Wave function significance; time independent equation

Significance of the wave function

• Ψ(x, t) is the complete description of the state of a particle (matter wave).
• |Ψ|² = Ψ*Ψ is the probability density: |Ψ|²dx is the probability of finding the particle in dx.
• Normalisation: ∫|Ψ|² dx over all space = 1.
• Ψ must be finite, single-valued and continuous, with continuous first derivative, so that probability is well defined.
• Expectation values of observables are obtained from Ψ, e.g. ⟨x⟩ = ∫Ψ* x Ψ dx.

Deduction of the time-independent equation

Start from the time-dependent equation:

iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + V(x)Ψ

1. If V depends on x only, use separation of variables: Ψ(x, t) = ψ(x) f(t).

2. Substitute and divide by ψ f:

   iħ (1/f) df/dt = −(ħ²/2mψ) d²ψ/dx² + V(x)

3. Left side depends on t only, right side on x only. So both equal a constant, the total energy E.

4. Time part: iħ df/dt = E f, so f(t) = e^{−iEt/ħ}.

5. Space part: −(ħ²/2m) d²ψ/dx² + Vψ = Eψ. Rearranged:

d²ψ/dx² + (2m/ħ²)(E − V)ψ = 0

In three dimensions: ∇²ψ + (2m/ħ²)(E − V)ψ = 0, or Ĥψ = Eψ.

This is the time-independent (steady-state) Schrödinger equation. Solving it with boundary conditions gives the allowed energies Eₙ and wave functions ψₙ. The full state is Ψ = ψ(x) e^{−iEt/ħ}, and |Ψ|² = |ψ|² does not depend on time, hence "stationary state".