Throwing a tennis ball through a wall

[vanesch]

A QM description of a tennis ball is not and cannot be made "simple". You are just applying a set of rules to where it was never meant to be applied. You might as well say "OK, I am now moving greater than c at a time 20 million years BEFORE the Big Bang. What do I see?" That is a mind game too, but it doesn't mean it has any reasonable answer.

There's a difference: the last proposition is totally meaningless, even in principle, but I object to your "cannot be made simple".
After all, when you use a quantum description of, say, a Calcium atom, then you take it that the nucleus is a single point particle, with just an x, y, z continuous position degrees of freedom, and eventually some discrete spin degrees of freedom.
Now, any nuclear physicist will tell you that that is a rather naive view of a nucleus, and that you have a complicated system in there, with complicated interactions between neutrons, protons and pions. And if you turn to a particle physicist, he will tell you that a proton made up of 3 quarks is a rather naive model, and that you have miriads of degrees of freedom in there, with gluons, a sea of virtual quarks and all that.

Nevertheless, for the purposes of atomic physics, it is sufficient to put all these internal degrees of freedom under the carpet, and just stick to some overall kinetical degrees of freedom: the position of the center of mass, and eventually a spin degree of freedom.
So I don't see why I'm not entitled to do the same to the object "tennisball". I abstract away its internal degrees of freedom (the atoms, the electrons, whatever), and I just keep some overall kinetical degrees of freedom: the position of its center of mass, and eventually a spin degree of freedom (which can take here very high values, to start looking like a classically spinning object).

I know that there is a difference with the nucleus case: we could say that we don't need the nuclear degrees of freedom because essentially only the ground state matters in the energy range we're exploring in atomic physics, because nuclear exitations are on much higher energy levels
So we limit ourselves to the "neighbourhood" of the nuclear grond state in the Hilbert space that does describe the nuclear degrees of freedom.
You could correctly argue that this is NOT the case for the ball-wall interaction, which could eventually be considered to be of the same order of magnitude as the exitations of the internal degrees of freedom. Granted. I already alluded to this before. So we're making errors here: we're excluding degrees of freedom which may have their say. In other words, we're making a very rough approximation... But that was granted!

 

[ZapperZ]

There's a difference: the last proposition is totally meaningless, even in principle, but I object to your "cannot be made simple".
After all, when you use a quantum description of, say, a Calcium atom, then you take it that the nucleus is a single point particle, with just an x, y, z continuous position degrees of freedom, and eventually some discrete spin degrees of freedom.
Now, any nuclear physicist will tell you that that is a rather naive view of a nucleus, and that you have a complicated system in there, with complicated interactions between neutrons, protons and pions. And if you turn to a particle physicist, he will tell you that a proton made up of 3 quarks is a rather naive model, and that you have miriads of degrees of freedom in there, with gluons, a sea of virtual quarks and all that.

But there is a difference here and with a tennis ball. A "nucleus" and an "atom" ARE quantum particles already! I don't have to put any effort to detect such properties. A nucleus and an atom are not "tennis balls".

I can even go a step further, and you should have seen me mentioned it - 10^11 electrons! I can consider that many electrons as a "quantum particle" when they are in a superfluid. Again, it has nothing to do with quantity. However, I can do that because I have the evidence that all of these are quantum particles, and I have a theoretical description on why they should behave quantum mechanically.

I have no such thing for a tennis ball. You cannot make it quantum mechanically simple because there have been zero instances where you can theoretically cause it to be simple, and empirically show it. it is different!

Zz.

 

[selfAdjoint]

Excuse me zapper, but aren't you rather contradicting yourself here?

In post #47 you said

Forget about tennis balls. I would be happy just to see a buckyball tunnels through something. We already can see that it CAN behave as a quantum particle under a certain condition.

But in #52 you're back to

But there is a difference here and with a tennis ball. A "nucleus" and an "atom" ARE quantum particles already! I don't have to put any effort to detect such properties. A nucleus and an atom are not "tennis balls".

There are big fairly complicated molecules (buckeyballs) that behave quantally and which we could get at least a sensible estimate of the probability of tunnelling through a given barrier. Is thiis or isn't it enough to show that the calculation problem for macroscopic objects, although technically beyond our present resources is not IN PRINCIPLE impossible?

