Busting Myths About Heisenberg's Uncertainty :: Episode 3

Aah! Finally, we come to our actual topic of discussion! But you see all that previous discussion (check out Episode 1 here) was necessary to build you up for this final stage. (If you have skipped all of the above, I would recommend you at least skim through them fast so that your concepts are all brushed up and shining clean.)

Busting Myths About Heisenberg's Uncertainty :: Episode 2

[For the New Readers: This is a continuation of the discussion started in Episode 1.]

What is the Wave-Particle Duality?

WARNING: PUT YOUR IMAGINATION CAPS ON! WE ARE ENTERING DANGEROUS TERRITORY.

You see this is the main source of confusion and misunderstanding in all of Quantum Mechanics. If we understand this well, we're half way through.

In the beginning of the last century, physicists came to notice that under certain very delicate situations, the electromagnetic waves – I said “waves” mark it! – seemed to behave like particles. Particles? But how can that possibly be? Waves and particles are totally opposite kind of things (you should be able to appreciate this now). How can delocalized waves which can pass through one another and interfere to add up or cancel each other, behave like localized particles which hit each other and bounce off? Well, no one knows! Not even today. Then how does it make sense at all? It doesn't – thinking classically.

Soon it was found that electrons behave in the same way, that is they are screwy exactly the same way as photons are. They behave like particles or waves or both or neither depending upon the experiment.

So what are these electrons? Particles? Waves? If waves, then waves of what? No one knows. Now we have given up on thinking about these questions on how crazy nature behaves. We think differently now. We think quantum mechanically. That is to say, we don't know and we believe, even nature doesn't know what is going on down at those tiny scales. But there is still a certain order to things, and that order is encapsulated in the theory of quantum mechanics.

You see the whole of quantum mechanics is based on three fundamental postulates: (1) The Superposition Principle, (2) The Measurement Principle and (3) Unitary Evolution.

Objects such as electrons or photons or any thing for that matter (I call them objects because I don't know whether to call them particles or waves or both or neither) have a certain wave-function associated with them that describes their state. I do not know what that wave is. I cannot physically measure it in any way possible. Trust me on this, I simply cannot. In an em-wave, I know the fields oscillate. In sound waves, pressure through air oscillates. In water waves, the water surface level oscillates. In this case, I do not know what shit oscillates. But I know this: the oscillations can have complex amplitudes (seriously?) and the absolute value squared of the amplitude is a measure of the probability distribution function describing the probability of existence of an object (as a particle) in a certain definite position for example. Now that is something I can measure! If there are several alternatives an object can take, there is an amplitude associated with each of these alternatives, and the state of the object is a vector sum of all these amplitudes. This is known as the Superposition Principle. It means nature does not know anything about which of the alternatives it should take (eg. which position it should occupy). It is in a superposition of all the alternatives. I don't know what that means though.

Difficult to understand? Well take the example of the double-slit experiment with electrons shot one at a time. If we think that the electron went through either one "OR" the other slit but not through both, we would run into very fundamental logical inconsistencies if at the same time we observed an interference pattern at the other end of the setup; but we do observe an interference pattern and therefore we must shun our intuition, our classical notion about a particle taking only one "OR" the other of all the available possibilities. Does that mean the particle goes through both the slits (splitting into half, going through both slits and interfering with itself), that particles in general "exist" in a fuzzy state taking up all the available possibilities at the same time? Well, we don't know. And we leave such questions for jobless philosophers to answer, because if we try to answer this question experimentally (like all physicists do), we find out that the particle goes through either slit1 or slit2, but not both. But having found that out, we do not run into any logical inconsistencies because nature leaves us without any interference and we're dumbstruck at her subtlety you see.

Two important fundamental principles of quantum mechanics are deduced from this experiment: 1) THE SUPERPOSITION PRINCIPLE which, as we have discussed earlier, states that there are amplitudes associated with any observation (the intensity or square of which is a measure of the probability of that observation to occur) and the resultant amplitude which describes the state of the system is a superposition of the amplitudes associated with all the possibilities that may exist for the system. 2) THE MEASUREMENT PRINCIPLE which states that a measurement of the state of the system reveals only partial information about it, reducing the resultant amplitude to just ONE of the total possibilities. This is also known as the collapse of the wave-function. It is only after a measurement is made that nature decides upon which of the alternatives she should take.

