do something else, you won't.

    Which is much more in-line with the definition of temperature used in the modern Kelvin scale.
Temperature (in Kelvin post 2019) is very much meant to be a measure of the average kinetic energy of the particles of a system.   (At least for simple systems it will be.  Whether our Kinetic Theory is adequately developed to determine the behaviour of particles in all systems like solids, liquids or a gas of photons is a different question).  If you put energy into a system that doesn't change the k.e.of the particles then, as Alancalverd stated, you won't change the temperature of the system.
    However the issue remains murky:   To the best of my knowledge, the requirement for a system to be in equilibrium hasn't disappeared in the new approach to defining temperature.  If you put energy into a system (e.g. to raise electrons to an excited state or change something else about the system that might be considered as a potential energy change instead of kinetic energy change for the particles) then you must wait until an equilibrium is re-established before the system has a well defined temperature.  When the energy of the system is re-distributed and the principles like the equi-partition of energy apply, it is very likely that the average k.e. of the particles will have increased.   (I said "likely" not guaranteed to always happen, changes of state might be one example where you can put energy into a system but there is no change in temperature).

Speculation about changes of state:
The actual shared property (temperature) is the average energy per degree of freedom.

[Taken from  https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4865254/  .  An article originally suggested by Bored Chemist ]

    How they (the community of scientists using the new definitions of the Kelvin scale) consider or determine temperature for systems that can show a change of state  (e.g. from gas to liquid)  should be interesting to see.  I haven't had time to look at this yet.  However, if the kinetic theory they use allows the gas to have, lets say 5 degrees of freedom (I chose 5 because that's easily explained by the models we have for a diatomic gas), while the liquid state only supports 4 degrees of freedom (which is reasonable because we assume particles in a liquid state have less freedom of movement and therefore less parameters describing their energy) then you can see that is possible for the system to have more total energy in the gas phase than the liquid phase but the   energy per degree of freedom,  i.e. the "temperature" can remain the same.  We have a possible explanation for latent heat, the system loses degrees of freedom during phase changes.
   Let's make it clear that I haven't had time to look at this yet, it just seems reasonable.   The new approach to defining a temperature scale (introduced since 2019) just wasn't in existence when I was studying thermodynamics.

Best Wishes.

8

New Theories / Re: what is temperature?

« on: 02/04/2022 22:30:34 »

Hi.

Dangerous is for Grizzly bears. Misleading implies intentional fraud.

    It wasn't my intention to be offensive.  Please don't assume it was.   What you said might confuse and in that sense "mislead" others like the OP who asked the question "what is temperature?"  I don't think you're trying to commit fraud or anything like that.

Best Wishes.

9

Just Chat! / Re: What is your main area of interest or expertise?

« on: 01/04/2022 14:49:39 »

Hi.

   I appreciate there are only about a dozen regulars but getting just three replies isn't much.  I wonder if you I can prompt another regular or two by just speculating on what their area of expertise is.  As before, I'm not going to speculate on anyone who isn't a moderator.

Colin2B   ---->   I'm deeply suspicious this is someone who is mentioned on (let's say near the top of) the "about" page for this website and seems to speak on the podcasts quite often.   They have claimed sailing interests but I wouldn't have thought they go racing in a sailing dinghy.  They've got a little bit of style so it might be a yacht of wooden construction and not the modern fibre glass rubbish.

ChiralSPO   --->  Physical Chemistry seems to be their specialty.  Fairly sure they have already declared using a model based on QM to predict the properties of elements as being one of their own research areas.

Evan_au  --->  There was a Naked Astronomy person from Australia.   However, they spend quite a lot of time on the Biology questions, which forces careful consideration.  I don't know that much about the Australian education system but I reckon they would have studied a broad range of science at University.   Their writing has a generally professional and formal style.  I'd say they've had some experience writing science articles for the public, perhaps as a journalist for some newspaper or someone running a science museum, something like that.   I suppose writing for the Naked Astronomy podcasts would fit that description.

Best Wishes.

10

New Theories / Re: what is temperature?

« on: 01/04/2022 01:46:55 »

Hi.