 

[ZapperZ]

Excuse me zapper, but aren't you rather contradicting yourself here?

In post #47 you said

But in #52 you're back to
There are big fairly complicated molecules (buckeyballs) that behave quantally and which we could get at least a sensible estimate of the probability of tunnelling through a given barrier. Is thiis or isn't it enough to show that the calculation problem for macroscopic objects, although technically beyond our present resources is not IN PRINCIPLE impossible?

Ah, but that's the whole point I'm trying to get across here.

The ability of buckyballs to produce interference effect is under an extremely RARE condition that we have to put them in. We must have ALL parts of the buckyball in coherence with each other so that the whole buckyball is really ONE single quantum object! It is then a large object but with a coherent wavefunction.

This is very much like a supercurrent. You have the 10^11 electrons, but they are all in ONE single coherent state. One could easily consider the whole glob of electron as a single object, the same way one considers a pair of entangled particle as being a single "macro" particle.

A buckyball at room temperature without being put in that state would not produce interference patterns, no matter how much you coax it. And note, tunneling processes are more difficult to achieve than the 2-slit interference experiments. So there's an extra whammy there.

But at least I can see how one can make each part of the buckyball coherent with each other. I have no idea how one does that to a tennis ball.

Zz.

 

[baryon]

Is it correct to say that there is no chance that any of the particles from the book would ever tunnel through the table and vice versa because neither object is radioactive? Pardon me for mentioning the weak force, but isn't that the force that dissolves the strong atomic bonds in nuclei? Based on what is known about the weak force, is it truly probable that this force would actually interact on the book or the table?
I'm not asking if the book could ever be driiven through the table or vice versa because this may be possible in a lab.
Also, I'm aware of Everett's theory but I do not believe we need introduce this to the conversation.

 

[Locrian]

Why is this thread back? In a thread prior to this, conversing with reductionists, I was presented with rocketship shoes and a diamond chair. In this thread I was presented with a point-like tennisball, which of course would fail you in a situation as complicated as a bumpy corner. I was actually hoping such inanity could slide its way into history.

 

[WhyIsItSo]

...but a macroscopic object are compose of the "quantum" particles.there must be a relationship between them. there must be a possibility that it will tunnel through the wall is we can give the ball a motion so that all particles of the ball will become a "wave-particle" with that energy...

But would it still be a tennis ball?

 

[jackle]

Inanity

Why is this thread back?...I was actually hoping such inanity could slide its way into history.

Asking questions (even meaningless ones) is how we learn. I don't see a problem with it.

 

[jackle]

This is probably irrelevant but I seem to remember all sorts of crazy things in my degree course. Massless springs come to mind.

 

[Farsight]

Vanesch, Zapper, anybody: how about a small "tennis ball" that was actually a Bose-Einstein condensate. Could that tunnel though a barrier?

 

[Crosstalk]

I was thinking that because the tennis ball is made of protrons, electrons, neutrons, and other things that can quantum tunnel by theirselves, there would be a chance (Amasingly small, though, although I can't give an estimate I would say on the order of 10^-100000000000000000) that all the particles in the ball would tunnel at once, and go right through the wall. There might also be an even smaller chance that that would happen and the particles would all rearrange themselves back into the ball, but my way off estimate for the probability of that would be about 10^-10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, but this estimate is probably higher than the actual chances.

 

[jackle]

Asking questions (even meaningless ones) is how we learn. I don't see a problem with it.

It has occurred to me that having said this, I might be opening the flood gates for people to assert all kinds of silly theories. I do approve of monitoring and checking threads in the interest of quality, but I can't seem to find the rules for this forum.

That aside, I'd like to think that nobody should be afraid to ask a 'silly' question.

In general I would appreciate an estimate more than the last post, when it is presented with the correct rigour. eg. What assumptions have we made? Where did the number 10^-(10^300 and something)? come from? Why do you think this? I think this thread is being ignored because it is too silly at the moment.

My guess is that there is always a chance for our tennis ball but that any estimate will be hopelessly wrong, even when attempting it seriously. There are probably many ways the tunnelling 'could' happen and the trick would be to estimate the most likely way as an approximation to the real thing.

 

[jackle]

the most 'likely' way

 

[jackle]

the 'real' thing

 

[SizarieldoR]

Sure. In theory it is possible.

