Expectation value is a very important topic in quantum mechanics, with the help of which we can get.
The idea of the position of a particle, first, let's take an example of the classical world, as you
can see in the diagram, let's say there are five stalls or food trucks within the range of five meters
at X is equal to one.
Let's say there are two people at X is equal to two.
There are three people.
At X is equal to three, there are two people at X is equal to four.
There are six people and at X is equal to five.
There are seven people.
So now if I want to calculate the probability of people standing at X is equal to four.
The answer is six by 20 and the probability at X is equal to one.
The answer is two by two and B, so 20 over here is the total number of people.
Now, if I take the average of X, it seems pretty simple like this.
Now, there is something called us weighted average whose equation looks like this.
Sigma X is equal to one, two, five.
X multiplied by B here, P is the probability of people standing at position X.
These are the discreet examples that we discussed.
And for discrete summation views, Sigma.
But for continuous system like this.
We use.
Integration.
God forbid.
Let's say someone has a viral fever and has to take injection from the medical doctor.
So the amount of viruses on the needle from point A to point B could be in thousands or more.
So this is the example of continuous system, but still we are in the classical world and we use integration
formally in the case of continuous system.
Now, let's enter the quantum world, weighted average in quantum world is called in expectation value.
So we already discussed earlier this example of identical test tubes where we located a quantum particle.
At a specific position.
If we take the same formula
now, we know that the probability of finding a quantum particle is Siwy multiplied by, say, conjugating
if we replace probability with PSI and its conjugate.
The final value looks like this.
This is the expected value of a quantum particle, which we can find across their best deal.
So basically, with the help of expectation value, we can track where exactly the quantum particle
could be hope, the concept of expectation value is clear.
Josh?