This might be helpful.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4865254/

   That is interesting and I've only scanned through it so far.  I'll read more later.
   As far as I can see the latest revisions to the Kelvin temperature scale were made in 2019 and that paper or article was written in 2016.  I think they (the article referenced) are talking about the modifications that were about to bring the standard to what it is now.

    I think there's a mis-print in the article (first paragraph under the title Gas Thermometry) :

Gas thermometry relies on the statistical-mechanical connection between the three kinetic degrees of freedom of an ideal gas and thermodynamic temperature: ½ m〈v²〉 = 3k_BT, where m is the mass of an atom and 〈v²〉 is the mean-square velocity of the atom.

   I'm fairly sure  that should be   ½ m〈v²〉 = (3/2) . k_BT.

   Anyway, if you (@Hamdani) are really interested in the best way to define temperature it might be well worth having a look at that article and the simpler overview of the situation in Wikipedia.

It (the triple point of water) (together with the definition of absolute zero) is the only fixed point on the whole of the Kelvin temperature scale.

   The latest definition of the Kelvin scale for temperature won't require the triple point of water as a fixed point.
Since May 2019, that value (the triple point of water) has not been fixed by definition but is to be measured through microscopic phenomena

[Quote from Wikipedia]

    Even absolute zero,  0 K,  doesn't really fit the description of what we would have called a "fixed point" in the old days.  No experiment has to be done or object set up to that temperature to calibrate the scale.   That value just falls out naturally from the theoretical definition of temperature from statistical mechanics.  You also have no choice about what numerical value you assign to this temperature,  it couldn't be -10  or  +10 because it's got to be proportional to the average kinetic energy of particles (which is supposed to be nothing).   
    You don't really have any freedom to set any fixed points and corresponding numerical values at those fixed points for the temperature.   For example, if you use the equation   ½ m〈v²〉 = (3/2) . k_BT,  to define temperature, T,  then the only way you can adjust the value of T for some pre-determined reference situation (like the average k.e. of the gas particles in an equilibrium mixture of ice, liquid and gas water, i.e. water at its triple point)  would be by changing the Boltzman constant.
    If I've read the article @Bored chemist  presented correctly,  the exact value of the Boltzman constant wasn't going to be fixed until 2018 (2 years after the paper was printed).  With the new value fixed, the triple point of water would be close to 273.16 Kelvin but there was no expectation that it would be exactly that value.

- - - - - - -
   There we go...  . I had no idea the way we think about temperature has changed that much and this recently.
Thanks again for the article @Bored chemist .

Best Wishes.

11

Physiology & Medicine / Re: Is sex better in the morning or evening?

« on: 24/03/2022 17:01:58 »

Dear Sir/madam,

This is just awful.
   Other forums would have removed the thread or moved it to their "Just Chat" or "Just Offensive" section.

   If you're going to answer a question like this, then sure you can have some humour but you should do it in the general background of discussing science.
     There are nocturnal / diurnal hormone flucations, there can be better times of the day for fertility.  There may be psychological effects etc. etc.

Yours faithfully,
   Disgusted of the British Isles.
(Not that it matters - but I'm also going to boycott this forum for a day and use Science Forums instead).

12

Plant Sciences, Zoology & Evolution / Re: How does natural selection work?

« on: 18/03/2022 21:25:35 »

Hi.

    The main details for natural selection have already been presented.    Most DNA alterations are just random mutations  etc.

    I haven't studied Biology for years but this was one of the more "funky" or convention-defying contemporary ideas that was discussed back in the day.   It's useful to suggest that your DNA doesn't always dictate what you do, sometimes it can be the other way round....

