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

| Property | Bohr (1913) | Schrodinger (1926) |
| ----------------------- | -------------------------------------- | -------------------------------------------------------- | ---- | --- |
| Description of electron | Particle on a definite circular orbit | Wavefunction $\psi$; probability density $| \psi | ^2$ |
| Quantum numbers | $n$ only | $n$, $\ell$, $m_\ell$, $m_s$ |
| Atoms predicted | Hydrogen and hydrogen-like ions | All atoms (with approximations beyond hydrogen) |
| Spectral features | Line positions only | Line positions, intensities, fine structure, Zeeman, ... |
| Quantisation | Postulated (angular momentum) | Emerges as standing-wave boundary condition |
| Conceptual basis | Semi-classical (orbits + ad hoc rules) | Fully quantum (wave equation + Born rule) |

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.

Examples in context

Example 1. Computing the hydrogen 1s orbital probability density at UNSW. The hydrogen 1s wavefunction is $\psi_{100}(r) = (1/\sqrt{\pi a_0^3}) e^{-r/a_0}$ with $a_0 = 5.29 \times 10^{-11} \text{ m}$. At $r = a_0$, $|\psi|^2 = (1/\pi a_0^3) e^{-2} = (1/\pi \times 1.48 \times 10^{-31}) \times 0.135 = 2.91 \times 10^{29} \text{ m}^{-3}$. Integrating the radial probability density $4\pi r^2 |\psi|^2$ shows the most probable radius is exactly $a_0$ (Bohr radius), but the expectation value $\langle r \rangle = 1.5 a_0$. There is no fixed orbit - only a probability cloud, replacing Bohr's definite circles with statistical "shapes" labelled by $n, \ell, m_\ell, m_s$.

Example 2. d-orbital splitting in a Lucas Heights $^{99m}$Tc complex. $^{99m}$Tc, used in NSW Health nuclear medicine scans, sits in a $4d^5$ configuration. The five d-orbitals (labelled by $\ell = 2$, $m_\ell = -2, -1, 0, +1, +2$) are degenerate in isolated atoms but split into $t_{2g}$ (lower, 3 orbitals) and $e_g$ (upper, 2 orbitals) in octahedral ligand fields, separation $\Delta_o \approx 2.5 \text{ eV}$. The energy spacing produces the absorption band that Tc-99m gamma-camera scans rely on. Schrodinger's quantum numbers $(n, \ell, m_\ell, m_s)$ predict exactly five d-orbitals, ten electrons maximum, matching the Pauli exclusion rule.

Try this

Q1. Name the four quantum numbers used to label an atomic orbital and state what each represents. [4 marks]

• Cue. $n$ (principal, energy/size); $\ell$ (azimuthal, shape); $m_\ell$ (magnetic, orientation); $m_s$ (spin, $\pm 1/2$).

Q2. State the Born interpretation of the wavefunction $\psi$. [2 marks]

• Cue. $|\psi|^2$ is the probability density of finding the electron at a given position; $\psi$ itself has no direct physical meaning.

Q3. Compare Bohr's model with Schrodinger's quantum-mechanical model of the atom. (a) State one similarity. (b) State two differences. (c) Explain one physical phenomenon (e.g. spectral fine structure) explained by Schrodinger's model but not Bohr's. [1+2+3 marks]

• Cue. (a) Both quantise energy levels with $n$. (b) Bohr: definite orbits, Schrodinger: probability cloud; Bohr: only 1 quantum number, Schrodinger: 4. (c) Fine structure requires spin and relativistic terms; Bohr predicts no splitting.