The Uncertainty Principle

    The word "Uncertainty" is the most common feeling we all face in our daily lives. It does not refer to any negative situation; rather, according to "The Cambridge Online Dictionary," uncertainty is a situation in which something is not known or certain. 

Let's consider the situation "You wrote an exam and waiting for your result." It's a good example of an uncertain situation in which you don't know about your result at a particular time. Simple, na. 

Let's think about the situation in a deeper sense. 

To think better let's make some changes in the situation now the situation is "You wrote an exam and waiting for your result. The result was out and suddenly your internet is not working". In this situation, you are unable to check the result online and also unable to contact anyone for the result, so the situation becomes that your friends know your result but you don't know.

The point here is at that particular time you don't know about the result, to know about the result you have to spend more time, but actually, your result will not change over time but knowing your result depends on time. Is it not interesting? 

Although one can say it's a piece of bad luck, all bad luck we face is also a valid situation, here point is not what situation we refer but in that situation what we are measuring is directly dependent on something, as our case "Knowing result depends on time" as time increase more chance will be to know the result and hypothetically if possible to decrease the time our knowledge about the result will become less.

In simple terms, it's an uncertain situation.

In a simple mathematical way, it is,

      Know the result (K) ∝ 1 / Time (t)

Also, we can write K*t ∝ 1

Now, to make it more general one can write K*t = a, where "a" is a proportional constant, simple math we all study at the elementary level.
Note - For your knowledge I mentioned again that "K" and "t" came because of the situation and constant "a" depends on the situation but is fixed for the situation.

Now I think you have a rough idea about uncertainty. How knowledge about one makes less knowledge about another. It's good to mention it's that we can always compare something with time, but depending on situation other things can follow same thing.

Uncertainty does not refer to that something is weird or absurd, rather tells you there is uncertainty of calculations overall, not a particular moment. Here the constant "a" plays an important role, it tells you how much uncertainty is there.

In our last blog, we discuss about superposition. let's recall the thing.

Scientists define superposition as the ability of a quantum system, it describes how a physical system can exist in multiple states until it's measured. 

We mentioned vectors ψ, in quantum mechanics, we call this wave function and denoted as ψ(x,t) where x refers to the position vector and t represents the time vector. This is a continuous function and it can be written as a superposition of functions that we saw last time. How do we get this ψ(x,t) by solving the Schrödinger equation we will discuss in future.

(Credit - Below example and description have been taken from the book - Introduction to Quantum Mechanics by David J. Griffiths, Second Edition, pages- 18-20)
 
Now imagine that you’re holding one end of a very long rope, and you generate a wave by shaking it up and down rhythmically (Figure 1.8). If someone asked you “Precisely where is that wave?” you’d probably think he was a little bit nutty: The wave isn’t precisely anywhere—it’s spread out over 50 feet or so. 

On the other hand, if he asked you what its wavelength is, you could give him a reasonable answer: it looks like about 6 feet.

By contrast, if you gave the rope a sudden jerk (Figure 1.9), you’d get a relatively narrow bump travelling down the line. This time the first question (Where precisely is the wave?) is a sensible one, and the second (What is its wavelength?) seems nutty—it isn’t even vaguely periodic, so how can you assign a wavelength to it? 

Of course, you can draw intermediate cases, in which the wave is fairly well localized and the wavelength is fairly well defined, but there is an inescapable trade-off here: the more precise a wave’s position is, the less precise its wavelength, and vice versa. 

This applies, of course, to any wave phenomenon, and hence in particular to the quantum mechanical wave function ψ(x,t). But the wavelength (λ) of ψ(x,t) is related to the momentum (p) of the particle by the de Broglie formula:

                       p = h / λ = 2πℏ / λ

where  ℏ = h / 2π  = 1.055 * 10^(-34) is the planck constant.

 Thus a spread in wavelength corresponds to a spread in momentum, and our general observation now says that the more precisely determined a particle’s position (x) is, the less precisely its momentum (p).

 Quantitatively, Δx * Δp ≥ ℏ/2

where is the standard deviation in x (uncertainty in x), and is the standard deviation in p.
Standard deviation is described as (Δx)^2 = <x^2> - <x>^2, where for any function the expectation value is 

This is Heisenberg’s famous uncertainty principle. 

 Please understand what the uncertainty principle means: Like position measurements, momentum measurements yield precise answers—the “spread” here refers to the fact that measurements made on identically prepared systems do not yield identical results.

You can, if you want, construct a state such that repeated position measurements will be very close together (by making a localised “spike”), but you will pay a price: Momentum measurements on this state will be widely scattered. Or you can prepare a state with a definite momentum (by making a long sinusoidal wave), but in that case, position measurements will be widely scattered. 

All the above is a logical explanation but for still long time most famous scientists have debated about its truthness. 

Before discussing let's see in simple words the gist of the Uncertainty principle, it says for wave function the more accurately you know about its position the less accurately know about how fast it's moving i.e., it's momentum and vice versa. 

Calculating simultaneously we can do different mathematical ways or calculate with 2 machines put together. 

I recommend three videos here that describe some such arguments.

Video1   Video2  Video3

So the answer is yes. Yes, we can do measurements in different ways but in equation  Δx * Δp ≥ ℏ/2 we see that ℏ is its limitations where ℏ = 1.055 * 10^(-34) is a very small quantity. This also named as Planck constant by the name of physicist Planck gave birth to a new era of science called Quantum Mechanics, i.e., how things behave at the quantum level.

That phenomenon is too small for us to observe and intuitively recognise, but although this change is small, it makes a big difference in our study. As I mentioned before our right-hand side constant plays an important role here.