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.

Beyond Euclidean Geometry

Euclidean Geometry ruled mathematics for 2 millennia. Even today a beginner in mathematics would believe that Euclidean geometry is perfect and it is impossible to prove it wrong. Why is it so? The answer is Euclidean Geometry is based on our common sense and our day-to-day experiences. However modern physics and mathematics is beyond common sense.

What does our common sense say?  Famous philosopher Aristotle in his book On heaven wrote “The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all.” Further astronomer Ptolmey came up with proof for this. His proof was simple: there can be only three lines mutually perpendicular to each other.

Bernhard Riemann, a student of the famous mathematician Gauss, came up with an alternative to Euclidean Geometry. He proposed Hyperspace. Before understanding hyperspace let us imagine a race of two-dimensional creature living on a sheet of paper. Their day-to-day experience will make them conclude that they live in a two dimensional world and as they don’t experience anything three dimensional, mathematicians in this flat world would conclude that only 2 dimensions exists and more than two dimensions is nonsense.

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Riemann imagined the same world of flatlanders but on a crumpled sheet of paper. What would these flatlanders feel now? Riemann concluded that flatlanders would still find their world flat as they themselves are crumpled. However when they try to walk in this crumpled world they would experience unseen mysterious force acting on them making them move right and left like a drunkard. However we know that this force experienced by the flatlanders is nothing but effect of the uneven terrain in which they live.

He then applied the same to our world. He predicted force to be a result of unevenness of the terrain of the three dimensional world we live in viewed from forth spatial dimension.

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Another thing proposed by Euclid is that sum of all the angles of a triangle is always 180^o. Riemann realized that this holds only in a flat surface. He realized that a surface can have positive curvature (like outer surface of a sphere) or negative curvature (a saddle) as well. Euclidean geometry didn’t hold in such surfaces.

Table 1

Type of plane	Parallel Lines	Sum of angles triangle
Flat Plane	Never meets	is exactly 180o
Positively Curved Plane	Never meets	is greater than 180o
Negatively Curved Plane	Always meet	is less than 180o

Riemann’s main aim was to mathematically explain how crumpled the sheet is. He was successful in this by the introduction of set of number defining how crumpled the space is at a given point. Let us take the example of four dimensional space. Riemann realized that to define a point at a point he needs 10 numerical values. This was represented in a 4x4 board as follow (note that g­₁₂=g₂₁ and so on. Therefore six values are redundant)

Table 2

g11	g21	g31	g41
g12	g22	g32	g42
g13	g23	g33	g43
g14	g24	g34	g44

These sets of numbers are called metric tensor. It is used to get all mathematical information needed to describe how curved or crumpled a point in N-dimensions.

A tesseract (4D of a cube) undergoing double rotation in 4D space.

What is "Time"?

            In the modern globalized world every person keeps thinking about time. Time is very important for all of us now. A famous saying in English "time and tide waits for no one" proves its importance. So is it for physics lovers. Time is a very interesting concept in physics and is also a very beautiful concept. Let’s start with the simplest thing about time, its unit.

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            SI unit of time is second. What is one second? One second can be defined as the time taken by light to cover 299,792,458 meters. As speed of light is 299,792,458 m/s, the time taken to cover that distance will be obviously, 1 second. Will one second be same for everyone? A common person will say yes, one second is one second for all how can it vary from person to person. If you think the same way sadly it’s wrong. Time varies with your velocity. As you approach the speed of light, you will feel the time to have slowed down. Why?

            Before understanding it let us understand few other concepts.

• 1.     Speed of light in a vacuum is same for all (3 x 10^8 m/s), irrespective of the velocity the observer is travelling. In the sense if a person is travelling at velocity of let’s say 1.5 x 10^8 m/s (for example), he would still measure the speed of light to be 3 x 10^8 m/s and a person at rest will measure the velocity of light to be 3 x 10^8 m/s.
• 2.     Law of physics remains the same in all inertial frame of reference.
• 3.     Speed of light is a cosmic speed limit; it is not possible to go faster than speed of light.

          
           Now let us look at a situation from two different frames of reference. Let’s say a person is travelling at velocity of 1.5 x 10^8 m/s, with a photon clock. In a photon clock a ray of light keeps getting reflected perpendicular between to reflecting surfaces, let's say mirrors. 


            Case (i)
            Let's say that the person who is travelling sees the clock. He will find the clock to be stationary and the ray of light will be getting reflected perpendicular to the mirrors, as per him.

            Case (ii)

            Now let's say another person is standing and watches this person with the photon clock moving. How will he see the ray of light in the clock? Is the clock stationary as per him? The answer is no. The clock is moving therefore by the time the ray reaches from one mirror the other mirror would have moved so the diagram will be like this:

            It is obvious that the distance covered by ray in Case (ii) is greater than that it Case (i). We know that distance covered is directly proportional to time taken. Therefore the observer without the clock will feel that the moving person is aging slowly. This phenomena is called time dilation. This phenomena is negligible at small scale however the when the velocity is huge this phenomenon is observed clearly. This gave rise to an interesting paradox called twin paradox.

