hydrogen atom orbitals explained
An electron in hydrogen does not travel a fixed path like a little planet. The state is set by a wave function, and what we see is |ψ|², the cloud of the probability of finding the electron. The wave function splits into a radial part R(r) that sets only the distance and an angular part Y(θ,φ) that sets the direction, and the shape is set only by the three integers n, l, mₗ. The first picture starts from the probability with distance from the nucleus, P(r) = r²R(r)².
How likely the electron is to sit at a given distance from the nucleus is what the radial distribution P(r) = r²R(r)² says. The ground 1s makes a hump near r ≈ 1, the Bohr radius a₀. Change n and l and the curve ripples, with n − l − 1 radial nodes, and at those places the probability of finding the electron is 0.
Distance alone cannot tell the directional shape of an orbital. That shape is set by the angular part |Y(θ,φ)|. An s with l = 0 is the same in every direction and round like a ball, a p with l = 1 stretches into a dumbbell of two lobes along an axis, and a d with l = 2 is a four-leaf clover. Two lobes of different color in a slice mean the sign of the wave, the phase, is opposite.
Put radius and angle together and the real picture appears. Scatter dots on a slice in proportion to |ψ|², and a dense place is where the electron is found often. This cloud is the probability distribution, not a little orbit the electron travels. Change the orbital and the cloud scatters again into that shape, and the same seed makes a new scatter reproducible.
Three integers name one orbital. They are the principal n, the azimuthal l, and the magnetic mₗ, and the rule is l = 0…n−1, mₗ = −l…l. So names like 1s, 2s, 2p, 3d appear, and the buttons block a value outside the allowed range on their own. The three integers set distance, shape, and direction together.
Last, look at the density as a contour of color. A bright place is where |ψ|² is large, and a dark band is a nodal surface where the density is exactly 0. 2s shows a round radial node ring, 2p shows a transverse nodal plane, and 3d shows both. Farther out the density fades smoothly to 0, and an orbital has no sharply cut boundary.