    There's a small chance that DNA transcription to produce proteins isn't always a one-way process.   There are viruses with RNA reverse transcriptase enzymes so that RNA can be converted to DNA.   This opens the door to the possibility that information may sometimes flow in that direction from RNA to DNA which might then get incorporated into the nuclear DNA of the cell.
     Lets take a simple model:   A host organism has a physically demanding environment in which it must survive and to do this it is using a lot of muscle and this exercise is building more muscle in the usual way.   This means there is a lot of RNA in the cytoplasm that encodes for muscle proteins.   If a virus gets on board, some of that RNA can be reverse transcripted into DNA which could end up getting incorporated into the nuclear DNA stores.  The host organism now has two (or more) copies of the gene encoding for the muscle protein, so they are likely to get a higher transcription rate of RNA encoding for that protein from then onward.  This means there has been a permanent change in the amount of muscle protein the organism will synthesize - but, more importantly, if that change has occurred in the germ cells of the organism then this change is passed on to future generations.
    The effect is minor, if it happens at all, but it's interesting.   It's the possibility that the actions of the organism during its life might influence the DNA.   The whole set of circumstances falling into place, like accidental incorporation into the nuclear DNA, would only be a one-in-a-million chance but that is precisely the sort of scale we're talking about for evolution over many years.  It probably isn't going to be the main route for evolution but it could be a boost to the process.

Best Wishes.

13

General Science / Re: Can matter be created and destroyed?

« on: 11/03/2022 03:03:08 »

Hi.

   Annhiliation of matter does happen.   It involves a particle and it's corresponding anti-particle interacting.

There are other threads already discussing the geneal ideas.   Here's a recent one:
https://www.thenakedscientists.com/forum/index.php?topic=83267.msg657398#msg657398

There are tons of other references.   Here's one for Wikipedia:
https://en.wikipedia.org/wiki/Annihilation

More generally there are many ways in which some rest mass can be lost.   Nuclear reactions often have products with less rest mass than the intial atoms.   The difference in rest mass being liberated as an equivalent amount of energy,  for example gamma rays or high velocity Beta particles carrying energy away.

If there's something more specific you wanted to know about, write more on this thread.   Although.... in general it's a good idea to focus threads in the main sections of the forum around a question.  Also you should expect to have more responses based on science and not on speculation or science fiction.   If you're just seeking more general discussion including lots of science fiction ideas, it might be better to move the discussion to the "Just Chat" section  or something similar.    (I'm not a moderator, I'm just guiding you to what seems sensible and usually recommended in my limited experience).

Best Wishes.


LATE EDITING:  The other thread I linked to was "Questions that antimatter to me".    The better one to link to would have been the follow-up which was titled  "Even more questions that antimatter to me",  sorry.

14

Physics, Astronomy & Cosmology / Re: Why don't gamma rays have a higher speed?

« on: 20/02/2022 15:35:57 »

Hi again.

....and the idea of a massless particle transferring momentum really does confuse people.

     I'm definitely with you, or at least sympathetic to, a lot of what you've said alancalverd.    It connects with something I was trying to get together and tidy up to make a post.   I'm just going to post it now, scruffy as it is...

    I would like to target something that @evan_au  mentioned first but lots of other people have also added to.

Massless particles (eg photons or the hypothetical graviton) which carry energy can only travel at "c"
(ES adjusted the spelling of the word "the")

    Go for it, evan_au, or anyone else reading this.  Present your proof or provide links to a reference etc.   

I've seen a lot of partial proofs and plausibility arguments but nothing that I'm completely happy with.   It doesn't mean it's wrong, I don't know everything - I'm just old enough and stubborn enough to question every little detail these days.

    A lot of arguments will start from this equation   

E²  =  m²c⁴ + p²c²       

[Eqn.1]

(with the usual meanings,   p = 3-momentum,  m = rest mass   etc.)

    ....because I would stop you there and ask exactly how you derived that equation to start with.    For massive particles that equation is easily obtained by considering the 4-momentum, P^μ, of a particle.    It's simple enough since a 4-velocity, U^μ = is well defined for particles with timelike paths and then we can define P^μ = m U^μ.   

      However, for particles that follow null paths we can't use the same tricks:  The 4-velocity is not defined since =  0   on a null path.   It seems common to just introduce a new definition for the 4-momentum of a photon,

P^μ  =   ( E/c   ,   V_xE/c² ,  V_yE/c² ,  V_zE/c² )      with  V_x = x-component of velocity (w.r.t. co-ordinate time) for the photon etc.
   ... but the question is how did you come to decide on putting those particular components into the 4-momentum? 

