Schrodinger Equation
The Schrodinger equation is the main key to physics for describing the quantum mechanical behaviour, It is an equation that describes how a wave function works in a physical system that evolves over time. Newton's second law (F = ma) is the main predection to what path a system will take over a surrent period following a set of conditions. In quantum mechanics, Schrodinger uses the Newton's law to create his equation. This equation helps to predict and study the quantum mechanical systems.
In order to understand the equation, let's see what each symbol mean. First we have the Psi: Simply, it is a quantum state. Which tells us the state of the quantum system, this means if we have a hydrogen atom then you can consider this as a quantum state and what the Schrodinger equation tells us is how this state should behave.
i = sqrt(-1), it is a complex number. h: Planck's constant = 6.62607004 × 10-34 m2 kg / s H: Hamiltonian operator, the basic funtionality of this symbol is that they operate in a quantum state. It can be expressed as the sum of the kinetic and potential energies and with operating it in the wave function, they shall produce the Schrodinger equation.
So basically, the equation shows us that the quantum state (psi) that changes by time someway depending on the total energy of the system. With this, there are a lot of limitations by simply calculating and using mathematics to solve the equation but we need a super computer to handle so many atoms which we currently don't have. In the future, maybe it will be easy to predict the quantum state of a system using high AI Intelligence in our advantage. Also, the foundation of the equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function ψ, which contains all the information that can be known about the system. In the Copenhagen interpretation, the modulus of ψ is related to the probability the particles are in some spatial configuration at some instant of time. Solving the equation for ψ can be used to predict how the particles will behave under the influence of the specified potential and with each other. Born rule is one of the key principles to quantum mechanics, because it shows us the link between the mathematical and the quantum field to make all predictions in quantum physics.
Increasing levels of wavepacket localization, meaning the particle has a more localized position.