Wave Mechanics

1 The Schrödinger equation In classical mechanics the motion of a particle is usually described using the time-dependent position ix(t) as the dynamical variable. In wave mechanics the dynamical variable is a wavefunction. This wavefunction depends on position and on time and it is a complex number – it belongs to the complex numbers C (we denote the real numbers by R). When all three dimensions of space are relevant we write the wavefunction as Ψ(ix, t) ∈ C . (1.1) When only one spatial dimension is relevant we write it as Ψ(x, t) ∈ C. The wavefunction satisfies the Schrödinger equation. For one-dimensional space we write ∂Ψ 2 ∂ i (x, t) = − + V (x, t) Ψ(x, t) . (1.2) ∂t 2m ∂x2