   The only "proof" or plasuibility argument I've ever seen for that definition of the 4-momentum of a photon is based on using some additional infomation about the behaviour of photons.  Something that you could observe in experiments on photons but is not otherwise obviously true for all massless particles.
    For example, assuming straight away that the momentum and energy are related  by  p = E/c.   Alternatively, you can derive that form for the 4-momentum under the assumption that the 3-momentum  in any spatial direction will be proportional to the velocity (w.r.t. co-ordinate time t and not proper time Tau) of the particle in that direction.   Another "proof" I've seen utilises a 4-wave vector for the photon (which assumed the photon had wave-like properties and propagated at speed c to begin with).   Anyway, all of these things are extra pieces of information you could obtain by experiment on light but I don't see why they're necessarily true for arbitary massless particles.

Best Wishes.

LATE EDITING:   This article,  https://www.wtamu.edu/~cbaird/sq/2014/04/01/light-has-no-mass-so-it-also-has-no-energy-according-to-einstein-but-how-can-sunlight-warm-the-earth-without-energy/
is quite a good one, it's easily readable by non-specialists, if anyone wants a general guide for why massless particles travel at the speed c.   It's typical of many Popular Science articles where E² = m²c⁴ + p²c²  [Eqn 1]  is the basis or starting point for the  argument.

15

General Science / Re: What is Quantum Tunnelling and how does Quantum Tunnelling work?

« on: 01/02/2022 19:41:13 »

Hi again.

   So let's see if there are any questions left over from Aeris.

Let's say, hypothetically speaking, a macroscopic object like a table or a chair undergoes Quantum Tunneling and essentially teleports from one location to another while I'm still looking at it. What would that look like? would the object just pop right out of and then immediately back into existence in the blink of an eye?

    Let's just look at a microscopic objects for a moment (before we speculate and generalise to macroscopic objects).
We're going to consider the typical situation with a square barrier potential between two regions of exactly the same potential  (region I and region III).
    Here's the diagram again:


We're going to fire a particle toward the barrier (from region I) which has Energy, E < V₀  exactly as before and we've already discussed the wave function and shown animated diagrams of this earlier.

    Firstly remember that the wave function is not the particle,  there isn't anything with properties like a particle until you go looking for it (make an observation to locate the particle).   The wave function just indicates the probability of finding the particle at a particular place.
   If you go back through the earier discussions and animated diagram you'll notice that the wave function isn't always 0 inside the barrier.   There is some time when the particle could be found there.

   Now this is where various sources of information (not just PopSci) are going to mess things up or mis-represent what seems to be shown by the mathematics.
   The amount of time over which the particle could be found "in the barrier" is controversial.   Some sources are going to tell you that it "instantly appears" on the other side of the barrier or just that "it will never be found in the barrier" etc.   I don't wish to use too much bad language but this is clearly utter bolderdash.

   Just go back and look at the animated diagram carefully if you want to, instead of doing the mathematics.   There are times when the real and imaginary components of the wave function are non-zero at a fixed x position inside the barrier.  So the square of the modulus of the wave function is not 0 at that time, this is the probability of finding the particle at that given point x in space and given time t.  I'm labouring the point here but we just need to be clear that the probability of finding the particle at that position x in the barrier is not 0 for some of the time.  The probability of finding of the particle in the barrier does not drop to 0 until some finite non-zero time has elapsed after the incident wave packet hit the barrier.   You will find many articles, some textbooks and countless discussions on Quora and similar websites that discuss qunatum tunneling as if it's instant.   Some of them are quite interesting and authoritative and discuss problems like the apparent breech of the speed of light   (If the particle could move from one side of a barrier of thickness, d>0, to the other side instantly than that is faster than light speed travel).   They're all good in their own way except that the fundamental premise they were based on was bollderdash.

    Anyway, the mathematics never implied that tunneling would be instant.

    Here's one article (from Scientific American) that describes an attempt to actually measure how long it takes a particle to tunnel through a barrier:
https://www.scientificamerican.com/article/quantum-tunneling-is-not-instantaneous-physicists-show/
   The details aren't too important, we just need to note that there may actually be some way of observing or measuring tunneling time in practice and it does not seem to be instantaneous.   It can be that the particle spends some time in the barrier.