 

[jackle]

...a 'chance'...

 

[Chaos' lil bro Order]

Sometimes I wonder if people take the ideas coming out of QM and incorrectly apply them to macroscopic objects for the sake of their own philosophical confirmation. Then I remember that I did exactly this earlier in this thread and ZapperZ showed my that I was wrong to do so. Then I see later in this thread people making the exact same mistake. I would point those people to Zapper's earlier comments to clear up any confusion they may be having.

 

[Chaos' lil bro Order]

Farsight

How could a tennis ball actually be a Bose-Einstein condensate? Isn't that too large of a hypothetical to hold any illustrative meaning? I'm very ignorant in this area, but is there any way, even theoretically that a macroscopic tennis ball could be treated as a Bose-Einstein condensate?

 

[Greenman]

I do think we must take QM seriously and classical mechanics as a prejiduce. The wavefunction does describe reality. However it is entirely logical that the 'predictions' of QM are 100% true yet not completely describe the entire system even in it's domain.

Could you elaborate on this please?

or could someone else explain what he means by that.

thanks :D

 

[vanesch]

Could you elaborate on this please?

or could someone else explain what he means by that.

thanks :D

Note that this thread was 3 years old...

 

[Haelfix]

As I understand quantum mechanics, my answer would be that indeed it is possible for a book to tunnel through a table. Just that the odds of this happening are like 10^10^10^10^10^10^...^10^-(some large number). So astronomically small relative to the timescales of both the universe and myriad other events from occurring, that its essentially for all physical purposes zero.

When I was taking my competency exam in grad school, one of the past questions was to derive an order of magnitude estimate for a molecule of hydrogen sitting on a table to tunnel through a 1 fm graphene wall with some given density. The accepted answer was ridiculously tiny, but also a notoriously controversial question b/c depending on what effects you take into account (eg what potential wall do you want to ansatz), the answer changes by hundreds of orders of magnitude!

 

[bjacoby]

As I understand quantum mechanics, my answer would be that indeed it is possible for a book to tunnel through a table. Just that the odds of this happening are like 10^10^10^10^10^10^...^10^-(some large number). So astronomically small relative to the timescales of both the universe and myriad other events from occurring, that its essentially for all physical purposes zero.

When I was taking my competency exam in grad school, one of the past questions was to derive an order of magnitude estimate for a molecule of hydrogen sitting on a table to tunnel through a 1 fm graphene wall with some given density. The accepted answer was ridiculously tiny, but also a notoriously controversial question b/c depending on what effects you take into account (eg what potential wall do you want to ansatz), the answer changes by hundreds of orders of magnitude!

Your answer, though reflective of common physics assertions, simply shows how far physics has moved away from science and toward magic and mysticism. Part of the problem is that mathematics has become worshiped as a substitute for science. Mathematics is regarded as more real than reality. Physicists commonly talk about fields and point charges and imaginary numbers as if these things actually has some real meaning in a physical world. Sorry mathematics is abstract. It does not need to agree with experiment. It only needs to be some self-consistent mental concoction.

So given such a mindset, we find hideous statements being passed off as physics. These include the mathematical extensions of QM such as tennis balls tunneling through walls and bowling balls being reflected from a table edge. I happen to know a physicist who actually calculated the bowling ball reflection. Sorry. You can roll them at a table edge FOREVER and they'll never ever come back. Why? Because the REAL ACTUAL energy functions at the edge of the table are not the ideal mathematical "wells" and "steps" that are so mathematically convenient. They turn out to be nice smooth falling off curves that could never produce a minor reflection let alone one on a bowling ball. Doubtless the same is equally true for real tennis balls.

So the bottom line is that all these stories of the "weirdness" of QM when transferred to daily life make great fodder to impress a class full of freshmen, but the truth is that it's nothing but Hollywood and just as much fiction as their latest movies. So the tunneling of a tennis ball may be "impossible" according to QM mathematical models, but a real tennis ball is different.

