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Schrodinger equation (view source)
Revision as of 23:48, November 14, 2008
, 23:48, November 14, 2008no edit summary
where <math>i</math> is the [[complex number|imaginary unit]],<br><math>\hbar</math> is [[Planck's constant]] divided by <math>2\pi</math>, <br><math>|\Psi\rangle</math> is the quantum mechanical state or [[wavefunction]] (expressed here in [[Dirac notation]]), and <br><math>\hat H</math> is the [[Hamiltonian]] operator.
where <math>i</math> is the [[complex number|imaginary unit]],<br><math>\hbar</math> is [[Planck's constant]] divided by <math>2\pi</math>, <br><math>|\Psi\rangle</math> is the quantum mechanical state or [[wavefunction]] (expressed here in [[Dirac notation]]), and <br><math>\hat H</math> 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. For the simplest case of a particle of mass m moving in a one-dimensional potential V(x), the Schrodinger equation can be written
<math>
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
</math>
===Eigenvalue problems===
===Eigenvalue problems===
One example of this type of eigenvalue problem is an electrons bound inside an [[atom]].
One example of this type of eigenvalue problem is an electrons bound inside an [[atom]].
==Examples==
==Examples for the time-independent equation==
===Free particle in one dimension===
===Free particle in one dimension===
In this case, <math>V(x)=0</math> and so we see that the solution to the Schrodinger equation must be
<math>\psi=Ae^{-ikx}</math>
with energy given by
<math>E=\frac{\hbar^2 k^2}{2m}</math>
Physically, this corresponds to a wave travelling with a [[momentum]] given by <math>\hbar k</math>, where k can in principle take any value.
===Particle in a box===
===Particle in a box===
===Electron in a hydrogen atom===
Consider a one-dimensional box of width a, where the potential energy is 0 inside the box and infinite outside of it. This means that <math>\psi</math> must be zero outside the box. One can verify (by substituting into the Schrodinger equation) that
<math>\psi=\sin(kx)</math>
is a solution if <math>k=n\pi</math> where n is any integer. Thus, rather than the continuum of solutions for the free particle, for the particle in a box there is a set of discrete solutions with energies given by
<math>E_n=\frac{\hbar^2 k^2}{2m}=\frac{\hbar^2n^2\pi^2}{2m}</math>
[[Category:Physics]]
[[Category:Physics]]