The Heisenberg Uncertainty Principle
Newton, realizing why the moon was free falling toward Earth, tried to work on mathematics, and realized that the mathematics of the year 1600 was not enough to understand the motion of a free falling moon, so it is attributed. also to Newton the creation of the principles of the Calculus. At 23 Newton was able to unify one of the four fundamental forces of nature, namely Gravity. Newton was able to explain many of the movements that happened on Earth and the celestial movements through the famous equation called "Universal Gravitation Equation"
Newton, realizing why the moon was free falling toward Earth, tried to work on mathematics, and realized that the mathematics of the year 1600 was not enough to understand the motion of a free falling moon, so it is attributed. also to Newton the creation of the principles of the Calculus. At 23 Newton was able to unify one of the four fundamental forces of nature, namely Gravity. Newton was able to explain many of the movements that happened on Earth and the celestial movements through the famous equation called "Universal Gravitation Equation"
S-entropy; k- Boltzmann constant; W- probability
Newton, realizing why the moon was free falling toward Earth, tried to work on mathematics, and realized that the mathematics of the year 1600 was not enough to understand the motion of a free falling moon, so it is attributed. also to Newton the creation of the principles of the Calculus. At 23 Newton was able to unify one of the four fundamental forces of nature, namely Gravity. Newton was able to explain many of the movements that happened on Earth and the celestial movements through the famous equation called "Universal Gravitation Equation"
Newton, realizing why the moon was free falling toward Earth, tried to work on mathematics, and realized that the mathematics of the year 1600 was not enough to understand the motion of a free falling moon, so it is attributed. also to Newton the creation of the principles of the Calculus. At 23 Newton was able to unify one of the four fundamental forces of nature, namely Gravity. Newton was able to explain many of the movements that happened on Earth and the celestial movements through the famous equation called "Universal Gravitation Equation"
The exponential term is expressed as follows:
We can express the exponential as follows:
In which we can reduce the equation to: (by the fundamental rule of the product of powers with the same base)
So proved so far why the "omission" of the sinusoidal trigonometric argument in the quantum wave function, and one more proof that its solutions are complex. This demonstration is just for you reader to understand the terms involved and not necessarily know how to calculate them.
The important thing to know about the quantum wave function is that it in itself does not represent a physical meaning as stated, what actually interests us about it is to define its "magnitude" or its "Maximum probability" which we mathematically say, the its "norm".
The wave function norm is necessary, because as we say its solutions are complex, they contain negative results. But mathematically and physically it is not possible to have negative probability. This imposition that we make to the wave function is precisely to be able to extract the positive results that make some physical sense.
Note that the function norm shows not the same product of Ψ(x,t), but a term with (*). This term is what we saw in review of complex function called "Conjugate Complex" and is expressed as follows:
We can say that the conjugate complex is the negative part of the complex solutions and it is precisely it that when we apply the norm the quantum wave function is nullified, in which we are left with the following expression:
Now we have a plausible physical solution, since the amplitude of the quantum wave can never be negative, because it contains an even power, this implies that we will always have a probability P(x) ≥ 0 (greater than or equal to zero). This does make sense because there is no way for a particle not to exist in space. In other words, the probability that a particle is P(x) = 0 is only valid in a specific region. of the space that we may be analyzing, but not for all space itself. This theoretical imposition that we made the quantum wave function is what we call Normalization of the wave function , where we theoretically require that the probability of the wave function must be between:
That is, the probability of finding any particle in a region of space will be maximum if |Ψ(x,t)|² = 1, equivalent to 100% and if | Ψ(x,t)|² = 0 the probability is zero, equivalent to 0%. In this way, the quantum wave function ceases to have a merely mathematical meaning and actually has a physical meaning according to this interpretation of the wave function norm. This probabilistic notion of the wave function was given by Max Born and is the most accepted until today by the scientific community of how to interpret the quantum wave function. We call this interpretation the "Copenhagen Convention". There are several other interpretations of the quantum wave function, but it is not convenient for us to see others besides this one being the most important in the time.