Hydrogen Atom Wavefunction Calculator

Enter the quantum numbers (n, l, m) and radius r (in Bohr radii a0) to compute the hydrogen radial wavefunction R_nl(r) and radial probability density, with energy level E_n and orbital name.

Input

Enter the principal quantum number n, azimuthal number l, magnetic number m, and the radius r in Bohr radii a0. The radial wavefunction and probability density are computed.

Principal quantum number n

Integer of 1 or more, sets the shell energy level

Azimuthal quantum number l

Integer from 0 to n minus 1, sets the orbital shape

Magnetic quantum number m

Integer whose magnitude is at most l, sets the orbital orientation

Radius r

Distance from the nucleus, in units of the Bohr radius a0

Result

Radial probability density P(r) = r squared |R_nl| squared

0.195367a0^-1

Radial wavefunction R_nl(r) = -0.110501 a0^-3/2

Orbital

Energy level E_n

-3.401423 eV

Radial nodes

Most probable radius

4 a0

The radial wavefunction R_nl(r) uses the normalized formula with associated Laguerre polynomials, with lengths in Bohr radii a0. The energy level is E_n = minus 13.605693 over n squared eV, and the number of radial nodes is n minus l minus 1.

How it works

Enter the quantum numbers (n, l, m) and radius r (in Bohr radii a0) to compute the normalized radial wavefunction R_nl(r) and the radial probability density P(r) = r squared times the absolute value of R_nl(r) squared.
R_nl(r) is built from the normalized formula using associated Laguerre polynomials. All lengths are in Bohr radii a0, so R_nl has units of a0^(-3/2) and the radial probability density P(r) has units of a0^(-1).
The energy level is E_n = minus 13.605693 divided by n squared, in eV, and the orbital name comes from n and the subshell letter (s, p, d, f and so on).
Quantum numbers must satisfy n at least 1, l between 0 and n minus 1, and the magnitude of m at most l. The number of radial nodes is n minus l minus 1.
The chart shows the radial probability density over radius, and the dashed line marks the radius r you entered.

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