The Hydrogen Atom – virtualdialog

What is an orbital?

An orbital is not a path along which an electron circles. It is a wave function ψ (“psi”), the solution of the Schrödinger equation for an electron in the field of the nucleus. The square |ψ|² gives the probability of finding the electron at a given place. In the view above, brightness is exactly this probability density.

The hydrogen atom is the only atom that can be solved exactly. That is why its orbitals are the yardstick for all other atoms and molecules.

Energy and oscillation: E = h · f

The wave function oscillates like any wave. The law that links energy and frequency is E = h · f. Here h is Planck’s constant, a constant of nature: h = 6.626 × 10⁻³⁴ joule · second. It is tiny, which is why we notice nothing of quantum effects in everyday life. The higher the energy of a state, the faster it oscillates: f = E / h. The same holds for light: a light particle (photon) of 2.4 eV oscillates at about 5.8 × 10¹⁴ Hz, which is light around 520 nm. Each orbital oscillates in itself at the frequency of its energy. Because only the phase of the wave turns and not the density, you do not see this oscillation in the picture. It becomes visible when you superpose two states (see below).

How to read the picture

Brightness means probability: bright means the electron is often found here.
Colour means the sign of ψ (blue positive, orange negative). The sign itself cannot be measured, but it decides how orbitals behave when they combine.
Nodes are surfaces where ψ is zero. The electron is never there, although it occurs on both sides.
“Bounded” shows the body in which the electron stays with 90% probability. “Soft” shows the whole cloud as a mist of dots.

The notation: 1s, 2p, 3d and the electron configuration

Orbitals have short names made of a number and a letter. The number is the shell n (1, 2, 3 …). The larger it is, the further out the orbital lies. The letter gives the shape: s (sphere), p (dumbbell), d (cloverleaf), f (even more complex). “2p” therefore means: second shell, dumbbell shape. A subscript names the orientation: 2p_z is the dumbbell along the z axis, 3d_z² the d orbital with a ring.

Electron configuration

Atoms with several electrons are described by listing which orbitals are occupied. The superscript number says how many electrons sit in that group: 1s¹ is hydrogen (one electron), 1s² is helium (two electrons). Sodium is written 1s² 2s² 2p⁶ 3s¹. In many atoms the inner shells look like those of a noble gas, so they are abbreviated: [Ne] 3s¹ means “neon core plus one 3s electron”. Each group has a maximum: s holds 2 electrons, p 6, d 10 and f 14 (two electrons per orbital, and there are 1, 3, 5 and 7 orbitals). In the ../../../en/atoms-and-molecules/gold/ chapter we use this to read the configuration of gold: [Xe] 4f¹⁴ 5d¹⁰ 6s¹.

Where do the orbital types occur in nature?

s: hydrogen and helium, of which the Sun and stars consist almost entirely. The single outer electron of the alkali metals is also an s electron: sodium glows yellow in street lamps and in table salt in a flame.
p: carbon, nitrogen and oxygen. Their directional p orbitals form the bonds from which water and the molecules of life are built.
d: transition metals such as iron (blood pigment, magnets), copper and gold. The d electrons provide colours, magnetism and the action of many catalysts.
f: rare earths such as neodymium (strong magnets), europium (red phosphors in screens) and erbium (amplifiers in fibre-optic cables).

The three quantum numbers: the address of an orbital

Every orbital has an “address” made of three numbers, much like a flat: building, floor, door. These numbers are called quantum numbers. They cannot be chosen freely; they are whole numbers with fixed rules.

n – the shell (the floor): 1, 2, 3 …: It sets size and energy. The larger n, the further out the orbital lies and the higher its energy (−13.6 eV / n²). At n = 1 the atom is about 0.1 nm across, at n = 2 about four times and at n = 3 about nine times as large. Also called the principal quantum number.
l – the shape (the type of room): 0 to n − 1: It sets the shape: l = 0 is s (sphere), l = 1 p (dumbbell), l = 2 d (cloverleaf), l = 3 f. The number l is also the number of nodal surfaces passing through the nucleus. A p orbital has one plane in which the electron is never found. Because l can be at most n − 1, the 1st shell has only s, the 2nd s and p, the 3rd s, p and d. Also called the azimuthal or angular-momentum quantum number.
m – the orientation (the direction of the room): −l to +l: It sets which way the orbital points in space, measured against a chosen axis (here z). There are 2l + 1 possibilities: one s orbital, three p orbitals (m = −1, 0, +1), five d orbitals, seven f orbitals. Intuitively, m says how many times the phase runs around the z axis (twice for m = +2, and the sign gives the direction of rotation). The real orbitals with names like p_x or d_z² are mixtures of +m and −m.

How many orbitals are there?

