Investigate the contribution of Schrodinger to the current model of the atom, including the probabilistic interpretation of the wavefunction and the concept of atomic orbitals replacing Bohr's fixed orbits

What this dot point is asking

NESA wants you to describe Schrodinger's wavefunction $\psi$ and the Born probability interpretation $|\psi|^2$, explain how the time-independent Schrodinger equation gives standing-wave solutions (atomic orbitals) with definite energies, identify the four quantum numbers and the standard orbital shapes (s, p, d, f), and contrast Schrodinger's model with Bohr's earlier picture.

The answer

The Schrodinger equation

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:

The unknown is $\psi(\vec r)$, the wavefunction. Solving it for the hydrogen atom (with $V = -k e^2 / r$) gives:

• the same energy levels $E_n = -13.6 \text{ eV} / n^2$ as the Bohr model,
• but as a consequence of a wave equation, not a postulate,
• with a wavefunction $\psi_{n \ell m}(\vec r)$ for each state that has a definite shape in space.

For multi-electron atoms the equation becomes too complicated to solve exactly, but accurate numerical methods give all the observed spectra and chemical properties.

Born's rule: $|\psi|^2$ as a probability density

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

the probability of finding the particle in a small volume $dV$ at position $\vec r$. The total probability integrates to 1:

This is the central conceptual shift in quantum mechanics: physical predictions are probabilities, not definite values. For an electron in an atom, $|\psi|^2$ gives the density of the "electron cloud" you see in textbooks.

Atomic orbitals

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

• Principal quantum number $n = 1, 2, 3, \dots$. Determines the energy and the average size of the orbital. Corresponds to Bohr's $n$.
• Orbital angular momentum quantum number $\ell = 0, 1, 2, \dots, n-1$. Determines the shape. Letters: $\ell = 0$ is $s$, $\ell = 1$ is $p$, $\ell = 2$ is $d$, $\ell = 3$ is $f$.
• Magnetic quantum number $m_\ell = -\ell, \dots, +\ell$. Determines the orientation in space.

The fourth quantum number, spin $m_s = \pm 1/2$, was added later (Uhlenbeck and Goudsmit, 1925) to account for fine structure. Each orbital can hold at most two electrons (one of each spin), the Pauli exclusion principle.

Shape gallery:

• s orbitals ($\ell = 0$): spherically symmetric. The $1s$ orbital has a single bright spot at the nucleus; the $2s$ has a node.
• p orbitals ($\ell = 1$): dumb-bell shaped, three orthogonal orientations ($p_x$, $p_y$, $p_z$).
• d orbitals ($\ell = 2$): five shapes, mostly cloverleafs in different planes plus one with a "doughnut".
• f orbitals ($\ell = 3$): seven still more complex shapes.

The energy of a hydrogen orbital depends only on $n$ (so $2s$ and $2p$ have the same energy). In multi-electron atoms, electron-electron interactions split this degeneracy; the orbital filling order ($1s$, $2s$, $2p$, $3s$, $3p$, $4s$, $3d$, ...) is the basis of the periodic table.

Comparison with the Bohr model

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.

What Schrodinger added

• Wave-mechanical foundation. A single equation predicts the atom's structure from the form of the Coulomb potential.
• Spatial distribution of electrons. Real, observable electron-density distributions explain bonding, molecular geometry, and the shapes of molecular orbitals.
• Selection rules and transition probabilities. Computed from $\psi$ for initial and final states; these give the relative intensities of spectral lines.
• Connection to chemistry. The periodic table follows from the orbital filling order under the Pauli exclusion principle.

What is still missing

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.

Common traps

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

Saying the electron is "smeared out". 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.

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

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

Treating Schrodinger as a small fix to Bohr. 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.

In one sentence

Schrodinger's wavefunction $\psi$, interpreted by Born as a probability amplitude with $|\psi|^2$ the probability density, replaces Bohr's definite orbits with three-dimensional atomic orbitals (s, p, d, f) labelled by four quantum numbers, giving a quantitative theory that explains the spectra and chemistry of all atoms.