# The Schrödinger equation

The Schrödinger equation is a mathematical equation that describes the evolution of a quantum mechanical system over time. It is a partial differential equation that governs the behavior of the wave function, which is a mathematical representation of the quantum state of a system. The equation is named after physicist Erwin Schrödinger, who formulated it in 1926.
The equation is written as:
iℏ ∂ψ/∂t = Hψ
In the bra-ket notation, the Schrödinger equation can be written as: iℏ ∂|ψ⟩ / ∂t = H|ψ⟩
Where:
i is the imaginary unit (equal to the square root of -1), ℏ is the reduced Planck constant, ψ (psi) is the wave function, t is time, H is the Hamiltonian operator, which contains information about the system’s energy and other physical properties.
The wave function, ψ, contains all the information about the system’s state, and the Hamiltonian operator, H, contains all the information about the system’s energy. The Schrödinger equation describes how the wave function changes over time, as the system’s energy changes. The solution of this equation is a complex function that describes how the system behaves over time.

The wave function collapse is a phenomenon that occurs in quantum mechanics when a quantum system is observed or measured. The wave function, which is a mathematical representation of the quantum state of a system, collapses from a superposition of states to a single state. In the standard interpretation of quantum mechanics, known as the Copenhagen interpretation, the wave function collapse is a random and instantaneous process that occurs when a measurement is made on a quantum system. The act of measurement causes the wave function to collapse from a superposition of states to a single state that corresponds to the measurement outcome. The probability of each outcome is determined by the amplitude of the wave function associated with that outcome. The Schrödinger equation is a mathematical equation that describes the evolution of a quantum mechanical system over time. However, it does not take into account the effect of measurement, which is why it is said that the Schrödinger equation describes the evolution of a closed system, whereas the collapse of the wave function is a description of the open system, that is, the system that is in interaction with the environment. The collapse of the wave function is not a physical process that can be observed, but rather a mathematical rule that describes the outcomes of measurements. The collapse of the wave function is not a physical process that occurs in the system being measured, but rather a change in our knowledge of the system.

The Schrödinger equation for the hydrogen atom is a specific form of the equation that describes the behavior of the electron in the hydrogen atom. It is given by:
-ℏ²/2m ∇²ψ(r,t) + V(r)ψ(r,t) = iℏ ∂ψ(r,t)/∂t
Where:
-ℏ²/2m is the kinetic energy operator, ∇² is the Laplacian operator, ψ(r,t) is the wave function of the electron in the hydrogen atom, V(r) is the potential energy of the electron in the hydrogen atom, which is given by V(r) = -e²/4πε₀r where e is the electron charge, ε₀ is the permittivity of free space and r is the radial distance from the nucleus, i is the imaginary unit (equal to the square root of -1), ℏ is the reduced Planck constant, t is time.
The wave function ψ(r,t) describes the probability density of finding the electron at a particular point in space and time. The equation describes how the wave function changes over time as the electron moves in the electric field of the nucleus. The solutions of the equation are the possible energy states of the electron in the hydrogen atom, and are known as the hydrogen atom wave functions.

The solutions of the Schrödinger equation for the hydrogen atom are represented by a set of mathematical functions known as the hydrogen atom wave functions or the hydrogen atom orbitals. These functions are represented by the symbol ψnlm(r,θ,φ), where n is the principal quantum number, l is the angular quantum number and m is the magnetic quantum number. Each wave function corresponds to a unique energy state of the electron in the hydrogen atom, known as the hydrogen atom energy levels. The principal quantum number, n, can take on positive integer values of 1, 2, 3, etc. The angular quantum number, l, can take on integer values from 0 to n-1. The magnetic quantum number, m, can take on integer values from -l to l.
The solutions of the Schrödinger equation for the hydrogen atom are used to explain the observed spectra of the hydrogen atom and the behavior of the electron in the hydrogen atom, including the energy levels and the probability density of finding the electron in a particular point in space.

Graphical representation of the hydrogen atom orbitals can be done in different ways, depending on the aspect you want to show. The most common representation is the probability density plot of the wave function, which shows the probability of finding the electron at a certain point in space. The hydrogen atom orbitals are represented by mathematical functions known as the hydrogen atom wave functions, which are denoted by the symbol ψnlm(r,θ,φ), where n is the principal quantum number, l is the angular quantum number and m is the magnetic quantum number. For example, the s-orbitals are represented by the wave function ψnlm = Rnl(r)Y00(θ,φ), where Rnl(r) is the radial wave function and Y00(θ,φ) is the spherical harmonic function. The radial wave function Rnl(r) gives the probability density of finding the electron at a certain distance from the nucleus, while the spherical harmonic function Y00(θ,φ) gives the probability density of finding the electron in a certain direction. The s-orbital are represented by a sphere with the nucleus in the center.
The p-orbitals are represented by the wave function ψnlm = Rnl(r)Ylm(θ,φ), where Rnl(r) is the radial wave function and Ylm(θ,φ) is the spherical harmonic function. The radial wave function Rnl(r) gives the probability density of finding the electron at a certain distance from the nucleus, while the spherical harmonic function Ylm(θ,φ) gives the probability density of finding the electron in a certain direction. The p-orbital are represented by a dumbbell shape, with a node in the center, where the probability of finding the electron is zero.
The d- and f- orbitals are represented by similar mathematical functions, and they have more complex shapes, with more nodes and lobes.

It’s worth noting that these representations are just a way to make the mathematical functions more intuitive, but the real behavior of the electron in the atom is described by the wave functions and their mathematical properties.

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