Now the question as to whether or not a tennis ball actually CAN pass through a wall under the right conditions (not simple QM statistical ones) is anther question all together. We do know that a wall is mostly empty space. There are hints (but nothing more) in stories and myths that saints and others can develop such an ability. Is it true or Memorex? Well who knows? Fact is one has to suggest that in some manner (but not a straight quantum mechanical one) it actually MAY be possible for a tennis ball to go through a wall, or the temperature in a room to be lowered, or a bowling ball to be reflected from a table edge. And the reason we must allow this possibility is the rule that says saying some physical phenomenon is "impossible" is the mathematical equivalent of saying "I'm a moron". Mathematics can express impossibility by definition. Physics has no such luxury.

I will leave the proof of this last theorem as an exercise for the interested student.

 

[DrChinese]

As I understand quantum mechanics, my answer would be that indeed it is possible for a book to tunnel through a table. Just that the odds of this happening are like 10^10^10^10^10^10^...^10^-(some large number).

Paraphrasing from the movie Dumb and Dumber:

So there IS a chance that I will end up married to Angelina Jolie...

 

[Greenman]

Note that this thread was 3 years old...

So does everything ZapperZ said 3 years ago still stand today?

 

[Greenman]

Does this mean that a tennis ball COULD INDEED go through a wall:

http://www.sciencedaily.com/releases/2009/09/090923151730.htm

 

[ZapperZ]

Does this mean that a tennis ball COULD INDEED go through a wall:

http://www.sciencedaily.com/releases/2009/09/090923151730.htm

The fact that these are josephson junctions under superconducting conditions means nothing to you? Notice my earlier argument that a tennis ball must have ALL parts of it in coherence with each other for the WHOLE tennis ball to tunnel through. So why do you think people go through all that trouble to do most of these experiments in the superconducting state?! I mean, look at the Delft/Stony Brook's Schrodinger Cat-state experiments. Why did they go through all that trouble to do their experiments in the superconducting state? Do you think they could maintain 10^11 particles in a coherent state if they don't go through such lengths?

Moral of the story: preservation of coherence is NOT EASY! Even after just ONE single interaction, that has been shown to be https://www.physicsforums.com/showpost.php?p=1498616&postcount=55"!

Zz.

 

[Greenman]

Sorry, I didn't mean to imply it was proof, but rather that *perhaps*? it is a good indication that quantum tunneling on objects such as a tennis ball have a nonzero *possibility*

In other words, that this is a step forward in favor of it being at least possible.

But on another note, I don't know if you got to read my message, but what do you think of Takagi's book on Macroscopic Quantum Tunneling?

 

[ZapperZ]

Sorry, I didn't mean to imply it was proof, but rather that *perhaps*? it is a good indication that quantum tunneling on objects such as a tennis ball have a nonzero *possibility*

In other words, that this is a step forward in favor of it being at least possible.

There is also a non-zero probability mount Everest could spontaneously appear right in front of you, or that a vase broken into a thousand piece could reassemble back into the original vase when you throw the pieces onto the floor. What's your point? When was the last time you saw this happening?

Arguing that something has a "non-zero" probability doesn't mean it will occur.

Zz.

 

[Greenman]

Touché, but I just wonder why so many intellectual people (people who dedicate their life to the respective field) talk about this phenomenon in such depth -

- personally for me what swayed my opinion was Takagi's book, which is why I'm dieing to know what you think of it...I assume you read it?

 

[ZapperZ]

Touché, but I just wonder why so many intellectual people (people who dedicate their life to the respective field) talk about this phenomenon in such depth -

Who?

If anyone tells you that a tennis ball can tunnel through a wall, ask them to show the experimental evidence for it! After all, the Standard Model predicts the existence of the Higgs. We don't just sit here and accept it. We go and hunt for it! This is why this is called physics, and not philosophy! The empirical component of it is necessary.

A tennis ball is a system that is made up of constituents that are not in coherence with each other! Period! Anyone who thinks that such a non-coherent system can actually tunnel together at the same time with equal probability needs to have his/her head examined, or better yet, needs to do some experimental work to have some connection to reality!

- personally for me what swayed my opinion was Takagi's book, which is why I'm dieing to know what you think of it...I assume you read it?

Nope! It would be foolish of me to review something I haven't read. Why do you hold it in such high regards?

Zz.

 

[Greenman]

It's not that they say a tennis ball can tunnel through a wall, but rather that macroscopic quantum tunneling for coherent objects is mathematically possible.

