Schrödinger equation – simple english wikipedia, the free encyclopedia gas zone edenvale


The Schrödinger equation is a differential equation (a type of equation that involves an unknown function rather than an unknown number) that forms the basis of quantum mechanics, one of the most accurate theory of how subatomic particles behave. It is a mathematical equation that was thought of by Erwin Schrödinger in 1925. It defines a wave function of a particle or system (group of particles) which has a certain value at every point in space for every given time. These values have no physical meaning (in fact, they are mathematically complex), yet the wave function contains all information that can be known about a particle or system. This information can be found by mathematically manipulating the wave function to return real values relating to physical properties such as position, momentum, energy, etc. The wave function can be thought of as a picture of how this particle or system acts with time and describes it as fully as possible.

The wave function can be in a number of different states at once, and so a particle may have many different positions, energies, velocities or other physical properties at the same time (i.e. "be in two places at once"). However, when one of these properties is measured it has only one specific value (which cannot be definitely predicted), and the wave function is therefore in just one specific state. This is called wave function collapse and seems to be caused by the act of observation or measurement. The exact cause and interpretation of wave function collapse is still widely debated in the scientific community.

where i {\displaystyle i} is the square root of -1, ℏ {\displaystyle \hbar } is the reduced Planck’s constant, t {\displaystyle t} is time, x {\displaystyle x} is a position, Ψ ( x , t ) {\displaystyle \Psi (x,\,t)} is the wave function, and V ( x ) {\displaystyle V(x)} is the potential energy, an as yet not chosen function of position. The left hand side is equivalent to the Hamiltonian energy operator acting on Ψ {\displaystyle \Psi } .

where the first equation is solely dependent on time T ( t ) {\displaystyle T(t)} , and the second equation depends only on position ψ ( x ) {\displaystyle \psi (x)} , and where E {\displaystyle E} is just a number. The first equation can be solved immediately to give

where e {\displaystyle e} is Euler’s number. Solutions of the second equation depend on the potential energy function, V ( x ) {\displaystyle V(x)} , and so cannot be solved until this function is given. It can be shown using quantum mechanics that the number E {\displaystyle E} is actually the energy of the system, so these separable wave functions describe systems of constant energy. Since energy is constant in many important physical systems (for example: an electron in an atom), the second equation of the set of separated differential equations presented above is often used. This equation is known as the Time independent Schrödinger Equation, as it does not involve t {\displaystyle t} . Interpretations of the Wave function [ change | change source ] Born Interpretation [ change | change source ]

There are many philosophical interpretations of the wave function, and a few of the leading ideas will be considered here. The main idea, called the Born probability interpretation (named after physicist Max Born) comes from the simple idea that the wave function is square integrable; i.e.

Where Δ x {\displaystyle \Delta x} is the uncertainty in position, and Δ p {\displaystyle \Delta p} is the uncertainty in momentum. This principle can be mathematically derived from the Fourier transforms between momentum and position as defined by quantum mechanics, but we will not derive it in this article. Other Interpretations [ change | change source ]