Heisenberg Uncertainty Principle

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     In 1927, Werner Heisenberg came up with his principle of uncertainty. He claimed subatomic world is not same as the macroscopic world we live in and Classical or Newtonian mechanics didn’t make sense in subatomic world. He along with many other physicists such as Max Planck, Erwin Schödinger etc. is considered to be the founder of Quantum Mechanics. Quantum Mechanics can be explained as a theory of subatomic world which behaves like Classical or Newtonian mechanics when applied to the macroscopic world.

     Heisenberg proposed that the process of observation itself disturbs the system; in other words we cannot determine any quantity without disturbing the system. This means that it is impossible to determine the values of physical quantities such as position and momentum without disturbing the system. If we have to measure the position of a particle, we have to disturb the system. If we disturb the system, we cannot find the momentum of the particle with great precision. This led Heisenberg to propose the following:

     “One cannot determine both the position and momentum of a particle simultaneously with any arbitrary precision. This has nothing to with the limitations of the instrument.”

Mathematically it can be written down as:

 ΔxΔp≥ħ/2

     Here, Δx is uncertainty in position of the particle and Δp is uncertainty in momentum of the particle. From the above relation we can understand that if we accurately know anyone of the two values (Δx or Δp) we will have no idea about the other value. This means if Δx=0 then Δp=∞.

Now we will discuss a proof for the Uncertainty principle. Heisenberg proposed that uncertainty principle can be proved true with help of classical optics. The following paragraph shows argument put forward by Heisenberg in support to his principle.

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     Heisenberg proposed a hypothetical microscope with where the electron at focus is illuminated by gamma ray photons. The resolving power of this microscope will be equal to the uncertainty of position of the electron. Mathematically: (here 2α is the angle made by the electron at focus with the lens.)

Δx=λ/2sinα

     Now we will relate the momentum of photons of gamma ray and that of the electron at focus. We know from classical mechanics that sum of momentum of photons and that of electron at focus is constant.

P=P_γ+P_e

     As photons hit the electron, momenta are going to be related. We can say that only if we know the momentum of photon accurately we can determine momentum of electron accurately. If we know the momentum of photon approximately, we can determine momentum of electron approximately. The relation between these uncertainties in momentum can be mathematically expressed as:

Δp_γ ∽ Δp_e

     Let’s assume that the photon which gets scattered after hitting the electron at focus enters the lens of microscope with some angle θ. As mentioned before, 2α is the angle made by the electron at focus with the lens. Then θ lies between angles – α and + α. Using mathematics we can say that the component of momentum along the axis of position of electron is a value which lies between –h(sinα)/λ and +h(sinα)/λ. This can be represented as:

Δp_γ=2h(sinα)/λ= Δp_e                     à(i)

We know that:

Δx= λ/2sinα                                    à(ii)

When we substitute (ii) in (i) we get the following:

Δx Δp ∽ h

The above equation is the mathematical representation of Uncertainty Principle.

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     We don’t experience Uncertainty in our day to day life because the value of ħ/2 is too small to have any significant effect to macroscopic objects. As mentioned in the very first paragraph of this post “Quantum Mechanics can be explained as a theory of subatomic world which behaves like Classical or Newtonian mechanics when applied to the macroscopic world.” Uncertainty principle’s effects become negligible in our macroscopic world.

     Though it one of the most accepted and popular principle in modern physics (Quantum Physics), question has been raised against it. Recently physicists have published papers trying to prove violation of uncertainty principle and a report on violation of Uncertainty principle is there in the link mentioned below:

https://www.sciencedaily.com/releases/2012/09/120907125154.htm

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Universe Full of Probabilities

Our universe is full of probabilities. We cannot be certain of most things in our world. We can only guess whether it will rain or not tomorrow. If you are living in Cherrapunji, then the probability that it will rain tomorrow will be high, let’s say 0.9. However still there is chance of it to not rain tomorrow (0.1). In other words, we can say that we are uncertain if an event will occur or not. The same uncertainty governs the modern physics.

                       

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A very famous physicist, Werner Heisenberg, came up with a principle which declared the existence of uncertainty and probabilities in physics. This principle is known as The Principle of Uncertainty. The theory implies that:

                 “It is impossible to determine the position and the momentum of a body simultaneously with any arbitrary precision. This has nothing to do with the limitations of the instrument.”

The above statement can be expressed mathematically as:

                                                        ΔxΔp≥ħ/2

                Δx= uncertainty in position

                Δp= uncertainty in momentum

                ħ= reduced Planck’s constant= 1.054 x 10^-34 J.s

This shows us that Δx and Δp are inversely related. If uncertainty in momentum increases, uncertainty in position decreases and vice versa. If we can accurately determine the position of a body i.e. Δx= 0 then, we will have no idea about the momentum i.e. Δp= ∞. The vice versa is also true.

Let us now discuss graph and what we can conclude from them:

Case (i)

Fig.1.

 

Here uncertainty in position is very less. We can easily say there is high possibility of finding the particle in the peak. However, what about momentum? The graph below shows us that we have no idea about momentum. The uncertainty in momentum is very high. Heisenberg Principle of Uncertainty in proved true.

Fig.2

If Δx↓ then Δp↑

Case (ii)

Fig.3

 

Here we don’t have a clear idea about the position of the body. Probability of finding the particle at a given position is the same at all points. The momentum of the same particle is shown in graph below and we see the uncertainty in momentum is very less i.e. we can make good predictions about the momentum of the particle.

Fig.4

If Δx↑ then Δp↓