Shell n	Shapes l	Orbital types	Number of orbitals
1	0	1s	1
2	0, 1	2s, 2p	4 (1 + 3)
3	0, 1, 2	3s, 3p, 3d	9 (1 + 3 + 5)
4	0, 1, 2, 3	4s, 4p, 4d, 4f	16 (1 + 3 + 5 + 7)

So the n-th shell has n² orbitals. Each holds two electrons with opposite spin (↑ or ↓, a fourth quantum number), which makes 2n² electrons per shell. This gives the structure of the periodic table.

In hydrogen all orbitals with the same n have the same energy. In heavier atoms the interaction between electrons lifts this degeneracy, which is why 2s lies lower than 2p there.

Why do orbitals have these shapes?

An electron in an atom is a standing wave, much like the vibration of a guitar string or a drumhead. Only certain vibration patterns fit into the space around the nucleus. The more nodes a pattern has, the more energy it carries. The number of radial nodes is n − l − 1 and the number of nodal surfaces through the nucleus is l. Together they make n − 1.

The picture of a vibrating plate. If you sprinkle sand on a vibrating metal plate, it collects on the lines that stay still (Chladni figures). Those are the nodes. Orbitals are such natural vibrations, only in the three-dimensional space around the nucleus. The comparison goes on: the plate’s displacement up or down corresponds to the sign of ψ (blue and orange). A long-exposure photo of the vibrating plate would be bright where it moves most and dark at the nodes. The electron’s probability density is distributed in the same way: large where ψ is large, zero at the nodes. So the sand lies where the electron is not. The higher the natural vibration, the more nodes and the higher the frequency, for the plate as for the orbital.

Superposition: when an atom emits light

Choose “Superposition” above to mix two states. This is allowed because the Schrödinger equation is linear. The result is no longer a still orbital but a cloud that oscillates at the frequency ΔE/h. If the centre of charge moves while it does so (for example 1s + 2p), the atom radiates light of that frequency like a tiny antenna. This is how the spectral lines of hydrogen arise, for example Lyman-α at 121.6 nm or Balmer-α at 656 nm.

Frequently asked questions

Where is the electron really?: It has no definite position until you measure it. The orbital only tells you with what probability you will find it where.
Doesn’t the electron orbit the nucleus like a planet?: No. Bohr’s planetary model was an important stepping stone but is outdated. A 1s electron, for example, has no angular momentum, so it cannot “circle”.
Why does the picture show signs if only |ψ|² can be measured?: The sign decides how waves combine: equal signs reinforce, opposite signs cancel. This is the basis of the chemical bond (see the next chapter).
How big is a hydrogen atom?: The mean distance between electron and nucleus in the ground state is 1.5 Bohr radii, about 0.08 nm. The Bohr radius is 52.9 pm.
Are there always two electrons in an orbital?: No. An orbital holds at most two electrons, and they must have opposite spin. Hydrogen has just one electron, which sits in the 1s orbital (1s¹). All the other orbitals you see here are empty in the ground state: they show where an electron could be. In helium there are two in the 1s (1s²).
Does the electron jump back and forth between the blue and the orange lobe?: No. Blue and orange are signs of the wave function: the two lobes oscillate in opposite phase, like the two halves of a standing wave. The electron is not in one lobe and then switching over; it is spread over both at once (as a probability), just never in the nodal plane between them. The density you see does not change in the process.
Does the density drop suddenly towards a nodal plane?: No, it falls steadily to zero and rises again on the other side. Exactly at the plane ψ is zero and grows from there linearly with the distance. The density, the square, therefore grows with the square of the distance: very flat, like a parabola floor. There is never a hard jump. That the pictures sometimes look sharp at planes is due to the display: the colour change from blue to orange looks like an edge, the brightness is compressed for better visibility, and the “Bounded” style deliberately cuts off at a fixed density (90% of the probability).

Think about it

Which orbital with n = 3 has two spherical shells as nodes?

The 3s orbital. The number of radial nodes is n − l − 1, so 3 − 0 − 1 = 2. Check it in the slice: you see two colour changes that appear as rings.

Why does 1s + 2p radiate light but 1s + 2s does not?

Only a cloud whose centre of charge moves back and forth can radiate. That is the case for 1s + 2p (Δl = ±1). For 1s + 2s the cloud just “breathes” spherically.

How many d orbitals are there on one level?

Five, because there are 2l + 1 orientations for l = 2. You see them under “Orientation” as soon as you choose n = 3 and l = d.

In the next chapter we look at a heavy atom: gold.

A question or an idea?

Something unclear, an error you spotted, or a chapter you would like to see? Write to me. Good questions and their answers will appear here later as an FAQ.

Write a message

Last updated: 2026-10-03