   Anyway, we can now try to generalise this to a macroscopic scale:  Basically, there's no reason to assume that the quantum tunneling of your chair from region I to region III would be "instant".   It's not going to go "poof" from here and just "poof" into existance over there.  It is very likely to spend some time "in the barrier" between the two places.   I'm not sure what this would "look like" but it could be a lot less impressive than teleporting.   Given a piece of space between the start and end point (i.e. a piece of space in the barrier), then the chair could be found there for a short amount of time during the tunneling.   I really don't know how to paraphrase this, applying QM to macroscopic objects is always just going to be speculation anyway.   It might look like the chair was just moving from the start position to the end position and occupies at least some but possibly all of the positions between those two points in passing.  A whole lot less impressive than teleportation, I'm sorry to dissapoint you.

Another long, post, sorry.    In summary the only important thing is that Quantum Tunneling isn't instant and may not look like teleportation.

Best Wishes.

LATE EDITING:   I probably should make it clear that the time it takes for a particle to tunnel through a barrier is controversial.   This was stated originally but it's worth stating again.  You should probably make your own investigations.  Personally, I'm sticking to what the Mathematics shows.

16

General Science / Re: What is Quantum Tunnelling and how does Quantum Tunnelling work?

« on: 01/02/2022 12:29:16 »

Hi.

If you state it as barrier potential, it's unclear what the dimension is.

    Sadly this terminology is used in the literature.  The "potential" is shown on the diagram in units of energy (for example Joules not Volts).  So it is the potential energy that the particle would have if it was physically located there.
      It's clear what you meant by the barrier energy but having just received the long post from Colin2B it would be wise to avoid saying anything that looks like E and carries the name "Energy" since that is usually reserved for the Energy of the particle.   The Energy of the particle is just called "the Energy", it almost always has the symbol E and it would be the total energy = potential energy + kinetic energy  under classical mechanics.

Best Wishes.

       

17

General Science / Re: What is Quantum Tunnelling and how does Quantum Tunnelling work?

« on: 31/01/2022 14:04:59 »

Hi.

If the barrier energy is increased to 40, will there be a portion of the wave packet passes through the barrier?

   Yes.    (In line with the previous post, it's the barriers potential you are talking about and not the energy of the particle).    The transmitted wave packet will have a smaller amplitude.

     http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html
Is actually a very useful link.   (It was found by Colin2B and used in his earlier post).   It offers an online calculator to determine transmission coefficients when a particle approaches a barrier of given height and width.
    The transmission coefficient  is approximately the ratio of (the square of) the amplitude of the transmitted wave to the (square of the) amplitude of the incident wave.   This decreases as the potential barrier height increases.

Best Wishes.

18

General Science / Re: What is Quantum Tunnelling and how does Quantum Tunnelling work?

« on: 31/01/2022 13:55:31 »

Hi again.

  This is going to get difficult for Aeris (or anyone else) to make sense of.   I'll spend a moment to try and pull the comments from Halc, Clin2B and myself together a bit.

Several of us picked this quote to represent something Aeris asked:

So it's kinda like the whole "Heat only moves from hot to cold" thing? The origin point of whatever is about to Quantum Tunnel needs to be in a state of higher energy than the end point? Is this right?

       However,  Halc  was always talking about the potential  when he first discussed the situation.    In classical mechanics (not qunatum mechanics) the potential energy is only one contribution to the total energy of the particle.
       Eternal Student was also careful to talk about the potential.
       Aeris was the only one to say "Energy" but most of us knew that they were talking only about the potential.
       Colin2B has spoken about the (total) Energy, E, of the particle.

Here's the overall summary:
   Halc suggested the tunneling would occurr from region I (where the particle started from) to region III (where the particle ends up) only if  the potential   in region I was higher than region III.
   Eternal Student (E.S. or "me") disagreed with that.
   E.S. also suggested that Halc's comment would be reasonable on a statistical basis  (so Halc doesn't have to feel too bad about suggesting it in the first place).
    Colin2B hasn't commented on that directly but once again has shown a diagram and hyperlink where the potentials in the two regions are exactly the same.

