quantum mechanics intro

"The ‘path’ comes into existence only when we observe it"
-Heisenberg

the theory of quantum mechanics states that with every possibility for an event in nature to take place, there is a quantity, called amplitude, associated with each alternative. furthermore, the amplitude associated with the overall event is obtained by adding the amplitudes of each of the alternatives. the probability that the event will happen is equal to the square of the absolute value of the overall amplitude. thus, if f1 and f2 are the amplitudes of the two possibilities for a particular event to take place, the amplitude for the total event is (as follows):

f = f1 + f2

and the probability for the event to occur is given by

P = | f1 + f2 |2

in the macroscopic world the total probability for an event to take place is given by

P = P1 + P2

the sum of the probabilities of each alternative. in quantum mechanics

P = | f1 |2 + | f2 |2 + f1f2* +f1*f2

= P1 + P2 + f1f2* + f1*f2

showing that the law of computing probabilities is not that of classical physics. the two additional terms are due to the interference of alternatives. if the event is interrupted before its conclusion, for example by determining if the event takes place through alternative 1, the amplitudes of all other alternatives can no longer be added to the total amplitude. the fact that the total probability follows from the knowledge of the amplitudes of all interfering alternatives forms the basis of what is called the Heisenberg uncertainty principle. the uncertainty principle asserts that there is a natural limit to the accuracy of any measurement. for instance the momentum of a particle cannot be precisely specified without loosing all information about its position, and vice versa. the uncertainty principle demonstrates that there are fundamental limitations to the use of concepts based on every-day experience.

however, the uncertain principle can be broken down into
1. Schrödinger equation
2. Simulation method
i will explore these methods in a continuation to this little intro.