Difficulty: Intermediate | Prerequisites: Part 1 of these notes (EM radiation, quantisation, photoelectric effect).
Now that you know energy is quantised and light has particle-like properties, this section explains how those ideas apply to atoms. Bohr built the first successful model of the hydrogen atom by assuming electrons orbit the nucleus only at certain allowed energies. Schrodinger then replaced Bohr's orbits with a full wave-mechanical treatment, giving us orbitals described by quantum numbers. This is where the shapes of electron clouds (s, p, d, f) and the rules for labelling every electron in an atom come from.
Bohr's model gives quantised energy levels for the hydrogen atom and correctly predicts its emission spectrum, but it fails for multi-electron atoms and wrongly treats electrons as particles in orbits. The Schrodinger equation replaces orbits with wave functions (orbitals), each labelled by quantum numbers n, l, and ml. A fourth quantum number, ms, describes electron spin. Together these four numbers uniquely specify every electron in an atom.
Bohr model
A model of the hydrogen atom in which the electron orbits the nucleus in fixed circular orbits with quantised energies. In simple terms, the electron can sit only on certain "rungs" of an energy ladder, not between them.
Ground state
The lowest energy state of an atom (n = 1 for hydrogen). Think of it as the electron's default position when it has not absorbed any extra energy.
Excited state
Any energy state above the ground state (n > 1). The electron reaches an excited state by absorbing a photon of exactly the right energy.
Emission spectrum / line spectrum
The set of specific wavelengths of light emitted when excited electrons drop to lower energy levels. Each element has a unique pattern, like a fingerprint.
Absorption spectrum
The set of specific wavelengths removed from a continuous spectrum when light passes through a gas. The missing wavelengths match the emission spectrum of that element.
Lyman, Balmer, Paschen series
Groups of spectral lines in hydrogen corresponding to transitions ending at n = 1 (UV), n = 2 (visible), and n = 3 (infrared), respectively.
Schrodinger equation
The fundamental equation of quantum mechanics, HΨ = EΨ, whose solutions give the allowed energy levels and wave functions of a system.
Hamiltonian (H)
The mathematical operator in the Schrodinger equation that represents the total energy (kinetic + potential) of the system.
Wave function (Ψ, "psi")
A mathematical function that describes the quantum state of a particle. It is a probability amplitude. In simple terms, it tells you the "shape" of the electron's behaviour in an atom.
Probability density (|Ψ|²)
The square of the wave function. It gives the probability of finding the electron at a particular point in space. Where |Ψ|² is large, you are most likely to find the electron.
Orbital
A wave function for an electron in an atom. A picture of an orbital shows the surface that encloses about 90% of the electron density.
Node
A region in an orbital where the probability of finding the electron is zero. As n increases, the number of nodes increases (n - 1 total nodes).
Principal quantum number (n)
A positive integer (1, 2, 3, ...) that determines the energy level and size of the orbital. Higher n means higher energy and greater average distance from the nucleus.
Angular momentum quantum number (l)
Takes integer values from 0 to n - 1. It determines the shape of the orbital: l = 0 is s (spherical), l = 1 is p (lobed), l = 2 is d (clover-shaped), l = 3 is f (complex).
Magnetic quantum number (ml)
Takes integer values from -l to +l. It determines the orientation of the orbital in space. For example, the three p orbitals point along x, y, and z.
Spin quantum number (ms)
Can be +½ ("spin up") or -½ ("spin down"). It describes the intrinsic angular momentum of the electron. Two electrons in the same orbital must have opposite spins.
Degenerate orbitals
Orbitals with the same energy. In hydrogen, all orbitals with the same n are degenerate (2s and 2p have the same energy). In multi-electron atoms, this degeneracy is broken.
Central idea: the electron circles the nucleus in only certain allowed circular orbits.
By convention, E = 0 at n = infinity (no interaction between nucleus and electron).
All allowed energies are negative (the electron is bound to the nucleus).
Energy levels get closer together as n increases.
The model correctly predicts hydrogen's emission spectrum but fails for atoms with more than one electron.