            Therefore we can say that one second is not the same for all. So our speed makes a huge difference at large scale. What are the other affects it has? We will discuss about it later.

Gravitation

 Let’s start with a famous story in science…

                Newton was sitting under an apple tree investigating the nature of force which makes moon go around the earth. All at a sudden an apple fell from the tree and this raised a question in Newton's mind ‘what made the apple fall down and not go up?’ This gave rise to a very important concept in physics called Gravitation.

                Now let’s discuss what Newton said about Gravity. Two bodies in this universe attract each other with a force which is directly proportional to the product of mass of both the objects and inversely proportional to the square of distance between them. Let’s put it in an equation:

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What’s that ‘G’? G is gravitational constant which is 6.67 x 10-11 Nm² kg^-2.

                Now let’s think about a situation. All at a sudden sun disappears! Sounds stupid isn’t it? As per Newton the planets will immediately start moving tangential to its orbit. This is because gravitational field will disappear and planets, which rotates around the sun due force of gravity, will be free to move and hence will move tangential.

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                However Einstein disagreed with this. As per Einstein the fastest possible speed in our universe is that light (which is approximately 3 x 10⁸ ms^-1) and is the same for everyone. Let's say there are two person, one at rest while the other one travelling at a very high speed with respect to the first person. Both of them will measure the speed of light as 3 x 10⁸ ms^-1 as it is not relative to the observer. Let’s call it cosmic speed limit. Even speed of gravity is slower than that light.

                We all know that it takes 8 min for light from sun to reach our earth.  Then we can say that gravity from sun take more than 8 min to reach earth. Then how will earth start moving tangentially immediately? So even if sun disappears the earth will be moving in its orbit for more than 8 min.

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                Einstein said that heavy objects such as planets and stars caused curve in space-time, considering space-time like a trampoline. Now let’s take the curve caused by our sun in the fabric of space-time. Planets follow the path formed by the curve caused by the sun.

                Now look at what will happen if sun suddenly disappears, with the new understanding of gravity proposed by Einstein. If sun disappears, the gravitational disturbance caused by this incident will form a wave, similar to that of ripple in a lake when a pebble is dropped in it. When this gravitational wave passes the earth we will find a sudden change is our orbit and only then the planet will start moving tangential to the orbit.

Spacetime

Space-time is defined as a mathematical model in which space and time are interwoven into one single continuum. The space-time of our universe in interpreted to that of a Euclidean space, with 4 dimensions, space consisting of 3-dimensions and time is taken as the fourth dimension. A unique position in a unique time in a space-time is defined as an event. Event is the basic concept of space-time.

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          All of us are familiar with the concept of planes i.e. 2-D plane, 3-D plane etc. A point in a 2-D plane is denoted as (x,y) and that in a 3-D plane is given as (x,y,z). Here x, y and z can be identified as the unique distance between the point and x-axis for x, y-axis for y and z-axis for z.

          An event in space-time can also be defined similarly as (x,y,z,t),  x, y and z here refers to the unique position of the point in the space while t denotes the unique time of the event. This way the coordinates specify where and when event occur. However the question which arises now is why do we need to include time and make it complex?

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          In classical mechanics, the non-relativistic model, time is considered as an independent and is treated as universal with constant passage irrespective of the motion of the observer. However in relativistic model, time is considered to be inseparable from the three dimensions of space as it has been noticed that time slows at higher speeds of a reference frame with respect to another reference frame. This phenomenon is referred as time dilation.

          However the number of dimensions required for analyzing our universe is a question among scientists. Speculative string theory predicts the number of dimensions to be 10 or 26, with M-theory predicting 11 dimensions, 10- spatial and 1- temporal). The widely accepted model is that of 3- spatial and 1- temporal, which is space-time. Increase in number of dimensions from four would have significant difference only in sub-atomic level.

Model Of Universe

First of all... Happy new year to all!

All of us have seen a bubble. It is something very common in nature. What is a bubble? A Bubble is an ellipsoid, which is usually a sphere. Its layer is made usually by a soap solution and encloses gas. You may think why I am talking about bubbles here in a blog dedicated to Multiverse. The answer is a gas bubble can be interpreted with Universe.

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Universe is also bubble-like which is expanding. Think of filling air into a balloon, it is something like that. The Universe is also expanding like that balloon in which we blow air. The balloon expands in a particular rate similarly Universe too expands at a given rate which is  74.3 plus or minus 2.1 kilometers (46.2 plus or minus 1.3 miles) per second per mega parsec (a mega parsec is roughly 3 million light-years). [value from www.space.com ]

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This interpretation of comparing Universe with a bubble was first done by Albert Einstein, one of the greatest scientist of all time. He called Universe as an expanding bubble. Later science developed and an idea of multiverse arose in which scientists believed that there is a possibility if existing of more than one bubble (universe). It was believed that Big Bang occurs when Two independent bubbles merges or a bubble splits into two independent bubbles. 

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We are saying that more than one universe exist but the interaction between universe is not what gives rise to the phenomena called big bang. We will discuss about it in upcoming posts. 

Meet you people soon in next post and would be pleased to see your comments and opinion about this topic.