Student guide
1. Orbitals
Choose a state such as 2p or 3d and compare its shape and nodes. The numbers mean: n sets the energy and overall size, l sets the shape (s, p, d, f for l = 0, 1, 2, 3), and the label after it (x, y, z, and so on) sets which way the shape points. Bright means likely, dark means unlikely, and black lines or rings are nodes where the electron is never found. Then switch to Superposition: one electron in two states at once. Watch the cloud slosh, and read the period in the panel.
2. Tunneling
Start with E = 1 eV, a 2 eV barrier and 0.3 nm width, and press Run. The electron does not have enough energy to go over the barrier, yet about 17% gets through. Change one slider at a time. Which change matters most? Then try an energy above the barrier: some of the wave still bounces back. Finally pick No barrier and watch the packet spread out as it travels.
3. Energy levels and light
Lift the electron to a high level, then drop it one step at a time and watch which lines appear in the spectrum. Only drops that end on level 2 give light you can see. Compare your lines with the four Balmer lines.
4. Size and Pauli
Drag the zoom slider from the whole atom down to the nucleus and notice how many times you must zoom. Then step through the first 18 elements and watch the boxes fill two at a time.
What the models leave out
The orbitals and energy levels are exact for the simple model of hydrogen (one electron, a point nucleus, no spin or relativity). Real hydrogen has tiny extra splittings. The tunneling tab is one dimension, not three. The pictures are slowed down enormously: real atoms move in femtoseconds.
A little history
Schrödinger, 1926. Building on Louis de Broglie’s idea that electrons behave like waves, Erwin Schrödinger, then a professor in Zürich, wrote his wave equation. His first paper, “Quantization as an Eigenvalue Problem,” reached the journal Annalen der Physik on 27 January 1926 and solved the hydrogen atom: the allowed energies came out of the equation itself.
Born’s probability reading, 1926. What is the wave? In a paper on collisions that summer, Max Born proposed in a note added in proof that the square of the wave’s size gives the probability of finding the particle. That reading, |ψ|² as probability, is what the brightness in these pictures shows. Born received the 1954 Nobel Prize in Physics for it.
Bohr and Schrödinger in Copenhagen, October 1926. Niels Bohr invited Schrödinger to Copenhagen. Schrödinger hoped his smooth waves would do away with sudden “quantum jumps”; Bohr and Werner Heisenberg argued the jumps were real. By Heisenberg’s later account the talks ran for days, and when Schrödinger fell ill Bohr kept arguing at his bedside. Schrödinger is reported to have said that if this damned quantum jumping was here to stay, he was sorry he had ever got involved. Afterward he wrote to thank Bohr warmly, but neither changed his mind.
Catching a quantum jump, 2019. At Yale, Zlatko Minev, Michel Devoret and colleagues watched an “artificial atom” (a superconducting circuit) jump between energy levels. They found a warning sign just before a jump, saw that each completed jump follows the same smooth path, and used that to catch and reverse jumps partway through. Their paper, “To catch and reverse a quantum jump mid-flight,” appeared in Nature in June 2019. In a sense both men had part of it: when a jump starts is unpredictable, as Bohr said, but the jump itself unfolds smoothly, as Schrödinger hoped.
A note on order: every hydrogen atom in the universe obeys the same equation and gives off the same colors. Scientists such as Kepler, Newton and Maxwell saw this kind of lawful order as a mark of a Creator whose world can be understood, and it is part of what makes physics possible at all.