Key limitation: it assumes a deterministic orbit (treats the electron as a particle on a defined path).
Atoms emit light only at specific discrete wavelengths, corresponding to electrons dropping from higher to lower energy levels.
Atoms absorb light only at those same wavelengths, corresponding to electrons jumping up.
ΔE = E_final - E_initial. If ΔE is negative, a photon is emitted. If positive, a photon is absorbed.
Hydrogen emission series:
Lyman series: transitions to n = 1, produces UV light.
Balmer series: transitions to n = 2, produces visible light (the coloured lines you see in hydrogen discharge tubes).
Paschen series: transitions to n = 3, produces infrared.
Schrodinger and de Broglie exploited wave-particle duality to describe the electron as a standing wave constrained by the nuclear potential.
The equation HΨ = EΨ, where H is the Hamiltonian operator (kinetic + potential energy).
Solutions give a set of quantised energy levels E_n and associated wave functions Ψ_n.
The wave function contains all information about the electron's state.
The Coulombic potential V(r) = -(Ze)(e)/r is the constraining potential that produces quantisation in atoms.
Ψ (the wave function) is a probability amplitude.
|Ψ|² (the square of the wave function) is the probability density: it tells you where the electron is most likely to be found.
An orbital is a wave function for an electron. Orbital pictures represent surfaces enclosing ~90% of the electron density.
Standing wave analogy: just as a guitar string can vibrate only in certain patterns (1 half-wavelength, 2 half-wavelengths, etc.), an electron around a nucleus can adopt only certain standing-wave patterns. The points of zero displacement are nodes.
n (principal): determines energy and size. n = 1, 2, 3, ...
l (angular momentum): determines shape. l = 0 to n - 1.
l = 0: s orbitals (spherical)
l = 1: p orbitals (two-lobed, with a node at the nucleus)
l = 2: d orbitals (four-lobed clover shapes, plus d_z²)
l = 3: f orbitals (complex, multi-lobed)
ml (magnetic): determines orientation. ml = -l, ..., 0, ..., +l. Gives 2l + 1 orbitals per sublevel.
ms (spin): +½ or -½. Two electrons per orbital, with opposite spins.
As n increases: orbital energy increases, more orbital types become available, the number of nodes increases (n - 1 total), and orbital size increases.
Electrons have intrinsic angular momentum (spin), behaving as though they are bundles of spinning charge.
Two possible spin states: spin up (ms = +½) and spin down (ms = -½).
Pauli Exclusion Principle: no two electrons in an atom may have the same set of all four quantum numbers (n, l, ml, ms).
Consequence: each orbital can hold at most two electrons, and those two must have opposite spins.
E_n = -2.178 \times 10^{-18} \; \text{J} \left( \frac{Z^2}{n^2} \right)Energy of an electron in orbit n of a hydrogen-like atom (Z = atomic number; Z = 1 for H). Energy is always negative (bound state).
\Delta E = E_{\text{final}} - E_{\text{initial}} = -2.178 \times 10^{-18} \; \text{J} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)Energy change for a transition between two levels. Negative ΔE means emission; positive means absorption.
|\Delta E| = \frac{hc}{\lambda}Relates the magnitude of the energy change to the wavelength of the photon emitted or absorbed. Rearrange to find λ: λ = hc / |ΔE|.
\hat{H}\Psi = E\PsiThe time-independent Schrodinger equation. H is the Hamiltonian; Ψ is the wave function; E is the energy eigenvalue.
V(r) = -\frac{(Ze)(e)}{r}Coulombic potential energy between the nucleus (charge +Ze) and the electron (charge -e) at distance r.
The discrete emission lines predicted by the Bohr model are the basis of spectroscopy, which is how astronomers determine the composition of distant stars, and why elements in the solar atmosphere produce dark absorption lines (Fraunhofer lines) in the Sun's spectrum. Quantum numbers and orbital shapes directly inform how chemists predict bonding, molecular geometry, and the properties of materials.
