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Schrodinger equation (view source)
Revision as of 16:04, September 19, 2016
, 16:04, September 19, 2016Added some stuff on normalisation of wavefunctions
<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>
<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>
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.
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
</math>
</math>
Note that solutions may be [[complex numbers|complex]] in nature.
===Derivation===
===Derivation===
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]].
===Interpretation of the Wavefunction===
The wavefunction is related to the probability of observing a particle at that location in space. More specifically, where the wavefunction has a greater value of <math>|\Psi|^2</math> at one location, a, compared to another, b, then the particle is more likely to be observed at a than b.
====Normalisation====
Wavefunctions are said to be "normalisable" if the following integral exists:
<math>\int^{\infty}_{- \infty} |\Psi(x, t)|^2 dx </math>
If that integral exists, then the wave function may be normalised by introducing a constant, <math>c</math>, such that
<math>\int^{\infty}_{- \infty} |c \Psi(x, t)|^2 dx = |c|^2 \int^{\infty}_{- \infty} |\Psi(x, t)|^2 dx = 1 </math>
Note that <math>c</math> may be complex. This will produce and new wavefunction, <math>c \Psi(x,t)</math>, which has the property that:
<math>P(a<x<b) = \int^{a}_{b} |c \Psi(x, t)|^2 dx</math>
This is the '''probability''' of observing the particle between points a and b.
For some wavefunctions, such as that of a free particle, the integral above does not exist and so they are said to be non-normalisable.
==Examples for the time-independent equation==
==Examples for the time-independent equation==
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