    Colin2B has commented that the (total) Energy, E, of the particle remains constant on both sides of the barrier.
    E.S. agrees with that and would actually strengthen it:   The Energy remains constant wherever the particle is found.   Even if you found it inside the barrier it would still hold.   

(This is going to take a moment to explain since a classical mechanics view of things would make it impossible for the potential V that the particle is experiencing at some place and time to be greater than the total energy of the system.  It would look like the kinetic energy had to be some negative value in order for the total Energy, E, to be less than V.  This might be a topic for later discussion since Aeris has already asked about conservation laws like the conservation of energy but for now I'm going to walk away and leave it there.   The Energy, E, remains constant).

Best Wishes.     

19

General Science / Re: What is Quantum Tunnelling and how does Quantum Tunnelling work?

« on: 30/01/2022 03:46:04 »

Hi.

Fascinating. I wonder though, what exactly constitutes as a potential barrier in this regard?

     Just about anything.    Quantum mechanics describes a situation where there is a wave function, usually written Ψ and a "potential" usually written V.   The potential is just a function of position and time.   If you like mathematics V = V(x,t).   That's all there is to it.   The potential is what will create your barrier.
   As Halc stated, the simplest systems to model involve something like a charged particle in an electric field, so V(x,t) is just the electrical potential at position x and time t.    However the potential can be anything you like,   it could be a hybrid of a potential due to the strong nuclear force plus some potential due to some electric field potential... whatever you're modelling at the time.   I've not seen gravitational potential being used but I don't see why it couldn't be.  The source of the potential hardly matters, it's just "the potential" which exists throughout space and time and the particle would experience this potential if it was there (and at that time).
      If we take your chair example and simplify the situation so that we are only asking the electrons in it to tunnell to somewhere else, then the empty space just to the left of the chair is "the barrier".   The chair has some electrons bound to its atoms.  Assuming the nucleii don't move, then we can find some approximation for the potential those electrons experience, most of it will be an elctrostatic potential.   The electrons have a certain energy and will usually be found in certain places where the potential is low, this will be around the atoms in the usual electron clouds.   (The exact solution is going to be impossible to calculate but very roughly we should get a solution like the sort of thing chemists draw for electron orbitals in molecules).   That potential has some value everywhere in space (and time) and indeed there is some small chance of finding a few electrons well away from their usual orbital cloud positions.     The usually empty space just to the left of the chair is a region of such high potential that a typical electron should rarely be found there.   Under classical (not qunatum) mechanics the electron should never be there because it would need an energy that is greater than it actually has (under classical mechanics the electron has effectively escaped from the atom which would demand that it has been given the full ionisation energy normally due for that atom and that electron starting from it's ususal orbital).   Anyway, the point is that the empty space just left of the chair is "the barrier", it's the region where the potential exceeds the energy, E, that is nominally available for the electron(s).
   
      Now, let's go back to considering a simpler system where there is only one particle and some potential V(x,t).   The system has a certain energy, E,  and in classical mechanics there would be no way that the particle you are modelling could be found or should exist in any place or time where the potential exceeds the Energy, E, that was available.   It simply does not have the energy to get there or exist there.   In Quantum mechanics things are a bit different:  The particle that is being modelled with the wave function, ψ(x,t),  can sometimes be found in a region where the potential exceeds the nominal amount of Energy, E, that was available.     Any region where the potential exceeds the energy, E, is called a "barrier".  In classical mechanics it would be very much a hard barrier, there's no way the particle can get beyond it or through it.   In Quantum mechanics a barrier has less effect on the wave function.   Instead of forcing the wave function all the way down to absolute 0 (which is saying "there's no chance of finding the particle here"),  the wave function will usually just fall off rapidly as you move into the barrier, typically you see an exponential fall off with distance and the wave function is often said to become "evanescent".
    An ideal or infinite barrier is one where the potential becomes infinite in value and it's a discontinuous jump (it's finite all the way for x < barrier location and then  precisely at x = barrier location  the potential jumps straight up to ∞ and stays there for x> barrier location).   There's probably no such barrier in the real world but it's a useful idea for the mathematics.   For an ideal or infinite barrier, the wave function is forced to absolute 0 value the moment you enter the barrier.  Only these ideal or infinite barriers act like true hard barriers in Quantum Mechanics.   Anyway, there's not thought to be any real life situation where a potential field shows behaviour like this, the potential may become very large and over a short amount of distance but it doesn't ever go vertical (making a discontinuous jump over 0 distance) and it never reaches an infinite value, just a large but finite value.   So... as far as quantum mechanics is concerned that's not a hard barrier, the particle can be found inisde that region (with a probability that falls off exponentialy as you progress further into the barrier).
    Quantum tunnelling is not just about finding a particle inside a region that should be a barrier.   Usually you have a situation where there are two regions of fairly low potential.   Two patches of space where the particle could easily exist since the system has Energy, E, greater than that potential.  If the barrier between these two regions is only finite (non ideal) then the evanescent wave function (recall that it's falling off exponentially with distance) will reach the other low potential region.  Once it get's there, it's fine it doesn't decay anymore.  The solution for the wave equation should give you perfectly satisfactory stationary states etc.    This is the sort of thing that happens when a particle is created in the first region and fired toward a barrier but there is a perfectly low enough potential region on the other side of the barrier.   What you get on the other side of the barrier is a perfectly stable wave function but just with much lower amplitude then the incident wave function that was sent into the barrier (most of the wave function is simply reflected back into the original patch - but some gets through).

