Werner Heisenberg   

The uncertainty principle

People often know that this principle has to do with the fact that you can’t know the place and velocity at the same moment; Position is measured at one moment in time while the velocity has to be measured over a period of time. This explanation is close to the truth but the actual quantum mechanics are even more beautiful. To understand the uncertainty principle, you need to know what uncertainty in quantum mechanics actually means.

An example: The average length of the Dutch male is 181,1 cm. This is a measured fact but if you were to pick any one man, you’ll understand that there is only a chance that he is indeed exactly 181,1 cm. The chance of him being 180 cm is greater than him being 170 cm and that chance is greater than the chance of him being 160 cm. In quantum mechanics this works on particles. You only have a chance that you will find the particle where your calculation says it is going to be. The further away from that place you look, the smaller the chance that you will find the particle there. Because there are billions upon billions of particles, there will always be a lot of particles that you will find at the maximum distance from their calculated position.

Heisenberg uncertainty principle is a follows:

In which:

 Δx is the uncertainty in position, expressed in metres. 

 Δp is the uncertainty in momentum. Momentum is the mass multiplied by the velocity.

 ħ is the reduced Planck’s constant and that’s just a number  1,05457.10-34 Js 

The uncertainty principle states that the uncertainty in position times the uncertainty in mass times velocity is always greater than a certain number. When the uncertainty in position becomes really small, the uncertainty in momentum has to become really large and vice versa.

Proof again nuclear electrons.

We can prove that electrons can’t be in the nucleus using the uncertainty principle. If an electron would be confined in the nucleus, we would know its position very accurately, the uncertainty would be small. Because the mass of the electron is also quite well defined, the uncertainty on the velocity must be great.

We’ll calculate how great.

Let’s take the nucleus of an uranium atom, it has a diameter of 11,7142 fm (femtometre) so the radius is 5,8571.10-15 m.

The uncertainty in the mass of the electron can’t be more than the mass itself, you can have no electron (mass zero) but you can’t have a negative mass. The mass of an electron is 9,10938291.10-31 kg. 

An uncertainty of 9,9.109 m/s in the velocity of the electron means that there must be electrons that actually have that velocity and that is where things go wrong. According to Einstein’s special theory of relativity, nothing can go faster that the speed of light in a vacuum, often rounded off to 3.108 m/s. Some of these nuclear electrons would travel at 32 time the speed of light! This proved that no electron can be found inside the nucleus.

Why are there protons and neutrons in the nucleus?

Protons and neutrons are much heavier:

Mass of a proton: 1,672621777.10-27 kg

Mass of a neutron: 1,674927351.10-27 kg

Opdracht

33.     Use calculations to show that protons and neutrons don’t violate Einstein’s relativity when they are located in the nucleus of an uranium atom.