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.
Thou. Must. Teach.
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