I suggest you read Takagi's book just to see what you think of it. He describes in detail how macroscopic quantum tunneling can be achieved and then provides possible systems for experimental observation of MQT.

He talks about how coherence can be maintained despite environmental factors and measurement issues.

 

[ZapperZ]

It's not that they say a tennis ball can tunnel through a wall, but rather that macroscopic quantum tunneling for coherent objects is mathematically possible.

Coherent object is a different beast. No one is arguing about that. But you are posting in a thread that are asking about a tennis ball! That is what I'm arguing.

The gymnastics that the Delft/Stony Brook experiments had to go through was to get that big, macro particle (10^11 particles) to be in coherence with each other. This is done! One has also done an interference experiments with buckyballs. Again, at very low temperature so that the thermal effects won't destroy the coherence. This is NOT the point of argument.

I suggest you read Takagi's book just to see what you think of it. He describes in detail how macroscopic quantum tunneling can be achieved and then provides possible systems for experimental observation of MQT.

He talks about how coherence can be maintained despite environmental factors and measurement issues.

But why? I mean, if he's saying that coherent object should be able to tunnel as a coherent unit, I'm not contradicting that! What exactly have I posted in this thread that is inconsistent with this?

Zz.

 

[Greenman]

I guess what I was getting at was that if the environment can be altered so as to have cohesion, then a tennis ball *could* go through a wall.

And Takagi describes how such an environment can be achieved and maintained.

Does this make sense?

 

[ZapperZ]

I guess what I was getting at was that if the environment can be altered so as to have cohesion, then a tennis ball *could* go through a wall.

And Takagi describes how such an environment can be achieved and maintained.

Does this make sense?

No.

Zz.

 

[Greenman]

Ok. I'll try to reiterate:

- Takagi describes how to create a cohesive environment.
- Quantum tunneling may occur at the macroscopic scale in a cohesive environment.

Therefore: If the tennis ball and the wall are in this cohesive environment, then quantum tunneling could occur.

Right?

 

[ZapperZ]

Read what I said as being the difference between physics and philosophy. I'm an experimentalist. You'll notice that all of my arguments are based on facts that have been verified experimentally, not just based on theoretical conjecture, or what is "possible".

Zz.

 

[Greenman]

Well, considering this is a theoretical question asking about the theoretical possibility, isn't it valid to answer with a logically-derived theoretical conjecture?

 

[ernestpworrel]

Why would anyone consider the possibility that a tennis ball could tunnel through a wall? QM doesn't even say that. All it says is that there is technically a probabilty of the ball tunneling through the wall, but it also says that there are numerous reasons the ball cannot tunnel through the wall. If you want to break the ball up into individual particles, then each particle has a good chance of tunelling through the wall but there is no real chance that the particles will coalesce into a ball with the same dimensions on the other side of the wall.

 

[ZapperZ]

Well, considering this is a theoretical question asking about the theoretical possibility, isn't it valid to answer with a logically-derived theoretical conjecture?

No, because we will then be back at Mount Everrest being theoretically possible to spontaneously appear in front of your face. After all, one can "logically derive" such a thing as well!

This is why there is such a brouhaha over the LHC and the catastrophic black hole swallowing our universe. People can't put into perspective the SCALE of possibility of things.

Zz.

 

[Greenman]

Yes, but then we go back to the fact that many very very smart people specialized in the subject have written books about the possibility. Books that are used to teach graduate level classes on Quantum Mechanics.

I haven't seen any books describing the math behind or environment needed to see Mount Everest spontaneously imploding on itself.

Surely then, this means there is a clear difference between the two?

 

[ZapperZ]

Yes, but then we go back to the fact that many very very smart people specialized in the subject have written books about the possibility. Books that are used to teach graduate level classes on Quantum Mechanics.

Where in a graduate level QM text does it tell you that a tennis ball call tunnel through a wall?

Zz.

 

[Greenman]

In Takagi's book. He describes how cohesive environments can be achieved and maintained and how macroscopic tunneling can occur in these environments.He actually uses a tennis ball and a wall as an example.

 

[ZapperZ]

In Takagi's book. He describes how cohesive environments can be achieved and maintained and how macroscopic tunneling can occur in these environments.He actually uses a tennis ball and a wall as an example.

And he has how much experimental evidence for this?