Take a break now! It's a lot to digest. Ruminate on what you have just read. Probably read the paragraphs again and then proceed. I would advise you read them twice and think about them thrice.

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Let us investigate a little bit more about what these wave-functions are. Read about the meaning of unitary evolution yourself. It just stresses on the fact that a particle, at any given instant of time, almost surely takes up at least one of the infinite plethora of possibilities. For example, I would expect the probability that an electron is somewhere there in the limitless vastness of space, to be equal to unity. A wave-function that ensures that, is called a normalized wave-function. The time evolution of wave-functions must preserve this normalization condition. The principle of unitary evolution is thus also known by the name 'principle of conservation of probability.' I will not talk about it now. You read it yourself. Cool? It's easy shit, but a very useful concept.

Think about this question. Where is the wave? Where does it float? Space obviously – the space around us, isn't it? Well, I would not shoot a “yes” right away at this. There is a certain amount of mathematical subtlety that you need to understand here. What is this space if I may ask? It is the set of all positions, is it not? Yes, of course! But I call that space by a special name: the position configuration space, because I consider the set of positions which would specify a configuration of the object I might want to study, in terms of the object's position or positions of the constituents of the object. Mostly when I say space I mean the position configuration space. But I can have other configuration spaces as well, for example the momentum configuration space which is the set of all momenta that an object can take. Here I will not have the x, y and z-axes, but perhaps p_x, p_y and p_z-axes. Suppose I know of a wave-function ø(x,t) in the position space and I ask, how does it look in the momentum space? What is the form of ø(p,t)? How do I do that? Do you know it? I will give you a hint! You have already come across this transformation several times, perhaps unknowingly but you have for sure. It obviously has something to do with position and momentum. Stop! Make a guess first before reading any further.

Well, don't be surprised when I tell you it's a Fourier transform. It comes from the famous de Broglie's relation which postulates that if the wave-function associated with a single object, ø(x,t) be an infinite plane wave of fixed wave number k₀ (remember this implies that the object is completely delocalized in space: it is in a superposition of all the infinite number of positions, equally weighted – close your eyes and imagine, my friend!), then the object has a fixed momentum given by p₀=ħk₀. Thus in the momentum space, the wave-function looks like a sharp spike at p=p₀ (it is a Dirac-delta function to be precise). You should be knowing from your study of mathematics that the Fourier transform of a plane wave is a Dirac-delta. In general, the transformation from the position configuration space to momentum configuration space and vice versa is accomplished by Fourier transforms. Period! There are reasons why it is a Fourier transform. It has to do with the superposition principle. It is a very beautiful picture actually. But I'll keep that tale for some other day. Right now, just get this straight. An object completely delocalized in the position space (thus meaning a plane wave associated with it) has a fixed definite momentum and conversely, an object completely delocalized in the momentum configuration space has a fixed definite position. An object partially delocalized in one of the configuration spaces would also be partially delocalized in the other configuration space.

But why do we need to look at it from another perspective? Is not position configuration space enough? Why do we need all these other configuration spaces?

Well, imagine this scenario. Say you have a nice wave-function ø(x,t) in the position space, in case of an electron for example. You would expect the probability distribution function ℙ(x,t) to be equal to the intensity of that wave, that is to say ℙ(x,t)=|ø(x,t)|². Now see that means, the object is not fixed at any position x but has a certain probability of being at any position x ranging from minus infinity to plus infinity. However, since you know the probability density ℙ(x,t), you can calculate the mean position x_mean very easily from the formula you should be knowing from your study of elementary statistics and probability: just integrate x.ℙ(x,t) from minus infinity to plus infinity. x_mean in general would be a function just of time. But keep in mind that the electron is not just there in the position x=x_mean, it is in fact everywhere until you make a measurement to find out where it is. You cannot really predict deterministically the outcome of such a measurement. The result would be random. However, if you make simultaneous measurements on a large number of identical systems, you would mostly find the electron to be near x=x_mean . Now suppose we want to know the momentum of the electron. What do we do? Since the mean changes with time, is the momentum proportional to the time derivative of x_mean? But we can't be sure about this, because x_mean is just the mean position. We have no conception about the idea of a classical position here. We are dealing with average values for God's sake. So how do we go about it?