    Wiki has a few good diagrams and animations for this:




    The more general situation that is considered to be Quantum tunneling doesn't involve "firing" a localised quantum packet at a barrier.    Instead you just have a situation where there is a particle with energy, E, and you place no other restrictions on it (don't try to fire it at the barrier, just leave it to exist wherever it can and solve the wave equations to see what you get).   The potential will represent a situation where there are two patches of space with a reasonably low potential and some region between them which is the barrier.  This barrier can even be an ideal or infinte barrier.  Anyway, the solutions you get indicate that the wave function will be non-zero (just to clarify it has time dependance so it might be 0 some of the time at a given place but it's not 0 all the time in that place) in both regions of low potential.   This indicates that there is some chance of finding the particle in either region.   If the barrier was ideal (an infinite square wall barrier) then the chances of finding the particle anywhere between those two regions is always 0,  yet somehow the particle can be found in either of the two regions.   You could find it in the first region today but the second region tomorrow etc.  It's NEVER going to be found in between the two, just in one or the other.   In this respect if we try to explain this as if the wave was a physical particle then we can imagine that the particle tunnels between the two regions in some amazing way,  it seems capable of disappearing from one and re-appearing in the other.    (However, it's never actually going to do that while you are observing the particle, under observation it stays in region 1 or region 2, it isn't going to relocate until the system is left unobserved.  This is a seperate issue, I hardly know why I mentioned it except that everyone always wants to "see" the thing disappear and re-appear - but this thing is more like a magician doing something behind the curtain, you can't see it happening but when it's done the doves have moved from their original cage to other cage).

    That's about all I know about Quantum mechanics and tunnelling.  I can put some equations down but I don't think anyone will want to see that.  Anyway they're in books and they explain it better than I can.

    I can already see that Wikipedia has sections about types of behaviour that they consider to be "Qunatum tunnelling" that I know nothing about.

This is a long post,  I'm signing off for now.

Best Wishes.

20

Physics, Astronomy & Cosmology / Re: Cant decide on a physics project

« on: 22/01/2022 23:21:18 »

Hi.

   Firstly, I know very little about Physics projects.   However, I would have thought people would like to know some of the following things:

1.   What is this project for?     Your own personal interest,  a school project   or something else.

2.   If it is for some academic establishment,  what sort of level or age is it intended for?
    Also if it ties in with any standard qualification could you indicate what that qualification is.

3.    It's nearly always best if you choose what interests you.  You'll do much better if it interests you.
       No matter what area you choose, there should always be the potential to develop the project to a level that is appropriate and in a direction that will keep you busy and interested.     For example, even if you're only interested in studying the inside of a ping-pong ball, there are ways you could extend this:   Would a static charge on the ball influence the interior of the ball?   Would the behaviour of the ball change if the interior wasn't air but was replaced with some other fluid?    etc.   

Best Wishes.