Proving the Uncertainty Principle

INTRODUCTION

Classical Physics and General Relativity are incompatible with small scale —smaller than 10-9 meters.

Quantum Physics or Quantum Mechanics (QM) explains the small scale universe, but, it is incompatible with large scale which is described by General Relativity (GR).

So, both QM and GR are incompatible with each other.

Heisenberg’s Uncertainty Principle states that we can’t measure the position and momentum of a particle with 100 percent certainty.

If we increase the certainty in one of these two, the certainty in the other quantity decreases.

This, sometimes, seems to be nonsense.

One argument against it can be that if we know the present position and momentum (that is, mass × velocity) of Earth, Sun and our Moon, we can find out when will the next solar or lunar eclipse will take place by using Newton’s laws.

But these big objects have very small uncertainty in their position and momentum, this is because uncertainty in position multiplied by uncertainty in momentum is always greater than half of reduced Planck’s constant and the value of half of reduced Planck’s constant is —

52 × 10-35, which is a very small value, so, objects far larger than an atom have a very small amount of uncertainty in their position and momentum.

This proof of uncertainty principle is ‘mathematical’ rather than ‘physical’.

So, how one can prove the uncertainty principle?

1. REASON OF UNCERTAINTY IN POSITION AND MOMENTUM OF A PARTICLE

So, one can thought that to see anything, light is needed, so to measure a particle’s position and momentum light will be needed.

We know that according to Planck’s quantum theory any energy including light energy can only be radiated in some indivisible units (particles) which are called quanta (singular – quantum) and light is radiated or made of the quanta called photons.

Suppose we want to measure position and momentum of an electron (a type of particle).

When we will ‘throw’ light on the electron, that is, when we will shoot photons on the electron, photons will collide with the electron and they will change the velocity of the electron and hence, momentum will also change and after the collision, the position of electron will also be changed.

So, this is one of the reasons that we can’t measure both the position and momentum of a particle precisely.

It can also be explained with the help of the wave-particle duality concept which is also based on Planck’s quantum theory. According to Planck’s theory, light is made of particles and we also know that light is also shows wave nature, so, light has a dual nature of wave and particle.

A French physicist, Louis de Broglie suggested that not only light showed a duality (of wave and particle) in its nature, but all other particles of matter, like electrons etc., behaves both as a waves and particle.

If this is true, then obviously we can’t measure both the position and momentum of a particle precisely, because wave is a transfer of energy in a medium or a field and when energy will be transferred, the medium or field will get ‘disturbed’, as a wave is nothing but transfer of energy, so, we can’t give a definite position to a wave. And because all particles behave as a wave also, so, now particles don’t have any absolute location or position.

This confirmation of uncertainty principle using wave-particle duality can also be understood in another way. With any particle, a wave will be associated, and in the ‘wave packet’, the particle can be located anywhere, so, there will be uncertainty in the position of particle.

2. DOUBLE-SLIT EXPERIMENT

Now a question arises, how one can prove the de Broglie’s hypothesis?

It can be proved using the two-slit experiment, initially performed by a British polymath and physician, Thomas Young to prove that light is a wave.

Before discussing this two-slit experiment, we should know what is ‘INTERFERENCE’.

Interference can be defined as the ‘interaction of waves’. This interaction can be ‘constructive’ or ‘destructive’.

When the waves are in same phase, that is, when peaks (or crests) of two waves ‘meet’, the two waves combine to become a single, but more stronger wave and when the waves are out of phase, that is, when peak of one wave meet with the bottom of second wave (or trough), the two waves cancel each other.

So, now we can discuss the two-slit experiment by Thomas Young.

Setup of the double-slit experiment : A light source, single slit, double slit and a screen is taken. The single slit is taken to make the light pass through the double slit. The light wave splits into two different waves when it passes through the double slit. The setup is something like this:

Observations : When a single light wave will split into two different waves,the two waves will interfere with each other and will create an ‘interference pattern’, something like this:

In the above image, light appears on the light-colored ‘spots’, that is, at the positions 1, 2, 3, 4, and 5 because of constructive interference and dark spots are formed due to destructive interference where waves cancel each other and light doesn’t appear on these spots.

Conclusion of Two-slit Experiment : Young’s two-slit experiment concludes that light is a wave because it interferes. If light showed only particle nature, then the light would appear only at the positions 2 and 4.

This surely, doesn’t prove de Broglie hypothesis, but it can.

According to de Broglie hypothesis matter particles also behave like wave, so, let’s take example of an electron.

To prove, de Broglie hypothesis the setup will remain the same, only the light source will be replaced by an ‘electron gun’, which will fire electrons and the screen will replaced by a ‘detector’ which will detect the electrons.

This experiment was suggested by an American physicist, Richard Feynman and this is called ‘Feynman’s thought experiment’, when this experiment was performed, the result were same as Feynman suggested and this led to the ultimate proof of wave-particle duality and de Broglie hypothesis.

And, even when this was performed using a single electron, there was still an interference pattern on the detector, so this meant that the electron would be passing through both slits at same time and interfering with itself!

This can happen because a single electron also shows wave nature so, the ‘electron wave’ will split and then interfere with itself, giving an interference pattern.

REFERNCES

1. Hawking, Stephen (2000). A Brief History of Time. New York: Bantam Books.
2. Hawking, Stephen (2005). A Briefer History of Time. London: Bantam Press.
3. Hawking, Stephen (2002). The Universe in a Nutshell.
4. Chen, Susan (2018, May 31). The De Broglie Hypothesis – Physics Research Project #1. Retrieved June 1, 2018, from Passion for STEM: https://passionforstem.wordpress.com/2018/05/31/the-de-broglie-hypothesis-physics-research-project-1/