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Schrodinger equation

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New page: The '''Schrodinger equation''' is a linear [[differential equation]] used in various fields of [[physics]] to describe the time evolution of quantum states. It is a fundamental aspect of ...
The '''Schrodinger equation''' is a linear [[differential equation]] used in various fields of [[physics]] to describe the time evolution of quantum states. It is a fundamental aspect of [[quantum mechanics]]. The equation is named for its discoverer, [[Erwin Schrodinger]].

==Mathematical forms==
===General time-dependent form===
The Schrodinger equation may generally be written

$i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle$

where $i$ is the [[complex number|imaginary unit]],<br>$\hbar$ is [[Planck's constant]] divided by $2\pi$, <br>$|\Psi\rangle$ is the quantum mechanical state or [[wavefunction]] (expressed here in [[Dirac notation]]), and <br>$\hat H$ is the [[Hamiltonian]] operator.

The left side of the equation describes how the wavefunction changes with time; the right side is related to its energy.

===Eigenvalue problems===
In many instances, steady-state solutions to the equation are of great interest. Physically, these solutions correspond to situations in which the wavefunction has a well-defined [[energy]]. The energy is then said to be an [[eigenvalue]] for the equation, and the wavefunction corresponding to that energy is called an [[eigenfunction]] or [[eigenstate]]. In such cases, the Schrodinger equation is time-independent and is often written

$E\psi=\hat H\psi$

Here, ''E'' is energy, ''H'' is once again the Hamiltonian operator, and $\psi$ is the energy eigenstate for ''E''.

One example of this type of eigenvalue problem is an electrons bound inside an [[atom]].

==Examples==
===Free particle in one dimension===
===Particle in a box===
===Electron in a hydrogen atom===

[[Category:Physics]]
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