Wave Lab

$\Psi$

Full time-dependent wavefunction (capital psi).

$n$

Principal quantum number, n = 1, 2, 3, …: the shell. Sets the energy.

$\ell$

Orbital angular-momentum number, 0 ≤ ℓ < n: the shape (s, p, d, f = 0, 1, 2, 3).

$m$

Magnetic quantum number, −ℓ ≤ m ≤ ℓ: orientation, the z-component of angular momentum mħ.

$R_{n\ell }(r)$

Radial part: how probability spreads with distance. Has n − ℓ − 1 radial nodes.

$Y_{\ell }^m$

Spherical harmonic: the angular shape (the orbital lobes). Has ℓ angular nodes.

$e^{-i{E}_{n}t/\hslash }$

Time evolution: a phase rotating at angular frequency Eₙ/ħ. Its magnitude is 1.

$i$

Imaginary unit, i² = −1.

$t$

Time.

|e^{−iEt/ħ}| = 1, so |Ψ|² never changes: a stationary state. Only superpositions of different Eₙ move.

$n\ge 1$

The radial series must terminate or R blows up like e^{+r}; termination forces an integer n.

$\ell \le n-1$

The terminating polynomial L^{(2ℓ+1)}_{n−ℓ−1} needs n − ℓ − 1 ≥ 0.

$|m|\le \ell$

Solutions in θ stay finite at the poles only if |m| ≤ ℓ.

$m\in \mathbb{Z}$

e^{imφ} must be single-valued after a full turn φ → φ + 2π, so m is an integer.

$E_n$

Depends on n alone (hydrogen's accidental degeneracy): n² states per shell, 2n² with spin.

$13.6\text{eV}$

The Rydberg energy, half the Hartree energy Eₕ = 27.211 eV.

$Z$

Nuclear charge in units of e (1 for hydrogen, 2 for He⁺).

$a_0$

Bohr radius 0.529 177 Å: the atomic unit of length.

$\rho$

Scaled radius (rho). Makes the formula the same for every shell.

$e^{-\rho /2}$

Exponential decay far from the nucleus: the electron is bound.

${\rho }^{\ell }$

Suppression near the nucleus by the centrifugal barrier ℓ(ℓ+1)/r².

$L_k^{(\alpha )}$

Associated Laguerre polynomial of degree k = n − ℓ − 1: supplies the radial nodes.

$!$

Factorial, k! = 1·2·…·k.

$\sqrt{\cdots }$

Normalisation constant chosen so ∫R²r²dr = 1.

${\int }_0^{\infty }$

Integral over all distances, from the nucleus to infinity.

$R^2$

Radial probability density per unit volume.

$r^{2}dr$

Volume of a thin shell is 4πr²dr; the 4π is carried by Y. Why 1s peaks at a₀, not 0.

$=1$

The electron is somewhere with certainty.

$\hslash$

Reduced Planck constant h/2π = 1.054 571 817 × 10⁻³⁴ J·s. Sets the scale of quantum effects.

$m$

Mass of the particle (for hydrogen, the electron mass mₑ = 9.109 × 10⁻³¹ kg).

$\psi$

Wavefunction (psi). A complex amplitude; |ψ|² is the probability density of finding the particle at a point.

$r$

Distance from the nucleus (origin).

$\theta$

Polar angle (theta), measured down from the +z axis: 0 at the north pole, π at the south.

$\varphi$

Azimuthal angle (phi), measured around the z axis from +x, 0 to 2π.

$\partial$

Partial derivative: the rate of change along one coordinate with the others held fixed.

$V(r)$

Potential energy. For hydrogen V = −e²/(4πε₀r): attractive, depends only on r.

$E$

Total energy of the stationary state (an eigenvalue).

$[\cdots ]$

The bracket is the Laplacian ∇²ψ written out in (r, θ, φ): the curvature of ψ in 3D.

$-\frac{{\hslash }^2}{2m}{\nabla }^2$

Kinetic-energy operator. The minus sign makes curvier waves carry more kinetic energy.

Because V depends only on r, the equation separates: ψ = R(r)·Y(θ, φ).