Every symbol explained
θ, φ & Eigenvectors
The angles, phases and special directions behind every equation on this site.
θ and φ: two angles find any direction
Think of Earth. θ (theta) is how far you have tilted down from the North Pole: 0° at the pole, 90° on the equator, 180° at the South Pole. φ (phi) is how far you have gone round, like longitude: 0° to 360°. Together with the distance r, they pin down any point in space.
- x = r sinθ cosφ
- 0.628
- y = r sinθ sinφ
- 0.527
- z = r cosθ
- 0.574
- x² + y² + z²
- 1.000
| Distance from the centre. Here r = 1. | |
| How much of you points up the z-axis: 1 at the top, 0 on the equator, −1 at the bottom. | |
| How far you are from the axis: 0 at the poles, 1 on the equator. It shrinks the circle you can go round. | |
| Split that circle into its x and y parts, exactly as on a flat unit circle. |
Phase angles: a point going round a circle
| Euler’s number, 2.718…, the base of natural growth. | |
| √−1. Multiplying by i turns an arrow 90°. That’s why an i in the exponent gives rotation, not growth. | |
| The phase angle: how far round the circle you are, in radians (2π = 360°). | |
| The shadow on the horizontal (real) axis. Trace it over time and you get a cosine wave. | |
| The shadow on the vertical (imaginary) axis: a sine wave, a quarter-turn behind. |
Multiplying phases adds their angles: eiα·eiβ = ei(α+β) = 105°. The size stays 1. This is why a wavefunction’s phase e−iEt/ħ can spin forever without changing |ψ|², and why m must be whole to close the rainbow ring.
Eigenvectors: the directions a matrix doesn’t turn
| A matrix: a machine that moves every arrow in the plane (stretch, turn, shear, flip). | |
| An eigenvector (‘own’ vector): an arrow that A only stretches, never turns. | |
| The eigenvalue: the stretch factor. λ = 2 doubles it, λ = −1 flips it, λ = 0 crushes it. | |
| How to find λ. For a 2×2 matrix it becomes λ² − (a+d)λ + (ad−bc) = 0, which factors into (λ − λ₁)(λ − λ₂) = 0. |
Turn v until Av (pink) lines up with it, and it turns green.
- Trace a + d = λ₁ + λ₂
- 4.00
- Det ad − bc = λ₁·λ₂
- 3.00
- λ₁
- 3.000
- λ₂
- 1.000
Symmetric matrices always have perpendicular eigenvectors, here at 45°, with λ = 3 and 1.
In quantum mechanics every measurable quantity is a matrix. Its eigenvalues are the only results you can ever measure, and its eigenvectors are the states that give them. Hydrogen’s energies −13.6/n² eV are the eigenvalues of its Hamiltonian.
The hydrogen wavefunction, one factor at a time
| The wavefunction. Its size squared, |ψ|², is the chance of finding the electron there. | |
| Three whole numbers. n is the shell (size and energy), ℓ the shape (s, p, d, f), and m which way the shape points. | |
| Depends only on distance r. It decides how far out the electron sits, and gives the n − ℓ − 1 radial nodes. | |
| Depends only on θ, the tilt from the z-axis. It makes the lobes and cones of p and d orbitals. | |
| Depends only on φ, the angle around the axis. It’s a pure phase that winds m times round the circle. | |
| Time. Also a pure phase, spinning faster for higher energy. Size 1, so |ψ|² never changes: a stationary state. | |
| The energy, an eigenvalue. Minus means bound; it gets closer to 0 (free) as n grows. | |
| Planck’s constant ÷ 2π = 1.055 × 10⁻³⁴ J·s, the quantum unit of angular momentum. |