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The wave function of an initially very localized free particle. In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). Wave functions are complex ...
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. [ 1 ] : 1–2 Its discovery was a significant landmark in the development of quantum mechanics .
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
It relates the scattered wave function with the interaction that produces the scattering (the scattering potential) and therefore allows calculation of the relevant experimental parameters (scattering amplitude and cross sections). The most fundamental equation to describe any quantum phenomenon, including scattering, is the Schrödinger equation.
The term "wave function" is typically used for a different mathematical representation of the quantum state, one that uses spatial coordinates also called the "position representation". [9]: 324 When the wave function representation is used, the "reduction" is called "wave function collapse".
= if and only if is exactly equal to the wave function of the ground state of the studied system. The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.
In the Schrödinger picture, the wave function or field is the solution to the Schrödinger equation; = ^ one of the postulates of quantum mechanics. All relativistic wave equations can be constructed by specifying various forms of the Hamiltonian operator Ĥ describing the quantum system .
Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function | = (), using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions, [6] [7] =!