There, you see! Because we know how to transform to the momentum configuration space, we can go about finding how the probability distribution function appears like in this space. Once we have the distribution over momentum space ℙ(p,t)=|ø(p,t)|², we can very easily calculate the average momentum p_mean using the same formula that we know from elementary statistics. Well it turns out that the average momentum is proportional to the time derivative of the average position (the proportionality constant being the electron's mass). That's an unexpected surprise but a pleasant surprise indeed. (It also turns out that the time derivative of the average momentum is equal to the average force – voila! Newton's law hidden in averages, another pleasant surprise!). Do keep in mind that these are just averages we are dealing with. In general, there will be a non-zero standard deviation from the mean.

So now you see the usefulness of different configuration spaces. If you want to find the mean of an observable quantity of your interest, just transform to that specific configuration space and then it is all very easy. For example, you want the average energy, go to the energy configuration space and compute the mean from the formula you know.

Take a deep breath now. Let me summarize by pointing out what we have learnt about wave-functions a.k.a. matter waves in this long discussion.

• Wave-functions do not mean point-like particles traveling in a sinusoidal path. That is a very wrong picture (and means you haven't understood the concept of waves well)! Matter-waves are mathematical waves of some uncanny shit which we cannot possibly imagine in a thousand years. However their intensities have a precise physical significance. They are probability distribution functions. E.g. a plane wave ø(x,t)=Aexp[i(kx–wt)] may be associated with an object but would never be physically observable. However, |ø(x,t)|²=A² is a physically measurable quantity – the probability distribution!

• Electrons, photons and all objects in this universe do not behave like particles, neither do they behave like waves. They behave like nothing we have ever had any direct experience with. We call them matter-waves. But we do not know what they are, how they look or what they do, but only that they exist because we have seen what happens to the world when they interfere. They do have observable consequences by which they make their presence felt – and that is how we study them! However, whenever we make any measurement, the wave-function collapses!

• Wave-functions need not be plane waves always. In fact, to tell you the truth, they never really are. Plane waves are non-physical wave-functions because they are non-normalizable (check it!). However, you should know from your knowledge of Fourier analysis that if you stack together a lot of different plane waves, you can make up any nice function you want – that includes good normalizable waves that represent physical systems in our world. E.g. The wave described by ø(x,t)=Aexp[–b|x|–iwt] (try to imagine how that function looks like) is a nice normalizable function but is not a plane wave.

UPCOMING: What's the fuss about the Uncertainty Rule?

Check it out in Episode 3.





 

For mathematical and technical details, contact me.

Busting Myths About Heisenberg's Uncertainty :: Episode 1

I have observed that there has been a lot of confusion and misunderstanding regarding the Heisenberg's Uncertainty Principle (and Quantum Mechanics in general) among undergraduate students and surprisingly also among some Indian textbooks following the old curriculum. I therefore take it as a challenge to clarify these misconceptions and present it here in this text. I'll make it very easy. Just try to use your gift of imagination along with me.

I will divide this entire thing into three episodes so that you are not intimidated by the length of this post.

Let us first set the basics right so that there is no miscommunication between us in the future. You can skip the topics which you think is butter and cheese for you. Go at your own pace.

[For those of you who want a more precise and mathematically coherent and rigorous description of the ideas I am about to discuss, please contact me and I will upload a post separately.]

Optional Advise: Put on some soft music in background and read through the material. Enjoy!

What are Waves?

Imagine a body moving in a zigzag path which is say like a sine curve. Does that constitute a wave? No, it doesn't. If something moves in a wavy path, it doesn't mean it constitutes a wave motion.

Imagine a pendulum swinging simple harmonically. If you plot the angular displacement against time, you will see a sine curve developing. Does that constitute a wave then? No, it doesn't. Every sine curve doesn't imply wave motion. I will tell you what exactly constitutes a wave motion.

