Function Repository Resource:

# HydrogenWavefunction

The position-space wavefunction of the hydrogen atom

Contributed by: Matt Kafker
 ResourceFunction["HydrogenWavefunction"][{n,l,m},a,{r,θ,ϕ}] gives the wavefunction for the hydrogen atom with quantum numbers (n,l,m) and Bohr radius a as a function of the spherical coordinates r,θ and ϕ. ResourceFunction["HydrogenWavefunction"][{n,l,m},a,{r,θ,ϕ},Z] gives the hydrogen-like wavefunction with nuclear charge Z.

## Examples

### Basic Examples (2)

The hydrogen ground state wavefunction:

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The squared magnitude of the wavefunction gives the probability distribution for finding the electron:

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### Scope (3)

Plot radial dependence of a few wavefunctions:

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Plot the polar dependence of one wavefunction at various radii:

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Plot the electron probability density for various wavefunctions:

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### Properties and Relations (4)

Verify the orthogonality property of HydrogenWavefunction:

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Verify the normalization property of HydrogenWavefunction:

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Verify that HydrogenWavefunction satisfies the time-independent Schrödinger equation:

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Show a change of nuclear charge:

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### Neat Examples (1)

Plot hydrogen orbital densities:

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## Publisher

Wolfram Summer School

## Version History

• 1.0.0 – 07 June 2021