Quick questions on Schrodinger's wavefunction and atomic orbitals: HSC Physics Module 8

In 1926 Erwin Schrodinger proposed a wave equation governing the de Broglie matter wave of a particle in a potential $V$. For a stationary state of definite energy $E$, the time-independent Schrodinger equation reads:

Max Born (1926) gave the wavefunction its physical interpretation. $\psi$ itself is complex and not directly measurable. The measurable quantity is:

The solutions for the hydrogen atom are labelled by three quantum numbers:

Bohr's success in hydrogen is recovered exactly: same energy levels and same Rydberg formula. But Schrodinger's model also explains why $2s$ and $2p$ exist as different angular shapes, predicts the ordering and filling of subshells, gives the chemical periodicity, and underlies essentially all of atomic, molecular and solid-state physics.

Schrodinger's equation is non-relativistic. The full theory of the electron requires the Dirac equation (1928), which automatically incorporates spin and predicts antimatter. Quantum electrodynamics (QED, 1948 onward) refines this further. At HSC level the Schrodinger picture with spin added is enough.

$\psi$ is a complex amplitude. $|\psi|^2$ is a probability density (probability per unit volume). The probability of finding the electron in a region is $\int |\psi|^2 \, dV$.

In stationary states the probability density is fixed in time, but each measurement still finds a localised electron. The smeared appearance is the distribution of many such localisations.

Orbits (Bohr) are paths. Orbitals (Schrodinger) are probability distributions. The shapes are 3D probability clouds, not trajectories.

Two electrons per orbital, with opposite spins. The Pauli exclusion principle is what makes electrons fill subshells rather than all collapsing into $1s$.

It is a fundamentally different conceptual framework: probabilistic instead of deterministic, wave equation instead of orbit postulates, and applicable to all atoms instead of just hydrogen.