The Schrödinger equation
We saw earlier something called the Wave Function. What is a Wave Function? Max Born was the physicist who gave the best interpretation of the wave function. He says that , the wave function is a mathematical function that allows us at a given instant of time to locate the probability of a particle being found in a certain place in space. To better understand this sentence, let's remember a concept of statistics, in case you haven't seen it yet don't worry, the concept is quite simple. Watch the video below and try to understand the concepts, don't worry about the names and nomenclatures mentioned in the video.
A probabilidade na matemática é a área que estuda as chances de ocorrência de um resultado, que são obtidas pela razão entre casos favoráveis e casos possíveis. No caso físico que queremos descrever aqui podemos assumir que estas chances de ocorrência são equiprováveis, ou seja, têm a mesma chance de acontecer em todo o espaço.
A estatística é o campo da matemática que relaciona fatos e números em que há um conjunto de métodos que nos possibilita coletar dados e analisá-los, assim sendo possível realizar alguma interpretação deles.
Well, as we said, the electron has dual behavior, and because it has wave characteristics as we saw in the Heisenberg principle, we cannot define the specific location of a wave in space, in other terms, measurements in Quantum mechanics are completely statistical and not deterministic (It is the result of a probability of finding the electron at a given location). Recalling the concept of frequency, let's imagine the double slit, the more electrons are thrown against it, an interference pattern will form as we have seen. After a certain time there will be lines that are more intense than others demonstrating that the greater number of electrons has a greater frequency of appearing at that point. A common graph of frequency analysis is the Gaussian, see the image below.
The concept of ''Normal'' in statistics refers to a more frequent value. Therefore, all results that are measured within this ''Standard'' will be more frequent. The physicist who managed to arrive at a mathematical model to define where the electron could be in a given instant of time was Erwin Schrödinger. The relationship he arrived at was...
Schrödinger's equation for one dimension (1D)
The above equation involves second order partial derivatives and advanced concepts of differential calculus. It is not up to the high school student to know how to solve such an equation, but to understand what its utility and what it describes. For didactic purposes, let's break down this equation specifically so that the student has a good understanding of its mathematical arguments, that is, for the curious. It is not up to us to know how to solve it here. That's why we'll dedicate a page specifically to talk about the details of this equation at the bottom of the page.
What exactly does this equation say? Schrödinger says that for a given location in space, what is the probability of finding the electron at that point? Well! To define what this equation gives us, Schrödinger says that the wave function of the electron and the other particles are in superposition, that is, the electron has the possibility of being everywhere at the same time, until it is measured, forcing it set 1 state possible of the infinite too many what we call " collapse of the wave function." A simple analogy given by Schrödinger is the well-known "Schrödinger's cat".
The idea is, imagine a box. And in this box you put a cat and a bottle of poison inside and close the box. After a while, what would have happened to the cat inside the box? According to quantum mechanics, before we look at the box, the cat is in a superposition state. That is, the only two possibilities of something having happened to him are superimposed (together, at the same time), that is, the two possibilities coexist. In the first case, either the cat takes the poison and dies, or it doesn't take the poison and stays alive.
When we decide to look at the box to know the state of the cat, as soon as we open it we will either find it alive or dead, that is, the act of observing the system forces this system to take one of the possible states. Which means that we collapse the superposition of the wave, we force nature to define a specific state. The Schrödinger equation allows us to predict where and what are the chances of a certain result appearing at a certain specific point in space. To understand better, let's see the following videos and animation.
Obviously Schrödinger was a guy who loved cats right? LOL. Before anyone despairs of the poor kitty, there are no records of experiments that have used cats for wave superposition testing. It was purely an idea and a mental exercise by the theorist to better understand the concept. Of course, for macroscopic objects this superposition phenomenon makes no sense, the idea of Schrödinger's cat is to make an analogy to a quantum system as if we were looking at a macroscopic system, that is, as if we we had become quantum, reduced to the same scale as the atom or smaller than it and see such phenomena. If the concept is still not very clear, I usually use a different analogy.
Imagine you go to a restaurant and the waiter gives you the menu. You place your order for example pasta, and wait for it. When the waiter arrives with the tray closed even before opening it, did the macaroni come to me as ordered? If we are going to bring an overlapping analogy, before we open the tray all possible orders that the restaurant provides (the menu orders) are possible to be in this tray (the orders are in an overlap, they coexist), until we decide to open the tray and check if the order is really the macaroni. Only when we open the tray will we see the various possibilities, just one! For when we observe, nature forces this superposition to be collapsed (broken) and defines only a specific state, that is, what we actually observe around (in everyday phenomena).
E aqui faço um adendo importantíssimo! CUIDADO!!! Quando nos referimos a um "observador" em Mecânica Quântica não tem nada a ver necessáriamente com um observador consciente (pessoa ou uma "entidade" qualquer). Na interpretação de Copenhague, o "observador" é qualquer entidade ou sistema que interaja com o sistema quântico e realize uma medição, causando o colapso da função de onda. Essa entidade não precisa ser consciente; pode ser qualquer dispositivo físico que interfira no sistema e produza um resultado mensurável. Por isso dize-se que o ato de tentar medir(interagir) com um sistêma quântico altera seus resultados. Sistemas quânticos são sistemas extremamente sensíveis e voláteis a qualquer tipo de perturbação. Isso ficará mais claro na sessão onde será explicado uma das "N" aplicações da Mecânica Quântica.