Look, I've given you examples of the Higgs, and the Delft/Stony Brook experiments. Especially in the Delft/Stony Brook experiments, they are doing nothing more than "textbook physics". Yet, these are still important experiments to show that, yes, a "macroscopic particle" CAN exhibit quantum properties. All they are doing is show how Leggett's scheme can, in fact, work! It was THE experimental verification. We didn't just sit back and applaud Leggett's cleverness at making such logical derivation out of what is essentially textbook physics!

You've jumped way past that already in your devotion to Takagi's book! This doesn't worry you at all? Forget about tunneling through a wall. Did you ever insisted to be shown this experimental evidence of a tennis ball being in such a coherent state FIRST? If there is one, please give me such a reference. Till then, I'm done with this discussion.

Zz.

 

[imiyakawa]

You can measure very accurately the position of the ball and its momentum simultaneously, and follow the trajectory of the ball every single step of the way. These are physical behavior that you cannot do with a quantum object.

According to professor Brian Greene, you can't. People just think they can because for all practical purposes a macroscopic measurement appears accurate, but it never really is.

 

[ArjenDijksman]

In Takagi's book. He describes how cohesive environments can be achieved and maintained and how macroscopic tunneling can occur in these environments.He actually uses a tennis ball and a wall as an example.

Maybe it would be better if he referred to more easily realizable experiments: http://link.aps.org/doi/10.1103/PhysRevLett.102.240401" by A.Eddi et al.

 

[ZapperZ]

According to professor Brian Greene, you can't. People just think they can because for all practical purposes a macroscopic measurement appears accurate, but it never really is.

It NEVER really is?

A "classical particle", by definition, obeys the standard classical laws. Now we can go on and argue that this is nothing more than a set of quantum laws that have undergone a gazillion decoherence, or we can argue that we are observing it using a "coarse-grained" measurements (do a search on here and you'll see that I've written several posts on this). It still does not detract from the issue that you have two separate domains here. If it is THAT easy to explain away the classical behavior, then someone should tell all those people who are working in the mesoscopic scale physics to stop wasting their time and go do something else!

Zz.

 

[boumbh]

It's just a curiosity and I'm searching for an estimation too... Like this kind of a stupid question:

I'm deliberate to apply Newtonian mechanics in a situation where it doesn't apply. On a flat surface (let's state that the Earth is flat and infinite). One train is traveling at 100km/h. An other is traveling at 100km/h on the top of the first train. The speed of the second train is therefore 200km/h. How many trains must I superpose so the top one reaches the speed of light?

This question has an answer even if it states a lot of aberrant assumptions. The answer is 10,792,529 trains (if I'm correct ). I know that to try out this experiment would require the train at the bottom to be very robust and very long (about one million km long to make it possible for all the trains to reach the speed of 100km/h), and a lot of gasoline. Also, the experiment would disprove this calculation since the principle of relativity should make a difference here.

We could also calculate the answer of this stupid question: assuming that my mother doesn't leave home, how much time must I drive my car at 100km/h, to catchup the age of my mother? (Due to time dilatation in the principle of relativity - which has been verified by experiment, so the result is expected to be a good approximation of the reality.)

I would like to know, for example, if I throw a pure graphite ball of 1/2 centimetre radius through a pure graphite wall of 1 millimetre width with a speed of 1m/s. Let's state the ambient temperature at 300K. We neglect the effect of air molecules, gravity, etc. What is the probability of the ball passing through the wall without damaging itself nor the wall? Is there a formula of the quantum principles that, outrageously applied to this case would give a result? Just to have a wrong approximation of an absurd fact, and then try to represent it.

For example : If an answer is 10^(-10^30), we can illustrate it:

If 6,000,000,000 people begins to write zeros right now with an average of 2 per second and per person, without sleeping, drinking nor eating (yes, they would die within 3 days, but), they would need 176 times the age of the universe to write down all the zeros of the number of time you need to throw a graphite ball etc. This is such an understandable illustration...

 

[pedrapgwilym]

As I see the basic nature of QT; every time you throw a tennis ball against a wall, some of it (a vanishingly small some of it) probably does 'tunnel' through the wall. You would have to keep at it for the remainder of the life of the Universe to accumulate enough of the ball on the other side of the wall to make it an observable quantity.