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Three quantum numbers label hydrogen atom states: n (principal, energy), ℓ (orbital angular momentum, shape), and m_ℓ (magnetic, orientation). Additionally, m_s = ±1/2 describes electron spin. Angular parts of wavefunctions are spherical harmonics Y_ℓ^m_ℓ(θ,φ), which determine orbital shapes. The quantum number n ranges from 1 to ∞, ℓ from 0 to n−1, and m_ℓ from −ℓ to +ℓ, giving multiple degenerate states at each energy level.
From your study of atomic orbitals — the s, p, d, f shapes and their nodes — you already know the visual vocabulary of quantum states. The quantum numbers are the systematic labeling scheme that connects those pictures to the mathematics of solving the hydrogen atom's Schrödinger equation in spherical coordinates. Solving the equation involves separation of variables: the wavefunction ψ(r, θ, φ) = R(r)·Y(θ, φ), where R(r) is a radial function and Y(θ, φ) is an angular function. Each separation introduces a quantum number that must take specific discrete values for the solution to be physically acceptable (normalizable and single-valued).
The principal quantum number n = 1, 2, 3, … comes from the radial equation and determines the energy: Eₙ = −13.6 eV/n². All states with the same n are degenerate in hydrogen — they share the same energy. The orbital angular momentum quantum number ℓ = 0, 1, 2, …, n−1 comes from requiring the angular solution to be well-behaved (square-integrable on the sphere). It determines the magnitude of the electron's orbital angular momentum: |L⃗| = ℏ√(ℓ(ℓ+1)). This is the quantum number you are really labeling when you say s (ℓ=0), p (ℓ=1), d (ℓ=2), f (ℓ=3). Finally, the magnetic quantum number m_ℓ = −ℓ, −ℓ+1, …, 0, …, ℓ comes from requiring the φ-dependence to be single-valued (the wavefunction must return to the same value after rotating 2π). It determines the component of angular momentum along any chosen axis: L_z = m_ℓ ℏ. A state with ℓ = 2 can have m_ℓ = −2, −1, 0, 1, or 2 — five choices, hence five d orbitals.
The angular functions Y_ℓ^{m_ℓ}(θ,φ) are the spherical harmonics — a complete, orthonormal basis for functions on a sphere. You can think of them as the "Fourier modes" of the sphere: just as sinusoids are the natural modes of a periodic line, spherical harmonics are the natural modes of a spherical surface. Their shapes directly give the orbital geometries you visualized earlier. Y₀⁰ is a constant (s orbital: spherically symmetric). Y₁⁰ ∝ cos θ (p_z: a lobe along the z-axis). Y₂⁰ ∝ 3cos²θ − 1 (d_z²: the double-lobe plus torus shape). The number of angular nodes equals ℓ, which is why higher-ℓ orbitals have more complex angular shapes. The real combinations of Y_ℓ^{m_ℓ} and Y_ℓ^{−m_ℓ} give the familiar d_{xy}, d_{xz} etc. orientations.
The electron's spin quantum number m_s = ±1/2 has no classical analog and does not come from the spatial Schrödinger equation — it arises from the intrinsic angular momentum (spin) of the electron, described by a separate two-component spinor. Combining all four quantum numbers (n, ℓ, m_ℓ, m_s), the number of distinct states at principal quantum number n is 2n² — two spin states for each of the n² spatial states. This counting, combined with the Pauli exclusion principle (no two electrons can share all four quantum numbers), directly produces the periodic table's shell structure: 2 electrons in n=1, 8 in n=2, 18 in n=3, each shell filling in the order you will study when you learn about orbital filling and the Aufbau principle.