The Quantum Atom
Probability clouds instead of orbits.
Lesson at a glanceShow the key points
- |ψ|²dV = probability of finding the electron in dV.
- States ψ_nlm: l = 0…n−1, m_l = −l…l; E_n same as Bohr (43.11).
- Ground state P(r) peaks at a₀, but no fixed orbit.
- Spin m_s = ±½; 2n² states per shell; Pauli: one electron per state.
Quantum mechanics replaced Bohr’s orbits with waves of probability. It reproduces his energies exactly, but tells a very different story about where the electron is.
The wave function
Light needs both pictures: the field wave E⃗(r⃗, t) and the photon. Where |E|² is large, photons are likely; yet a detector always catches a whole photon at one spot, never a fraction. An electron is described the same way, by a wave function ψ(r⃗, t), found by solving Schrödinger’s equation. ψ may be spread over a large region, but the electron is always detected whole at a single point; the probability of finding it in a small volume dV is |ψ|²dV. All its properties, energy included, are contained in ψ.
The hydrogen atom in quantum mechanics
Its solutions ψn l ml are labelled by three numbers:
- n, the principal quantum number: 1, 2, 3, …
- l, the orbital angular momentum quantum number: 0, 1, …, n − 1.
- ml, the magnetic quantum number: −l, …, +l (2l + 1 values).
No orbits, only probabilities
The ground state is
The chance of finding the electron between r and r + dr is P(r)dr, with P(r) = (4/a₀³)r²e−2r/a₀. It peaks at r = a₀, where Bohr put the electron, but the electron can be found closer or farther. For n = 2, l = 0, P(r) has two humps, near a₀ and near 5.4a₀, while Bohr would put every n = 2 electron at exactly 4a₀.
Spin, shells and Pauli’s principle
Every electron also has a built-in spin angular momentum, whose component along any direction is +h/4π or −h/4π, described by a fourth quantum number ms = ±½. A set (n, l, ml, ms) is one quantum state. Each n has 2n² states, forming a major shell: K (n = 1), L (n = 2), M (n = 3)…
Values of l are written with letters s, p, d, f for l = 0, 1, 2, 3. States with the same n and l form a subshell: 1s, 2s, 2p, 3s… In atoms with many electrons the energy depends on both n and l, and electrons fill the states from the lowest energy upward.
| Shell | n | Subshells | Maximum electrons (2n²) |
|---|---|---|---|
| K | 1 | 1s | 2 |
| L | 2 | 2s, 2p | 8 |
| M | 3 | 3s, 3p, 3d | 18 |
| N | 4 | 4s, 4p, 4d, 4f | 32 |
- 1An electron is described by a wave function ψ; |ψ|²dV is the probability of finding it in dV.
- 2Hydrogen: states ψ_nlm with n, l (0…n−1), m_l (−l…l); energies match Bohr (43.11).
- 3Ground state P(r) peaks at a₀ but there is no fixed orbit.
- 4Spin m_s = ±½; 2n² states per shell (K, L, M…); Pauli: one electron per state.
- 5Subshells s, p, d, f fill from lowest energy.
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