Quantum state

The quantum state

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The quantum state

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One bright morning, you find yourself on the terrace of a skyscraper, absorbing the grand view of the city beneath. As breathtaking as this view appeared, your stomach lurched every time you gazed straight down. Walking people and scrambling cars, all appeared like swarms of ants crawling busily.

How high is the building, you wonder. A random thought kindled your curiosity. A question: is there a simple way you can know what the height of the building is, perhaps without having to use a measuring tape? The question remained in your mind long after the experience.

It was when you shared this thought with a physicist friend that you learnt from him that there indeed is a way to compute the height of the building, standing on its terrace. All you have to do is drop a stone from the top (nowhere near the people beneath, of course) and record the time that elapsed between the moment you dropped the stone and the moment it hit the ground, with a stop clock. And Sir Isaac Newton has a formula ready for you to plug in this value of time and obtain the distance through which the stone fell, which essentially is the height of the building.

Average velocity is defined as the total displacement over total time. \[\bar{v}={\Delta x\over\Delta t}\]

It also can be computed by finding the mean of the initial and the final velocities, \[\bar{v}={{u+v}\over2}\]

Plugging the two equations together,

\[{\Delta x\over\Delta t}={{u+v}\over2}\]

Rearranging and using the equation $v=u+a\Delta t$ (which is nothing more than a rearranged definition of acceleration), we have,

\[\Delta x=ut+{1\over 2}a{\Delta t}^2\]

For the stone, the displacement $\Delta x=h$,the height of the building, the initial velocity $u=0$, since the stone is just dropped from rest, and the acceleration $a=9.8\, \mathrm{m/s^2}$, the acceleration due to gravity. Hence, if you measure $\Delta t$, the time for which the stone fell through, you can calculate the height of the building.

In calculating the height of the building here, what you have actually done is: given you know where the stone was at a specific time (when you dropped it), you have calculated where it will be at a later time (when it hit the ground). The difference between these locations of the stone is just the height of the building.

Alternately, if you know where the object was at any specific time, how fast and in what direction it moved then and under what forces it has been moving, you gain the complete knowledge of how the position of the object has been changing and will continue changing during its entire journey. This essentially unfolds the entire path of the object before your eyes. And this gives you tremendous prediction powers with regard to the object’s motion. Using this, you can find out where the object will be at any specific time, how fast and in what direction it would be moving then and so on. In other words, you know everything there is to know about the object’s motion once you’ve figured out its trajectory.

Writing the previous equation in a more elaborative manner,

\[x(t)=x_0+u(t-0)+{1\over 2}a(t-0)^2\]

Where $x_0$ is the initial position of the object at the initial time $t=0$. Hence if you know where the object was at some time $t=0$ and how fast it moved then $u=0$, and given that the forces causing the motion are known $a={F\over m}$, then all you must do is put in the value of the time and wait for the equation to tell you all the past, present and future positions of the object. \\

Further, using this function, we can compute what the velocity of the object will be at any instant of time, \[v(t)={\mathrm{d}x(t)\over\mathrm{d}t}=u+at\]

Is there anything more you’d like to find out about the object’s motion?

But now, say, the stone you dropped was so small that you couldn’t see it when it hit the ground so far below. But you’re certain the stone did hit the ground. You could then perhaps expect to be able to hear the sound of the stone hitting the ground. Or ensuring you dropped it in water (say there was a swimming pool right below), you could note the time you saw the stone splash and ripples beginning to spread out. As you can see, not being able to see the stone directly didn’t cripple you. You still could find indirect means of measuring the position of the stone at the time it hit the ground.

The problem we faced with the tiny stone dropped from the top of the skyscraper is a rough example of the issue we have with the tracking of microscopic particles such as electrons. Partcularly, electrons are as small as small can get. They have no known structure. They’re just point particles. Infinitely smaller than that small stone we dropped from the skyscraper. Even protons and neutrons, the other two gardencvariety types of particles that make matter, are a trillion times smaller than a millmeter. The point is, there’s no hope of being able to see these micro particles the way we see stones.

