Wave Particle Duality

 I will try to give a rough idea about wave particle duality to people who do not know what it is. All you need to know is what a function is.

 

What do we mean by a particle? We have this picture of a small ball-like thing, or a point with some mass  – that’s somewhere or the other all the time. That’s the defining thing about particles – they have a position at all times. So we can describe a particle’s position by a continuous function x(t). That’s the partcle picture.  But for real life objects, can  we desribe them that way?  Does such a function exist at all times? How do we know?

 

Now we can’t tell what the position of anything is without looking at it. This ‘looking at it’ is called measurement. So, when we do a position measurement we get a position. But what about before the measurement? There is no way of knowing what x was, or even if there was a position at all. Simply put, we can’t describe the position by x(t) at all times.

 

 Then what is the right way of dealing with real life objects, like electrons and all?  That forms the substance of Quantum Mechanics. QM says that, between  measurements, all we can say about a variable like position is to give a function f(x,t), called the position wave function. What is the significance of the wave function? If you square it, it gives the probability. So the  values of f(x,t)-square at various points will be the probabilities of the position being found there.

 

 How does the wave come in?  The wave function of an object varies with space and time in a particular way,  but that depends on the surroundings it finds itself in. There’s an equation that describes this variation. If we know the surroundings the object is in, we can find out how the wave function will vary.   Under certain circumstances it varies like a wave.

 

What does that mean? Imagine a wave.  It moves along with its crests and troughs, now rising now falling. That’s the same thing that happens to this function. If you graph f(x) with x for a certain time, you will see crests and troughs, and these crests and troughs move forward with time, just like waves do. That’s roughly the idea.

 

  As the wave function varies like a wave, the probability of finding the electron gets to places where only a wave could reach, and particles couldn’t. There’s this phenomenon called interference, shown by waves.

 

Suppose there are two sources of some kind of wave, placed some distance apart. They flow in crests and troughs. Now suppose they meet at a point where they both have crests. So the two crests will add and make a bigger crest. But suppose a crest meets a equal trough. Then they get cancelled and you get neither crest nor trough. If you take a cardboard with two holes in it, place it before some wall and throw light on ir you will see:  Alternate patches of light and darkness on the wall. The two holes act like two sources, sending in the light waves. You get Light where crests meet, dark where crest meets trough.

 

Now do it with electrons? What do you get, You get the same pattern. Some places on the wall the electrons will hit, at someplaces  no   electrons will it.. That’s because the wave function is like a wave, it shows interference. It has its crests and troughs. If electrons were particles, we would not get this pattern.

 

 So electrons behave in their own way, which is neither particle nor wave.

 

( I may restructure and edit this post a bit.  Meanwhile, any comments regarding inaccuracies or unclearness of presentation will be particularly welcome)

 

Back to School

Attending the 4th AKR School on General Relativity  at SINP was a pretty interesting, if exhausting experience. We learned a real lot about GR and Cosmology in the past two weeks, stuff that I'm still trying to digest. 

 The daily schedule was  like this : from 10:15 to 1:30, lectures. Then Lunch. Again from 2:30 to 5:45 tutorials where we worked out problems based on what we learned. It was pretty cool. But it was also pretty tiring, given that SINP is about one and a half hours from me home.

It was also interesting to meet new people. I made some new friends : Irshad from Delhi U, Himadri from Baroda U, Soumi from Brabourne, Dibakar da and Sumit from NBU, Deeptimoy da from TIFR and the gang from SXC. Best of luck to all these people! 

I understand that when speaking of such serious matters as GR and Cosmo, one shouldn't digress to topics of food. But being the shameless foodie that I am, I can't help mentioning that the food was excellent. I can rave about the Ilish machh and Pathar mungsho for days. Sadly, Irshad and Himadri found Bengali preparations too sweet and had a tough time, food-wise. It scares me to think what will happen to me when I step out of my little state, into the big bad world of non-Bengali  khana. Aaaaargh!!!

There, there. Have some mishti doi and don't worry about it. 

About Kashmir

Most Indians have this idea that the whole Kashmiri freedom struggle thing had been a Pakistani conspiracy. The naive Kashmiri is fooled by the sly Pakistani agents into rebellion against India. This, however, is not true. There was a genuine rebellion by Kashmiris against India and most people in the valley still want independence.

If anyone is interested about what was the matter and what is the matter, these are some links which I've found informative and worth reading: 

 Kashmir: The Unending War by Pankaj Mishra

 

Kashmir: A View from India by Tavleen Singh

I have no idea how the problem can be solved, but I think that people need to realise that there is a problem and why it is there. 

About Heisenberg's Uncertainty Principle

Most of us have done sums about falling particles and how long it would take them to fall and things like that. Take this problem for example: If a ball falls from the top of a building (on earth) of height 20 metres at 12 o'clock, where is it at 12:02? The noticeable thing about the question is this : we are saying that if you know the position of the ball at some time, and the environment it is in ( is there something pulling or pushing it? in this case it's earth, pulling it through gravity) , you can tell exactly where you'll find it later. There will be a certain position where it will invariably turn up. It won't , for example, say ' no thank you' and go back to the top of the building. Or that's what we thought.



