Before we derive anything today here is an interesting quote that the fortune command showed me :
MATH AND ALCOHOL DON’T MIX!
Please, don’t drink and derive.
– Mathematicians Against Drunk Deriving
Today in VERITAS we shall prove a fundamental law of Physics. We shall also see how different the world of elementary particles is from the world of our everyday experience.
Lets take 2 kids and lets give each one a white ball. Each kid knows which ball he has and can identify his ball. Now lets suppose the kids are not allowed to mark their balls with colors. i.e the balls look identical. Now if the kids throw their balls at each other they can identify which ball belongs to whom by following carefully the trajectory of the motion of the balls.
Now lets play with elementary particles(like electrons). All electrons look exactly the same(no way to color them). And there is no way to identify one electron from the other bacause we cannot follow the trajectory of their motion. Note that to follow the trajectory of motion of a particle we need to know its position and speed at every instance of time. This is not possible according to Heisenberg’s uncertainty principle.
So we come to the conclusion that elementary particles loose their identity when they are close to each other. Now lets make a even more remarkable statement: It is not that we humans cannot distinguish between these particles… Nothing in nature can distinguish between these particles!!!!
In Quantum Mechanics( the Physics of elementary particles) all the properties of a particle are denoted by a state. So when we have to say that particle 1 is in the state A we write it like this : |A>. So the statement that particle 1 is in state A and particle 2 is in a state B can be written as :
In our notation the particle on the left side is always particle 1 and one on the right is particle 2. Now the attentive reader will point out that we are distinguishing between the particles … and that is not done! Yes that is right. if |A> |B> is possible then |B> |A> is possible too. So what we are saying is that particle 1 can be in state A or in the B state and the same with particle 2.
Now we know that both |A> |B> and |B> |A> are possible. There is a relation between these 2. There are 2 kinds of particles : The Bosons (with integral spins. eg photons) and Fermions(with half integral spins eg electrons) . Now the relationship between |A> |B> and |B> |A> is different for both these:
For Bosons(eg photons) |A> |B> = |B> |A>
For Fermions(eg electrons) |A> |B> = – |B> |A>
So for Fermions the particles interchange with a – sign. This brings us to the fundamental Physics theorem which we wanted to prove:
Let the two particles be in the same state A . Then the statement looks like |A> |A> . Now notice that this state |A> |A> CAN ONLY BE TRUE FOR BOSONS! This is because |A> |A> can never be equal to -|A> |A>. So two fermions ( eg electrons) can NEVER be in the same state because they need to interchange with a – sign and that is not possible for something like |A> |A>. This is a fundamental principle of nature and is called Pauli’s Exclusion principle.
All of Chemistry is based on this principle. In a atom no two electrons can be in the same state. That means that they have to have different values of any or some of the following properties :
magnetic quantum number
This decides how the periodic table would be filled.
See how strange the Physics of elementary particles is. It violates the thinking that we have gained out of our everyday experiences.