In fact, not being able to see the stone from the skyscraper was essentially a limitation of our eyesight. But not being able to see the micro particles is way beyond the ability of even the most powerful microscopes. Hence, while dealing with the micro particles, we must get used to not relying on our sight always and being able to find alternate, more indirect ways of keeping track of them, something like what we did when we cleverly dropped the tiny stone into the pool and watched the ripples it made.

An important point must be stressed here – when dealing with stones, we relied on observations in order to obtain numerical values for the required quantities (such as the position of the stone or the time it took to fall), but throughout the process, we had faith that the stone would be behaving the way it ought to even when we aren’t observing it. Such faith has been strongly encoded in our minds through experience.

Even evolutionarily, when say our ancestors saw a lion charging towards them at a distance, it was very necessary that they imagined the lion continuing to charge towards them even when they didn’t actually see it while running for their lives. They may not have been able to keep their eyes on the beast throughout the chase. It was almost like an imaginary video they ran in their minds which quite often helped them escape the lion. On the contrary, if our minds were built so that we questioned the lion’s chase every time we looked away from it while we ran and believed that the chase is real only when we saw the lion, we as a species wouldn’t have made it this far.

Hence we shouldn’t blame ourselves when we tend to treat the micro particles in the same way. Once we detect a particle at a certain position, which say is drifting constantly under no forces, we run our imagined videos of the particle continuing to drift even when we have no evidence for it. But the problem is such an imagination almost always leads us to wrong conclusions for the particles. Let’s look at it more closely. What exactly do we mean when we say we detected a particle’s position? As we’ve made it clear, we cannot just see the particles and say the particle was here or there at some time. We therefore need some kind of detector that must be placed in the path of the particle and which say produces a click sound if the particle hit the device. Yes, we have detected the particle’s position. But would that not have disturbed the fragile particle? The sensor is certainly made of hundreds of millions of particles. Hence, the detection of the particle by the sensor must be looked at as an interaction between the single particle we were after and the hundreds of millions of particles of the sensor. Disturbance is inevitable. If only turning back and looking at the chasing lion disturbed the chase and caused the lion to run away in a different direction altogether. The lion is anything but fragile.

What we have discussed above is one example of the difficulty we face and hence a strange way of thinking that we must get used to when dealing with micro particles. To be specific, when concerend with falling stones, we had an intuition of how the stone would fall and also Newtonian mechanics enabled us to find the trajectory of the stone – the complete information about the positions, speeds and direction of travel of the stone. And from this information we could compute everything there is to know about the stone’s motion. This includes the time it took for the stone to fall, the speed with which it smacked the ground, how quickly did the stone gain speed as it fell and most importantly, at what height the stone was at any instant during the fall. Hence, once the equation fo the trajectory of the stone is found, everything about its motion is known.

But when considering micro particles, as we saw, such complete information seems elusive. In trying to know the position of the particle at any instant, we placed a sensor in its path, and in doing so we changed the speed of the object (perhaps even made it zero). So, the conventional way of measuring the position and speed of the particle are no longer avaliable to us. We therefore cannot obtain an equation for the trajectory of the particle.

So it seems that all information about the particle remains hidden from us. We however have faith in the fact that if not in the conventional way, there exists some way in which the information can be extracted. It is this faith that motivates us to define soemthing called the quantum state of the particle. Much like the equation of the trajectory of the stone encompassed all information of the ston’s motion and hence the information about its position, speed and direction of motion was called the classical state of the stone, the quantum state is assumed to somehow contain all the information that can be possibly extracted about the particle’s motion.

Mathematically, we expect the quantum state to be a complex function of space and time. We generally use the greek alphabet $\psi$ to represent quantum states.

It is with this faith that the quantum state contains information about the micro particle or in general a system (containing micro particles) that we begin our journey into quantum mechanics – the best way we know of knowing everything there is to know of the microworld.

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