It turns out that we just can't have enough information to tell you exactly where the ball can be. If you do an experiment with a hundred different balls, it turns out that we get different results, even if all the controllable conditions are kept just same. In one experiment you get 1 (say), in the other 2 or something. These differences are usually small, which is why for large objects like a ball we can neglect it. But the smaller our object gets, the more these differences matter.
But are these things totally random? Can we make any predictions at all? Yes, we can. Tell me where it was initially, and what the environment is, and I can tell you is the probability of finding it at different places. There'll even be a probability of it jumping up again ( though I admit none of its actually saying ' no thank you') . Not as good perhaps as telling you exact locations, but that's the best that can be done. And it's not just true of positions, but any physical quantity that's there : momentum, angular momentum etc. The theory which provides the scheme for working out these probabilities is Quantum Mechanics.

Now to uncertainty. Uncertainty is just the standard deviation (or sometimes half the s.d)calculated from these probabilities. Larger the uncertainty, larger is the range of values we are likely to get for whatever it is that we are measuring. Smaller the uncertainty, smaller is this range of likely results. That's why it's called uncertainty, duh!

So what's theHeisenberg's Uncertainty Relation? It is a result in Quantum Mechanics which says that the product of position uncertainty and momentum uncertainty of our object is always greater than a certain number. So if you find (through experiments) that the standard deviation in position values is small for our object, then it always turns out that the standard deviation in momentum values will be large. It is a statistical rule; narrower the probability distribution for position, wider the pd of m, and vice versa.

So this tells us that there must be something funny between position and momentum; by knowing the uncertainty in one, I can say something about the uncertainty in the other. There is something funny between the two, they are 'observables that do not commute with each other'.

How the uncertainty principle follows from this commutation relation, I'll write in another post.

Our (cancelled) plan for Mahalaya

' Let's go somewhere this Mahalaya!', said Mota , flailing his arms around as he does when excited. 'Very good', the rest of us answered, 'But where?' ' Diamond Harbour?' suggested Mota. 'How aboout Bandel?' Debashish offered. 'I know!', interrupted Mota, and from the fact that he had started to wave his arms around we knew he was onto something. ' Let's just go to the station and board some train, any train, and go to some strange location.' Shoumik, Debashish and I, too moved to speak, sipped our teas and nodded approval. But Suvo had an idea. And when Suvo has ideas, we willingly lend our ears. After all, he was the chap who had ragged innocent passengers on a train by asking first ' is this S-7?' and when answered in the negative 'is S-7 to be found between S-6 and S-8?' - all with prefect innocence and devastating effect. So the motley crowd that had gathered at Mama's after a long and grueling class listened attentively.

' If we want to go just anywhere', said Suvo, ' we can go there for free. And legally too.' He looked at our puzzled faces and grinned in his mischievous way. ' Simple.' he explained, 'We travel by Chetona.' For those of you who don't know what Chetona is, it is not a train you can board by buying a ticket. Au contraire, you board it by not buying a ticket. To make things clear, if you are caught traveling without a ticket, they put you in Chetona (presumably all shackled and tied up) and take you to some arbitrary location whence the guard shouts ' download!' and they dump you there and leave. Suvo's suggestion was that we travel by Chetona willingly and I took to the idea immediately.

' Well said, laddie!' I said, ' I can totally picture the scene. We'll board one of those posh, uppity trains, filled with all those cigar smoking rich thugs and their bejewelled hot wives. Then I will walk suavely to the ticket examiner and strike a polished, refined conversation with him. We'll discuss Plato and Aristotle and Spinoza's Ethics, for TEs I know are philosophical people. Perhaps we'll agree on Descarates, but have a slight disagreement about Leibniz. While we're at Nietzsche I will tell him, with complete nonchalance, 'Speaking of philosophy, did I tell you that I don't have a ticket? None of my friends do either.' You are saying that the TE would take it in his stride and tell us that Nietzsche used to do the same thing? You underestimate the Public Servant! They're men of action. When they meet a chap traveling without a ticket, they don't shrug and say ' Some people have all the tickets, others have none, and I am wearing a silly uniform. It's all absurd. ' and leave it at that. They take the chap by the scruff of his neck and put him in Chetona. And that's what will happen to us. What do you guys say?'

'It is a great idea', said Debashish, ' And you are forgetting one thing. The added pleasure of traveling with the real scums of the society, the lowliest of criminals : the ticketless travelers themselves!'

Needless to say, everyone thought it was a great idea. It was decided that this is how we'll travel at Mahalaya.

Next day, we learned that the holiday has been canceled.

The LHC won't destroy the earth

Suddenly, everyone is talking about the LHC, and how it will recreate the big bang (it won't) and how micro black holes will destroy the earth. A few days back, my uncle told me that the world is about to end. Then friend and fellow blogger ad libber was asking me about it. Yesterday, it was all over TV, and they were saying the same thing.

Much as I would like the earth to be destroyed ( exams being very close ) , experts in the field assure us that no such things are about to happen. In fact, some of them seem to have gone blue in the mouth from continually having to reassure the public. So people, please get the message: LHC isn't going to destroy the earth. See the CERN safety report or this .

For what the LHC will do, check this out.