Last Updated on April 26, 2025 by Sushanta Barman
Interactive Plotter for Hydrogen Atom Wave Functions
How do I use this wave function plotter?
- Select Quantum Numbers
Choose valid values of quantum numbers \( n \), \( \ell \), and \( m \) from the dropdown menus. - Generate the Plot
Click the “Plot Orbital” button to create a 3D isosurface visualization of the selected orbital. - Understand the Visualization
The plot displays both the radial and angular components of the hydrogen atom wavefunction. - Save the Plot
Click the “Save Plot as PNG” button to download the generated plot as an image.
Theoretical Background: Hydrogen Atom Wave Function
The quantum mechanical wave function of a hydrogen atom is the solution to the time-independent Schrödinger equation for an electron bound to a proton via the Coulomb potential. Owing to the spherical symmetry of the problem, the wave function is most naturally expressed in spherical coordinates \( (r, \theta, \phi) \), and it separates into radial and angular parts:
\[\psi_{n\ell m}(r, \theta, \phi) = R_{n\ell}(r) \cdot Y_{\ell}^{m}(\theta, \phi),\]
where, \( n \) is the principal quantum number (\( n = 1, 2, 3, \dots \)), \( \ell \) is the orbital angular momentum quantum number (\( \ell = 0, 1, \dots, n – 1 \)), and \( m \) is the magnetic quantum number (\( m = -\ell, \dots, +\ell \)). The function \( R_{n\ell}(r) \) represents the radial part of the wave function, which depends only on the radial coordinate \( r \), while \( Y_{\ell}^{m}(\theta, \phi) \) is the spherical harmonic function describing the angular dependence in terms of the polar and azimuthal angles \( \theta \) and \( \phi \), respectively.
Angular Part: Spherical Harmonics
The angular dependence of the hydrogen atom wave function is described by the spherical harmonics:
\[Y_{\ell}^{m}(\theta, \phi) = N_{\ell m} \cdot P_{\ell}^{|m|}(\cos\theta) \cdot e^{im\phi},\]
where, \( P_{\ell}^{|m|} \) denotes the associated Legendre polynomial, \( N_{\ell m} \) is the normalization constant that ensures the condition \( \int |Y_{\ell}^{m}|^2 \, d\Omega = 1 \), and \( d\Omega = \sin\theta \, d\theta \, d\phi \) represents the differential solid angle element in spherical coordinates.
The spherical harmonics determine the shape and symmetry of the orbitals, such as lobes and nodal planes.
Radial Part: Laguerre Polynomials
The radial function \( R_{n\ell}(r) \) describes the probability amplitude of finding the electron at a distance \( r \) from the nucleus. It takes the form:
\[R_{n\ell}(r) = N_{n\ell} \cdot \left( \frac{2r}{n a_0} \right)^{\ell} \cdot e^{-r / (n a_0)} \cdot L_{n – \ell – 1}^{2\ell + 1} \left( \frac{2r}{n a_0} \right),\]
where, \( a_0 \) is the Bohr radius, \( L_{n – \ell – 1}^{2\ell + 1} \) represents the associated Laguerre polynomial, and \( N_{n\ell} \) is the radial normalization constant.
This part determines the radial nodes and extent of the orbital in space.
Interpretation
The square modulus of the wave function, \( |\psi_{n\ell m}(r, \theta, \phi)|^2 \), gives the probability density of finding the electron at a particular location. The shapes of hydrogen orbitals—such as spherical \( s \)-orbitals, dumbbell-shaped \( p \)-orbitals, or the more intricate \( d \)- and \( f \)-orbitals—are direct manifestations of the angular part, while the radial distribution governs how far these structures extend from the nucleus.
I am a senior research scholar in the Department of Physics at IIT Kanpur. My work focuses on ion-beam optics and matter-wave phenomena. I am also interested in emerging matter-wave technologies.