Imagine an infinitely long abacus board with just one bead in each vertical column (with infinitely so many columns next to each other). First get that picture right! Once you have that picture in your mind, know that a bead can move vertically up and down a column, but cannot jump into an adjacent column. Now settle each of the beads down to the bottom of the columns they are stuck to, so that you see a nice horizontal string (infinitely long mind you!) of beads packed next to each other. That we call an axis, say the x-axis with each bead representing a point in the axis, at this moment. Now rearrange the beads in such a way that you can make out another pattern, not like the original straight horizontal string anymore but like a sine curve now. So if you attach an axis along the columns that hold the beads, call it y-axis, then the pattern of the packing of the beads next to each other can be represented by the equation $ y(x) = A \sin(x) + B $. Now keep in mind that this equation represents a 'pattern' – just a damn pattern – of the arrangement of beads in a sequence at this instant of time, nothing more, nothing less. Get ready, we are close!

Suppose at the next instant of time, I make a rearrangement again and the new pattern looks as if the previous pattern has moved slightly to the right. Then again at the next instant, I rearrange the beads so that the pattern shifts slightly more to the right. If I do so at every instant of time making rearrangements of the beads continuously for a long time, I can see a continuous motion of the sine curve towards the right. Lo and behold! This is a wave, my friend! Yes this, – this is the very definition of a wave! (It mathematically translates to say that $ y(x,t) = A \sin(x – v t) + B $, where $v$ is the speed at which the sine curve appears to move towards the right).

So think! What is it that is moving towards the right? Is it the beads? No! Obviously not. It is the pattern. Beads only move up and down the columns remember? They cannot possibly move sideways. So as time progresses and I see a wave motion, what happens to the individual beads? How am I rearranging them every instant of time? Well, if you think about it for a little while, you will know that I am making each of the beads oscillate in their columns simple harmonically, with successive beads differing in their oscillation by just the right phase.

So what have we learnt about wave motion? Three things! First of all, there should be a lot of beads that can go up and down a column. Secondly, wave motion is the motion of a pattern in the arrangement of beads along their respective columns. And thirdly that when a wave passes through a string of beads, they execute simple harmonic oscillations in their respective columns in such a way that the pattern we call wave appears to move. (In general, the beads can oscillate in an-harmonic ways too). So nothing physical moves along with the waves. It's just the motion of patterns. But wait, it needs energy to set up oscillations in the beads. One needs to do the rearrangements at every instant of time. Who does that? So it must be that there is energy in the wave. And the energy in the wave does the rearrangements! Whoever first sets up a wave motion, puts up some energy in making the beads oscillate up and down, and as the pattern a.k.a. the wave moves further down the line, it takes the energy with it making other beads on its way oscillate up and down, thus making it possible for itself – a pattern – to progress through space with time. Thus we can have the following definition for a wave.

“Wave motion is the motion of energy through space.”

Let us take the example of electromagnetic waves in empty space. The array of beads is analogous here to the electric and magnetic fields set up in space. In the absence of any em-waves, their values are zero everywhere. However, the fields suddenly start to oscillate up and down in a particular direction when an em-wave gushes through them at the speed of light, and continue to do so at the frequency of the wave as long as the wave persists. It takes up energy to set up non-zero electric and magnetic fields in empty space. The wave carries that energy. We see the manifestation of that energy in the oscillation of the fields.

Clear, are we?

Think about all the properties of waves (delocalization, superposition etc.) that you have read about in textbooks already. Now is the right time.

What are Particles?

The idea of particles should not be a difficult topic for any of us to understand. But anyway, for those of you who want to read about it, particles are localized indestructible lumps of energy that behave exactly the same way as bullets shot from a gun would do.

There! I am not gonna repeat that again.

UPCOMING: The Mother of QM: The Wave-Particle Duality!

Read about it in Episode 2.

Applications of Group Theory ~~ A Brief Intro!

Group theory you see has many interesting applications in varied fields of science and technology because group theory is inherently a theory of symmetry and structure. I will not go through the details of any of these applications but rather discuss how I, as a physicist, personally use and appreciate group theory on daily basis.