Schrödinger's Equation Analysis
Initial Approach
In order to give a clearer mathematical notion of what Eq.Schrödinger tells us, let's do a basic analysis of the equation, after all this is the central theme of the site and not to leave the subject too vague in relation to the previous page and not too in-depth on this page, because to understand the completeness of the equation it is required the notion of Differential and Integral Calculus , Multivariable Calculus , Partial Differential Equations and Complex Numbers.
The purpose here is not to know how to solve the equation itself, but to make an approach basic of Calculus said above, without delving into the concepts in it itself and delving a little more into pertinent concepts of elementary algebra and pre-calculus, to stay as clear possible we will be giving examples of each concept to facilitate the understanding of what we want to give about the explanation of Eq.Schrödinger.
1) What is Differential and Integral Calculus?
Differential and Integral Calculus, which we will just call Calculus onwards, covers all the others Calculations mentioned above. the calculation emerged in the need to create a more accurate tool and analytics about some phenomena that were being studied in antiquity, one of those who contributed to its creation was Isaac Newton, as we mentioned in the Review of Mechanics and another also that contributed for Calculus along with Newton was Leibniz.
One of these needs was used in the study of celestial bodies by Newton in his theory of Gravitation and his 3 Laws of Mechanics. The problem was... How can I measure bodies that are in constant motion relative to each other simply using algebra? Think about how complicated it would be to describe the movement of a star if it changes its position every day and every day you have to recalibrate equipment, redo the following calculations... Conventional algebra simply did not handle these problems and many others when it was a motion of any body, not with the same precision and in some cases it had no solution, which was exactly what Newton wanted to do. Newton then began to work on mathematics and found something fantastic and have an extremely geometric and easy-to-see character.
The first concept that we are taught in the Calculus course is the concept of Limit ! In a didactic way I call this operator "Aproximometer".
I know what you must be thinking right now... "Limit?🤨Operator?😣. Calm down! Let's start by explaining what an operator means in a generic way in mathematics. Understand operator simply as an algebraic tool for manipulating values, and in some cases "parameters". In algebra we have 4 well-known elementary operators that you are already more than familiar with.
a) We have the SUMoperator , and yours notation is (+): For example:
We take a value any A and any other value B and we operate them (manipulate them...) by a sum rule (associate/join) which gives us another value C as a result. example
OR
The same analogy can be made with the rest of the operators you already know, operator (Subtraction (-) ; Division (÷) and Multiplication (×)) and their respective rules for "handling" values and parameters.
Unlike the elementary algebra operators, the Limit operator does not operate with numbers, but with values of functions ! Yes!😂 Let's see this concept again from now on, one of the concepts if not the most important in all of Calculus.
b) Functions: I believe that many of you when you started to study this topic in mathematics did not take it as seriously as you should. Let's make a good brushstroke on the concept of function and important concepts to be understood within the subject. Starting with the Domino idea of a function (𝔻).
I know for now review this apparently it will make no sense with Eq.Schrödinger's explanation. Believe me, it will do a lot, because we don't have the objective of knowing how to calculate numerically the Eq. But to understand its implications and what it is actually describing, and like it or not, the whole intuitive concept revolves around the Domain of the wave function that we talk so much about. For this we have to have a clear notion of this concept to understand factors that they are in the expression, which posteriorly we will talk about one of the topics about Complex Numbers.
So let's start defining some important concepts:
● What is Domino? Domain of a function is nothing more than a set of values that satisfy the solution of a given function.
● What is a Function? It is a mathematical representation that we make of an association of values from one numerical set to another, when manipulated or not to a set of values. operations, such as (addition, division, root, etc.).
Let's give some examples of why reviewing these concepts have importance. To start, let's see 3 types of functions to be able to exemplify the application of domain study.
● Linear Function (Eq.straight) :
This is one of the simplest functions we learned in school, the first degree function (line equation). We know that (a) and (b) are constants, that is, values that do not change and the only parameter that can have different values is our variable (x). Now, in order to analyze the domain of this function, we have to look if the function requires any algebraic restrictions. In this specifically there are no restrictions because any value we put in (x) will have a solution. We will use the most common notation to represent domains.
That is, (x) is contained in all real numbers. As an example, let's assume a = 1 and b = 0 and show this function on the graph. For any value of (x) [axis on the horizontal] we will have a value associated with the axis (y) [axis on vertical].
● Rational Function : The function rational is another type of function that we have to remember because there is a constraint in which we have no solution.
The rational function is nothing more than a function composed of two other functions divided one by the other, with the restriction of h(x) ≠ 0, since there is no solution for divisions with 0. To understand, let's see the example:
Note that f(x) is composed of two other functions. g(x) = x+3 and h(x) = x-3, remembering that being a division we cannot have a division by 0. So to find the domain of this function (that is, the range of values in which the function itself has a solution) is...
This means that the only value that h(x) cannot take for the denominator to respect the constraint of a rational function, that is, that we cannot divide by 0, (x) can take on any value except 3. It has to be different from 3. Writing the domain, we have:
The analysis of the domain of a function helps us to realize that exactly by restricting the existence of solutions for the rational function, we know that exactly at the value 3 there is no solution. What does it mean to have no solution? See the graph itself, at the point x=3 where the function h(x) would go to zero, the own function f(x) does not touch that point. That is, there is no solution for f(x) when x=3, not in the set of Reais (ℝ) at least.
The last function that we are going to see and not least, is the most important function perhaps that we would have to understand.
● Root Function : The function source also is one of several functions that have a certain constraint on existence. As you may have already seen, there is no square root with negative numbers, only positive ones.
So to determine the domain of the square root function, we would be with:
The point we want to emphasize this function is that the square root function, as it is an inverse of the quadratic function, does not admit negative values, because any number raised to the square or to any even exponent will ALWAYS be positive in the set of Reals (ℝ) ). The problem is that square roots can accept negative numbers, but it is not part of the set of real numbers (ℝ), because there is no consistency for such a solution in it, so over the years mathematicians noticed the beginning of another set mathematician who admits negative numbers in roots, the set of Complex Numbers (ℂ). We will further explain the concept in the pages what is the definition of complex numbers.