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The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations. [4]: II:268 The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions.
The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality.
Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension Wavefunction: ψ, Ψ To solve from the Schrödinger equation: varies with situation and number of particles Wavefunction probability density: ρ = | | = m −3 [L] −3: Wavefunction probability current: j
The quantum wave equation can be solved using functions of position, (), or using functions of momentum, () and consequently the superposition of momentum functions are also solutions: = + The position and momentum solutions are related by a linear transformation, a Fourier transformation. This transformation is itself a quantum superposition ...
The Schrödinger equation describes the space- and time-dependence of the slow changing (non-relativistic) wave function of a quantum system. The solution of the Schrödinger equation for a bound system is discrete (a set of permitted states, each characterized by an energy level ) which results in the concept of quanta .
Beyond the equations of motion, other aspects of matter wave optics differ from the corresponding light optics cases. Sensitivity of matter waves to environmental condition. Many examples of electromagnetic (light) diffraction occur in air under many environmental conditions.
The time-independent Schrödinger equation for the wave function is ^ = [+ ()] = (), where Ĥ is the Hamiltonian, ħ is the reduced Planck constant, m is the mass, E the energy of the particle. The step potential is simply the product of V 0 , the height of the barrier, and the Heaviside step function : V ( x ) = { 0 , x < 0 V 0 , x ≥ 0 ...
The Schrödinger equation applies to the new Hamiltonian. Solutions to the untransformed and transformed equations are also related by U {\displaystyle U} . Specifically, if the wave function ψ ( t ) {\displaystyle \psi (t)} satisfies the original equation, then U ψ ( t ) {\displaystyle U\psi (t)} will satisfy the new equation.