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Mathematics at the rescue

Human being invented something very useful for everything is difficult to understand: mathematics.

Can you imagine the size of the earth? Hmmm it's hard. But if I tell you that the earth diameter is 12742km you can understand that. You know what is 1km and what time it takes to walk this far and you can extrapolate to this distance.
Nevertheless you never “experienced” such huge distance, just like you cannot experience quantum physics.

Mathematics helps us to perceive what is beyond our understanding. Thanks to mathematics we can understand the incomprehensible. And it will help us to apprehend quantum mechanics!

And don't be frightened with formulas, this is no big deal!

Probabilities

We saw with Schrodinger's cat that quantum behaviors are based on probabilities. At least this is the way we perceive it.
This chapter to remind you basics of this branch of mathematics (that I always hated to be honest).

Let's consider the following example: you flip a coin and get the result: heads or tails.
You have 50% chances to get heads and 50% chances to get tails.

The basic rule of probabilities is that the sum of events percentages must be 100%. But in formulas we don't use percentages, we use numbers called probability amplitude.

In the case of tossing a coin, we have two amplitudes:

$p_{h}$ to get heads,
$p_{t}$ to get tails.

With probabilities amplitude, the sum of squared absolute values of probability amplitudes must be 1 .
Said differently: $delim{|}{p_{h}}{|}^2 + delim{|}{p_{t}}{|}^2 = 1$ . You see? Mathematics helps!

Because we have the same chance to get heads or tails, $p_{h} = p_{t} = 1/sqrt{2}$ . Let's check:

$delim{|}{1/sqrt{2}}{|}^2 + delim{|}{1/sqrt{2}}{|}^2 = 1/2 + 1/2 = 1$

Imagine now you find a trick to change the chance to get heads or tails. Let's say 70% chances to get heads and 30% chances to get tails. We would have:

$p_{h} = sqrt{0.7} = 0.8366$
$p_{t} = sqrt{0.3} = 0.54772$

Let's check: 0.8366^2 + 0.54772^2 = 1
Computed amplitudes are correct.

Superposition

Like Schrödinger's cat, quantum particles are in superposition between the generator and the detector. That is to say dead and alive.
The same for particles: when superposed it is both |0> and |1>.
This state is described with the following equation:

delim{|}{psi}{}> = alpha delim{|}{0}{}> + beta delim{|}{1}{}>

With:

the particle state,
the amplitude of ,
the amplitude of .

Between the generator and the detector the particle is in a quantum state balanced by probabilities between |0> and |1>. When particle hits the detector we read the result and get |0> or |1>.

If the generator is calibrated to launch particles with perfect balanced probabilities to get |0> and |1>, here is the state of particle between the source and the detector:

delim{|}{psi}{}> = 1/sqrt{2} delim{|}{0}{}> + 1/sqrt{2} delim{|}{1}{}>