Students often think the Bohr model's circular orbits are correct. They are not. Electrons occupy orbitals (probability clouds), not orbits (defined paths). The Bohr model is historically important but physically wrong for anything beyond hydrogen.
Students often confuse "orbital" with "orbit." An orbital is a wave function (a probability distribution in 3D space). An orbit is a fixed circular path. They are fundamentally different concepts.
Students often think |Ψ| directly gives probability. It does not. You need |Ψ|² (the square) for probability density.
Students sometimes believe that an electron in a p orbital is moving in a figure-eight path. The lobes represent regions of high probability, not a trajectory.
⚠️ Calculating ΔE for hydrogen transitions and converting to wavelength is a very common exam question. Know which n is initial and which is final.
⚠️ Identifying valid sets of quantum numbers: if asked "which of the following is a valid set (n, l, ml, ms)?", check that l < n, |ml| ≤ l, and ms = ±½.
⚠️ Know which emission series (Lyman, Balmer, Paschen) produces which type of radiation (UV, visible, IR).
⚠️ Be able to state the Pauli Exclusion Principle and explain why an orbital holds at most two electrons.
⚠️ Know orbital shapes: s is spherical, p has two lobes, d has four lobes (plus d_z²), f is complex. Relate shapes to l values.
True or false: in the Bohr model, the electron can exist at any distance from the nucleus.
Fill in the blank: the Balmer series corresponds to electron transitions ending at n = ______.
True or false: for hydrogen, the 2s and 2p orbitals have different energies.
Fill in the blank: the maximum number of electrons in a sublevel with l = 2 is ______.
True or false: two electrons in the same orbital can have the same spin quantum number.
Answers: 1. False (only certain allowed orbits). 2. 2. 3. False (they are degenerate in hydrogen). 4. 10 (five d orbitals, 2 electrons each). 5. False (Pauli Exclusion Principle requires opposite spins).
Q: At what wavelength will emission from n = 4 to n = 1 for the hydrogen atom be observed?
A: ΔE = -2.178 × 10⁻¹⁸ J (1/1² - 1/4²) = -2.178 × 10⁻¹⁸ (1 - 1/16) = -2.178 × 10⁻¹⁸ (15/16) = -2.04 × 10⁻¹⁸ J. Then λ = hc / |ΔE| = (6.626 × 10⁻³⁴)(3.00 × 10⁸) / (2.04 × 10⁻¹⁸) = 9.74 × 10⁻⁸ m = 97.4 nm. This is a Lyman series line (UV).
Q: What is the longest wavelength of light that will ionise hydrogen from its ground state (n = 1)?
A: Ionisation means n_final = infinity, so ΔE = -2.178 × 10⁻¹⁸ (0 - 1) = +2.178 × 10⁻¹⁸ J. Then λ = hc / ΔE = 9.13 × 10⁻⁸ m = 91.3 nm. Longest wavelength corresponds to minimum energy for ionisation.
Q: Which of these is a valid set of quantum numbers: (n=2, l=2, ml=0, ms=+½)?
A: Invalid. For n = 2, l can be 0 or 1 only (l must be less than n). l = 2 requires n ≥ 3.
Q: How many orbitals exist in the n = 3 shell?
A: n² = 9 orbitals total. One 3s, three 3p, and five 3d.
This connects to Part 1 (EM radiation and quantisation) because the Bohr model uses E = hν to predict emission wavelengths. It connects forward to Part 3 (electron configurations and periodic trends), where the quantum numbers and orbital shapes covered here determine how electrons fill orbitals and why elements in the same group behave similarly.
Bohr model, hydrogen atom, energy levels, emission spectrum, absorption spectrum, line spectrum, Lyman series, Balmer series, Paschen series, Schrodinger equation, Hamiltonian, wave function, psi, probability density, orbital, node, quantum numbers, principal quantum number n, angular momentum quantum number l, magnetic quantum number ml, spin quantum number ms, s orbital, p orbital, d orbital, f orbital, Pauli Exclusion Principle, electron spin, degenerate orbitals, Coulombic potential, hydrogen-like atom, General Physics Ch 12