Wave function
In quantum mechanics, the Wave function, usually represented by Ψ, or ψ, describes the probability of finding an electron somewhere in its matter wave. To be more precise, the square of the wave function gives the probability of finding the location of the electron in the given area, since the normal answer for the wave function is usually a complex number. The wave function concept was first introduced in the Schrödinger equation.
Mathematical Interpretation
The formula for finding the wave function (i.e., the probability wave), is below:
[math]\displaystyle{ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{x},\,t)=\hat H \Psi(\mathbf{x},\,t) }[/math]
where i is the imaginary number, ψ (x,t) is the wave function, ħ is the reduced Planck constant, t is time, x is position in space, Ĥ is a mathematical object known as the Hamiltonian operator. The reader will note that the symbol [math]\displaystyle{ \frac{\partial}{\partial t} }[/math] denotes that the partial derivative of the wave function is being taken.
Wave Function Media
The wave function of an initially very localized free particle.
Traveling waves of two free particles, with two of three dimensions suppressed. Top is position-space wave function, bottom is momentum-space wave function, with corresponding probability densities.
Scattering at a finite potential barrier of height V0. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. E > V0 for this illustration.
Still of first frame of animated gif
The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.
Continuity of the wave function and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.
