Assign every symbol
The equation, and only then the example it is evaluated on. This is a spin in a constant field along z. Nothing is left as a letter without a value or a matrix.
i ℏ d|ψ⟩/dt = H |ψ⟩
H = (ℏ ω / 2) σ_z
| i | Imaginary unit. i² = −1. Not a variable. |
|---|---|
| ℏ | h / 2π. h = 6.62607015×10⁻³⁴ J·s exactly. ℏ = 1.054571817×10⁻³⁴ J·s. |
| t | Time in seconds. The parameter. Not an operator. |
| |ψ(t)⟩ | c↑(t)|↑⟩ + c↓(t)|↓⟩. Two complex numbers, |c↑|² + |c↓|² = 1. |
| d/dt | Ordinary derivative of those two numbers. No spatial derivative in this example. |
| H | (ℏω/2) σ_z = diag(+ℏω/2, −ℏω/2). The two diagonal entries are the two energies. |
| ω | Real angular frequency, s⁻¹, fixed by the magnet. Not an outcome. |
| σ_z | Matrix diag(1, −1). Eigenvalue +1 on |↑⟩ = (1, 0), eigenvalue −1 on |↓⟩ = (0, 1). |
| |ψ(0)⟩ | |+x⟩ = (1/√2)(|↑⟩ + |↓⟩). So c↑(0) = c↓(0) = 1/√2. |
Componentwise, after cancelling ℏ, which is not zero: i ċ↑ = (ω/2) c↑ and i ċ↓ = −(ω/2) c↓. That is the equation in numbers. No probability has entered.