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Every s, p and d orbital you draw in chemistry is a solution of one equation, written in 1926 by an Austrian physicist who later argued with almost everyone about what it meant. Knowing how Erwin Schrodinger got there, and what he got wrong, makes orbitals easier to understand. This post covers the history and then works through the energies and node counts that exams ask for.
Who was Erwin Schrodinger?
His Nobel Prize biography says he was born in Vienna in 1887 and studied at the University of Vienna from 1906 to 1910. There he "acquired a mastery of eigenvalue problems in the physics of continuous media", the maths of vibrating strings and drums. The same biography notes that he disliked memorising data and learning from books. He served as an artillery officer in the First World War and then moved through several universities before settling at Zurich.
At Zurich, during the first half of 1926, he found his equation. The biography says it came from his dissatisfaction with the quantum condition in Bohr's orbit theory and his belief that atomic spectra should be set by an eigenvalue problem, the same kind of problem that fixes the notes a string can play.
What did the 1926 papers do?
Schrodinger published a series of papers titled "Quantisierung als Eigenwertproblem" (quantisation as an eigenvalue problem) in Annalen der Physik. Later that year he summarised the theory in English in Physical Review. The abstract lists the wave equation, its application to the hydrogen atom, and "the intrinsic reason for the appearance of discrete characteristic frequencies".
That last phrase is the key idea. In Bohr's model, the rule that only some orbits are allowed was simply imposed. In Schrodinger's equation, only certain energies give acceptable waves around the nucleus, just as only certain wavelengths fit on a guitar string. The Nobel Prize summary says the equation "accurately calculated the energy levels of electrons in atoms". He shared the 1933 Nobel Prize in Physics with Paul Dirac.
What does the wave function mean?
Here Schrodinger and most of his colleagues parted ways. The abstract of his Physical Review paper says the wave function "physically means and determines a continuous distribution of electricity in space". Max Born proposed a different reading in a 1926 paper on collisions: the square of the wave function gives the probability of finding the particle. Born received the 1954 Nobel Prize in Physics "especially for his statistical interpretation of the wavefunction".
Schrodinger never liked it. His Nobel biography says he "disliked the generally accepted dual description in terms of waves and particles, with a statistical interpretation for the waves". His famous cat, from a 1935 article in Die Naturwissenschaften, was meant to show how strange the probability reading looks when applied to everyday objects.
Worked example: energies and nodes from the wave equation
You will not solve the equation in school, but you use its results all the time.
Energy levels of hydrogen. The solutions give Eₙ = −13.6/n² eV.
n = 1: −13.6 eV
n = 2: −3.40 eV
n = 3: −1.51 eV
An electron dropping from n = 3 to n = 2 releases 3.40 − 1.51 = 1.89 eV. The wavelength is λ = hc/E = (6.626 × 10⁻³⁴ × 3.00 × 10⁸) / (1.89 × 1.602 × 10⁻¹⁹) ≈ 6.56 × 10⁻⁷ m, or 656 nm, the red line of hydrogen.
Nodes. As NCERT Class 11 Chemistry states, an orbital has n − 1 nodes in total: l angular nodes and n − l − 1 radial nodes.
2s (n = 2, l = 0): 1 radial, 0 angular.
3p (n = 3, l = 1): 1 radial, 1 angular.
4d (n = 4, l = 2): 1 radial, 2 angular.
Common mistakes:
Calling an orbital an orbit. An orbital is a region of probability, not a path.
Dropping the minus sign. Bound-state energies are negative; the energy released is the difference.
Counting nodes at the nucleus or at infinity. Radial nodes are the spheres in between where ψ = 0.
Mixing up ψ and ψ². Graphs of ψ can dip below zero; graphs of ψ² cannot.
The JEE (Main) 2026 syllabus asks for "atomic orbitals as one-electron wave functions" and the variation of ψ and ψ² with r for 1s and 2s orbitals. Check the bulletin for your exam year.
What happened to Schrodinger later?
He succeeded Planck in Berlin in 1927, left Germany in 1933 after Hitler came to power, and after several moves became director of the School for Theoretical Physics at the new Dublin Institute for Advanced Studies, where he stayed until 1955. His 1944 book What is Life? looked at biology through a physicist's eyes.
What can a student take from Schrodinger's way of working?
Old tools solve new problems. The eigenvalue maths he learned as a student in Vienna became the key to the atom. Learn your maths well even when it seems unrelated.
Ask why a rule holds. Bohr's allowed orbits worked, but Schrodinger wanted the reason, and finding it gave a better theory.
You can be right about the equation and wrong about its meaning. Keep checking your interpretation against evidence.
Orbitals are easier to grasp when you can turn them around, and the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read. For exam practice, see our guide to hydrogen spectrum and Bohr model problems.
Try this today: work out the radial and angular nodes for 3s, 3d and 5f, then check that each adds up to n − 1.
