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↓

Double-Slit Experiment

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Sir Isaac Newton and Christiaan Huygens developed two different theories of light. While Newton's theory, called Corpuscular Theory of Light, describes light as a beam of particles (corpuscles). On the other hand Huygens' theory considered light to be a wave. These two theories are the base for the dual nature of electromagnetic radiations. These two surely are one of the most important theories in physics but the aim of this post is to discuss about an experiment conducted Thomas Young in 1803.

The Double-Slit Experiment made most scientist of 19th century to believe that light is a wave.

Let us understand the experiment now. Light is made to pass through slits of very small length. Before we look at the result of this experiment let us understand what will happen if the same experiment is conducted with waves of water, which we know is a wave, and a ball, which we know is a particle.

Case (i) Waves of water

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Let's say we drop a stone in a pond and it's obvious that we will observe ripple. Let us imagine Double-Slit Experiment is conducted with these ripples. All of us know the wave will undergo diffraction in both the slit. This will look like two different waves each emerging from one slit. Let us consider this as two different waves. These two waves will undergo interference. Interference is a phenomena of waves where two waves superpose to be either constructive (where the amplitude of resultant wave is greater than the amplitudes of the original waves) or destructive (where the amplitude of resultant wave is lesser than the amplitudes of the original waves). We can conclude two things from double slit experiment of water waves:

1. Not localized

2. Interference exist

Case (ii) Balls

We know that ball is a particle. Let us assume the slit is large enough for the ball to pass through. However the ball cannot pass through both the slits like wave. So the ball will either pass through the first or second slit. Moreover the ball will not undergo diffraction as it’s a particle and not a wave. It will just go and hit the screen. A small difference in the location, where the ball hits the screen is possible as it is very probable for the ball to just touch the edge of slit and hence change its path by small angle. However still its position can be said to be localized in a particular region. So we can conclude that:

1. It is localized

2. Interference is not noticed.

Now let’s look at the result of Young’s Double-Slit Experiment.

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In the above figure we notice the interference pattern. This gave rise to a belief among the physicists that light is a wave and Newton’s belief of it being a particle was a mistake. (However in the next century Albert Einstein’s research in photoelectric effect [E = hv] once again led to the confusion whether light is a wave or particle. Later it was concluded that light has dual nature of both wave and particle)

The future universe

"What is going to happen?” is something everyone of us are interested in. All of us want to know what is future. By All I also mean science lovers. Science lovers are interested in finding the future of universe, I mean the ultimate fate of universe.

Our scientists are working on it. According to them the Ultimate fate can be determined by three values:

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          1. The rate at which our universe expands. This is called Hubble’s constant. It was named after famous astronomer Hubble who actually measured the expansion of the universe.

               Here we must also discuss about Metric Expansion of Space. It refers to the increase in distance between two distant parts of the universe with time. We can say the scale itself changes with time.

          2.  Density Parameter Ω (Omega) or the average density of matter in our universe.

               Ω = Ω­_m + Ω_rel + Ω_λ

                    Total Density Parameter is the sum of  total mass density of ordinary and dark matter and total density of relativistic particles such as photon and neutrinos and effective mass density of dark energy.

          3.  Lambda or the cosmological constant of our universe. This term was introduced by Albert Einstein in 1917. 

One thing we must understand is that omega which is density of matter can also be defined in terms of gravity (as the gravitational force is determined by mass of the body).  Now let us look at few possible scenarios taking lambda to be about zero:

          1.  Omega greater than one

          This means that there is sufficient matter in universe to generate enough energy to reverse the cosmological expansion of universe. This means universe will start shrinking. Let's name this big crunch.

          It has also been observed that if this is the case the curvature of universe is negative, something similar to a saddle and space and time are infinite. Such a universe is called open.

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          2.  Omega less than one

          This means that there isn't sufficient matter in our universe to reverse the cosmological expansion. This means universe will expand forever. As universe keeps expanding the temperature will start falling. Hence the ultimate fate of universe will be big freeze or big chill.

          It has also been observed that if this is the case the curvature of universe is positive, something similar to a sphere and space and time are finite. Such an universe is called closed.

          3.  Omega exactly equal to one

          If omega is equal to one then universe is flat and will expand forever. However the matter present in universe is just sufficient enough to maintain the temperature, not leading to big freeze.